{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Sphere $\\mathbb{S}^2$\n", "\n", "This notebook demonstrates some differential geometry capabilities of SageMath on the example of the 2-dimensional sphere. The corresponding tools have been developed within\n", "the [SageManifolds](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 9.3 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 10.4, Release Date: 2024-07-19'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display math formulas using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $\\mathbb{S}^2$ from the manifold catalog" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The 2-sphere, with predefined charts and embedding in the Euclidean 3-space, can be obtained directly from SageMath's manifold catalog:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^{2}\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^{2}$" ], "text/plain": [ "2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2 = manifolds.Sphere(2)\n", "S2" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3\n" ] } ], "source": [ "print(S2)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,(\\theta, \\phi)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,(\\theta, \\phi)\\right)$" ], "text/plain": [ "Chart (A, (theta, phi))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.spherical_coordinates()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi\$" ], "text/latex": [ "$\\displaystyle g = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi$" ], "text/plain": [ "g = dtheta⊗dtheta + sin(theta)^2 dphi⊗dphi" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.metric().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $\\mathbb{S}^2$ defined from scratch as a 2-dimensional smooth manifold" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the purpose of introducing generic smooth manifolds in SageMath, we shall not use the above predefined object. Instead we shall construct $\\mathbb{S}^2$ from scratch, by invoking the generic function `Manifold`:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "S2 = Manifold(2, 'S^2', latex_name=r'\\mathbb{S}^2', start_index=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first argument, `2`, is the dimension of the manifold, while the second argument is the symbol used to label the manifold.\n", "\n", "The argument `start_index` sets the index range to be used on the manifold for labelling components w.r.t. a basis or a frame: `start_index=1` corresponds to $\\{1,2\\}$; the default value is `start_index=0` and yields $\\{0,1\\}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The function `Manifold` has actually many options, which are displayed via the command `Manifold?`:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "# Manifold?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By default `Manifold` constructs a smooth manifold over $\\mathbb{R}$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(S2)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^2$" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\mathbb{S}^2$ is in the category of smooth manifolds over $\\mathbb{R}$:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}$" ], "text/plain": [ "Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.category()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Category of smooth manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(S2.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At the moment, the real field $\\mathbb{R}$ is modeled by 53-bit floating-point approximations, but this plays no role in the manifold implementation:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Real Field with 53 bits of precision\n" ] } ], "source": [ "print(S2.base_field())" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.base_field() is RR" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coordinate charts on $\\mathbb{S}^2$\n", "\n", "The function `Manifold` generates a manifold with no-predefined coordinate chart, so that the manifold (user) **atlas** is empty:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\right]$" ], "text/plain": [ "[]" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce some charts. At least two charts are necessary to cover the sphere. Let us choose the charts associated with the **stereographic projections** to the equatorial plane from the North pole and the South pole respectively. We first introduce the open subsets covered by these two charts: \n", "$$ U := \\mathbb{S}^2\\setminus\\{N\\}, $$ \n", "$$ V := \\mathbb{S}^2\\setminus\\{S\\}, $$\n", "where $N$ is a point of $\\mathbb{S}^2$, which we shall call the **North pole**, and $S$ is the point of $U$ of stereographic coordinates $(0,0)$, which we call the **South pole**:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To find the method to create an open subset, we type `U = S2.` to get the list of possible methods by autocompletion:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "#U = S2." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset U of the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "U = S2.open_subset('U')\n", "print(U)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset V of the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "V = S2.open_subset('V')\n", "print(V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As an open subset of a smooth manifold, $U$ is itself a smooth manifold:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Join of Category of subobjects of sets and Category of smooth manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(U.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We declare that $\\mathbb{S}^2 = U \\cup V$:

" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "S2.declare_union(U, V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **stereographic chart** on $U$ is constructed from the stereographic projection from the North pole onto the equatorial plane: in the [Wikipedia figure](https://en.wikipedia.org/wiki/Stereographic_projection) below, the stereographic coordinates $(x,y)$ of the point $P\\in U$ are the Cartesian coordinates of the point $P'$ in the equatorial plane.\n", "\n", "![stereographic projection](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Stereoprojzero.svg/241px-Stereoprojzero.svg.png)\n", "\n", "We call this chart `stereoN` and construct it via the method `chart`:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [], "source": [ "stereoN. = U.chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The syntax `.` in the left-hand side implies that the Python names `x` and `y` are added to the global namespace, to access to the two coordinates of the chart as symbolic variables. This allows one to refer subsequently to the coordinates by these names. Besides, in the present case, the function `chart()` has no argument, which implies that the coordinate symbols will be `x` and `y` (i.e. exactly the characters appearing in the `<...>` operator) and that each coordinate range is $(-\\infty,+\\infty)$. As we will see below, for other cases, an argument must be passed to `chart()` to specify each coordinate symbol and range, as well as some specific LaTeX symbol." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*Note:* the notation `.` is not standard Python syntax, but a \"SageMath enhanced\" syntax. \n", "Actually the SageMath kernel preparses the cell entries before sending them to the Python interpreter. The outcome of the preparser is shown by the function `preparse`. In the present case:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)\n" ] } ], "source": [ "print(preparse(\"stereoN. = U.chart()\"))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another example of preparsing:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|Integer(2)**Integer(3)|\$" ], "text/latex": [ "$\\displaystyle \\verb|Integer(2)**Integer(3)|$" ], "text/plain": [ "'Integer(2)**Integer(3)'" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "preparse(\"2^3\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The chart created by the above command:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(U,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(U,(x, y)\\right)$" ], "text/plain": [ "Chart (U, (x, y))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (U, (x, y))\n" ] } ], "source": [ "print(stereoN)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle x :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( -\\infty, +\\infty \\right)\$" ], "text/latex": [ "$\\displaystyle x :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( -\\infty, +\\infty \\right)$" ], "text/plain": [ "x: (-oo, +oo); y: (-oo, +oo)" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinates can be accessed individually, either by means of their indices in the chart ( following the convention `start_index=1` set in the manifold's definition) or by their names as Python variables:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle x\$" ], "text/latex": [ "$\\displaystyle x$" ], "text/plain": [ "x" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN[1]" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y is stereoN[2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinates are SageMath symbolic expressions:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|<class|\\verb| |\\verb|'sage.symbolic.expression.Expression'>|\$" ], "text/latex": [ "$\\displaystyle \\verb||$" ], "text/plain": [ "" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(y)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\text{SR}\$" ], "text/latex": [ "$\\displaystyle \\text{SR}$" ], "text/plain": [ "Symbolic Ring" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Stereographic coordinates from the South Pole\n", "\n", "We introduce on $V$ the coordinates $(x',y')$ corresponding to the stereographic projection from the South pole:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [], "source": [ "stereoS. = V.chart(\"xp:x' yp:y'\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this case, the string argument passed to `chart` stipulates that the text-only names of the coordinates are xp and yp (same as the Python variables names defined within the `<...>` operator in the left-hand side), while their LaTeX names are $x'$ and $y'$." ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(V,({x'}, {y'})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(V,({x'}, {y'})\\right)$" ], "text/plain": [ "Chart (V, (xp, yp))" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At this stage, the user's atlas on the manifold is made of two charts:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right)\\right]$" ], "text/plain": [ "[Chart (U, (x, y)), Chart (V, (xp, yp))]" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To complete the construction of the manifold structure, we have \n", "to specify the transition map between the charts `stereoN` = $(U,(x,y))$ and `stereoS` = $(V,(x',y'))$; it is given by standard inversion formulas:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & \\frac{x}{x^{2} + y^{2}} \\\\ {y'} & = & \\frac{y}{x^{2} + y^{2}} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & \\frac{x}{x^{2} + y^{2}} \\\\ {y'} & = & \\frac{y}{x^{2} + y^{2}} \\end{array}\\right.$" ], "text/plain": [ "xp = x/(x^2 + y^2)\n", "yp = y/(x^2 + y^2)" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_to_S = stereoN.transition_map(stereoS, \n", " (x/(x^2+y^2), y/(x^2+y^2)), \n", " intersection_name='W',\n", " restrictions1= x^2+y^2!=0, \n", " restrictions2= xp^2+yp^2!=0)\n", "stereoN_to_S.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the above declaration, 'W' is the name given to the chart-overlap subset: $W := U\\cap V$, the condition $x^2+y^2 \\not=0$ defines $W$ as a subset of $U$, and the condition $x'^2+y'^2\\not=0$ defines $W$ as a subset of $V$.\n", "\n", "The inverse coordinate transformation is computed by means of the method `inverse()`:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & \\frac{{x'}}{{x'}^{2} + {y'}^{2}} \\\\ y & = & \\frac{{y'}}{{x'}^{2} + {y'}^{2}} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & \\frac{{x'}}{{x'}^{2} + {y'}^{2}} \\\\ y & = & \\frac{{y'}}{{x'}^{2} + {y'}^{2}} \\end{array}\\right.$" ], "text/plain": [ "x = xp/(xp^2 + yp^2)\n", "y = yp/(xp^2 + yp^2)" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_to_N = stereoN_to_S.inverse()\n", "stereoS_to_N.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

In the present case, the situation is of course perfectly symmetric regarding the coordinates $(x,y)$ and $(x',y')$.

\n", "

At this stage, the user's atlas has four charts:

" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right)\\right]$" ], "text/plain": [ "[Chart (U, (x, y)),\n", " Chart (V, (xp, yp)),\n", " Chart (W, (x, y)),\n", " Chart (W, (xp, yp))]" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us store $W = U\\cap V$ into a Python variable for future use:

" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [], "source": [ "W = U.intersection(V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly we store the charts $(W,(x,y))$ (the restriction of $(U,(x,y))$ to $W$) and $(W,(x',y'))$ (the restriction of $(V,(x',y'))$ to $W$) into Python variables:

" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(W,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(W,(x, y)\\right)$" ], "text/plain": [ "Chart (W, (x, y))" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_W = stereoN.restrict(W)\n", "stereoN_W" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_W is S2.atlas()[2]" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(W,({x'}, {y'})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(W,({x'}, {y'})\\right)$" ], "text/plain": [ "Chart (W, (xp, yp))" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_W = stereoS.restrict(W)\n", "stereoS_W" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Coordinate charts are endoved with a method `plot`. For instance, \n", "we may plot the chart $(W, (x',y'))$ in terms of itself, as a grid:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 18 graphics primitives" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_W.plot()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "More interestingly, let us plot the stereographic chart $(x',y')$ in terms of the stereographic chart $(x,y)$ on the domain $W$ where both systems overlap. We split the plot in four parts to avoid the singularity at $(x',y')=(0,0)$ and\n", "ask for the coordinate lines along which $x'$ (resp. $y'$) varies to be colored in purple (resp. cyan):" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 72 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph = (stereoS_W.plot(stereoN, ranges={xp:[-6,-0.02], yp:[-6,-0.02]},\n", " color={xp: 'purple', yp: 'cyan'}) \n", " + stereoS_W.plot(stereoN, ranges={xp:[-6,-0.02], yp:[0.02,6]},\n", " color={xp: 'purple', yp: 'cyan'})\n", " + stereoS_W.plot(stereoN, ranges={xp:[0.02,6], yp:[-6,-0.02]},\n", " color={xp: 'purple', yp: 'cyan'})\n", " + stereoS_W.plot(stereoN, ranges={xp:[0.02,6], yp:[0.02,6]},\n", " color={xp: 'purple', yp: 'cyan'}))\n", "graph.show(xmin=-1.5, xmax=1.5, ymin=-1.5, ymax=1.5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Spherical coordinates\n", "\n", "The standard **spherical coordinates** $(\\theta,\\phi)$ are defined on the open domain $A\\subset W \\subset \\mathbb{S}^2$ that is the complement of the \"origin meridian\"; since the latter is the half-circle defined by $y=0$ and $x\\geq 0$, we declare:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset A of the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "A = W.open_subset('A', coord_def={stereoN_W: (y!=0, x<0), \n", " stereoS_W: (yp!=0, xp<0)})\n", "print(A)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The restriction of the stereographic chart from the North pole to $A$ is

" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,(x, y)\\right)$" ], "text/plain": [ "Chart (A, (x, y))" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_A = stereoN_W.restrict(A)\n", "stereoN_A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We then declare the chart $(A,(\\theta,\\phi))$ by specifying the intervals $(0,\\pi)$ and $(0,2\\pi)$ spanned by respectively $\\theta$ and $\\phi$:

" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,({\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,({\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (A, (th, ph))" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher. = A.chart(r'th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "spher" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)\$" ], "text/latex": [ "$\\displaystyle {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)$" ], "text/plain": [ "th: (0, pi); ph: (0, 2*pi)" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The specification of the spherical coordinates is completed by providing the transition map with the stereographic chart $(A,(x,y))$:

" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & -\\frac{\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) - 1} \\\\ y & = & -\\frac{\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) - 1} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & -\\frac{\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) - 1} \\\\ y & = & -\\frac{\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) - 1} \\end{array}\\right.$" ], "text/plain": [ "x = -cos(ph)*sin(th)/(cos(th) - 1)\n", "y = -sin(ph)*sin(th)/(cos(th) - 1)" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher_to_stereoN = spher.transition_map(stereoN_A, \n", " (sin(th)*cos(ph)/(1-cos(th)),\n", " sin(th)*sin(ph)/(1-cos(th))))\n", "spher_to_stereoN.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also provide the inverse transition map:" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " th == 2*arctan(sqrt(-cos(th) + 1)/sqrt(cos(th) + 1)) **failed**\n", " ph == pi + arctan2(sin(ph)*sin(th)/(cos(th) - 1), cos(ph)*sin(th)/(cos(th) - 1)) **failed**\n", " x == x *passed*\n", " y == y *passed*\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] } ], "source": [ "spher_to_stereoN.set_inverse(2*atan(1/sqrt(x^2+y^2)), atan2(-y,-x)+pi)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The check is passed, modulo some lack of trigonometric simplifications in the first two lines." ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\frac{1}{\\sqrt{x^{2} + y^{2}}}\\right) \\\\ {\\phi} & = & \\pi + \\arctan\\left(-y, -x\\right) \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\frac{1}{\\sqrt{x^{2} + y^{2}}}\\right) \\\\ {\\phi} & = & \\pi + \\arctan\\left(-y, -x\\right) \\end{array}\\right.$" ], "text/plain": [ "th = 2*arctan(1/sqrt(x^2 + y^2))\n", "ph = pi + arctan2(-y, -x)" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher_to_stereoN.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The transition map $(A,(\\theta,\\phi))\\rightarrow (A,(x',y'))$ is obtained by combining the transition maps $(A,(\\theta,\\phi))\\rightarrow (A,(x,y))$ and $(A,(x,y))\\rightarrow (A,(x',y'))$ via the operator `*`:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & -\\frac{\\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ {y'} & = & -\\frac{\\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) - \\sin\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & -\\frac{\\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ {y'} & = & -\\frac{\\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) - \\sin\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\right.$" ], "text/plain": [ "xp = -(cos(ph)*cos(th) - cos(ph))/sin(th)\n", "yp = -(cos(th)*sin(ph) - sin(ph))/sin(th)" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_to_S_A = stereoN_to_S.restrict(A)\n", "spher_to_stereoS = stereoN_to_S_A * spher_to_stereoN\n", "spher_to_stereoS.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, the transition map $(A,(x',y'))\\rightarrow (A,(\\theta,\\phi))$ is obtained by combining the transition maps $(A,(x',y'))\\rightarrow (A,(x,y))$ and $(A,(x,y))\\rightarrow (A,(\\theta,\\phi))$:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\sqrt{{x'}^{2} + {y'}^{2}}\\right) \\\\ {\\phi} & = & \\pi - \\arctan\\left(\\frac{{y'}}{{x'}^{2} + {y'}^{2}}, -\\frac{{x'}}{{x'}^{2} + {y'}^{2}}\\right) \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\sqrt{{x'}^{2} + {y'}^{2}}\\right) \\\\ {\\phi} & = & \\pi - \\arctan\\left(\\frac{{y'}}{{x'}^{2} + {y'}^{2}}, -\\frac{{x'}}{{x'}^{2} + {y'}^{2}}\\right) \\end{array}\\right.$" ], "text/plain": [ "th = 2*arctan(sqrt(xp^2 + yp^2))\n", "ph = pi - arctan2(yp/(xp^2 + yp^2), -xp/(xp^2 + yp^2))" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_to_N_A = stereoN_to_S.inverse().restrict(A)\n", "stereoS_to_spher = spher_to_stereoN.inverse() * stereoS_to_N_A \n", "stereoS_to_spher.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The user atlas of $\\mathbb{S}^2$ is now

" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right), \\left(A,(x, y)\\right), \\left(A,({x'}, {y'})\\right), \\left(A,({\\theta}, {\\phi})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right), \\left(A,(x, y)\\right), \\left(A,({x'}, {y'})\\right), \\left(A,({\\theta}, {\\phi})\\right)\\right]$" ], "text/plain": [ "[Chart (U, (x, y)),\n", " Chart (V, (xp, yp)),\n", " Chart (W, (x, y)),\n", " Chart (W, (xp, yp)),\n", " Chart (A, (x, y)),\n", " Chart (A, (xp, yp)),\n", " Chart (A, (th, ph))]" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us draw the grid of spherical coordinates $(\\theta,\\phi)$ in terms of stereographic coordinates from the North pole $(x,y)$:

" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "image/png": 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Kz0T9ZMkCLXTbNliFypVD43qNFIMmRNVMQgIEZ9WqSDc4cwYpLKlSKT0z18HfH0JBM7VZj+YPtZ127aCV1qtH9NlnRJ07wySu4fZoQlSt3LoFn9SECeiBeOEC6t9q2EZAABLmHz6U9TAzZ84kDw8PGjJkiKzHkZ137yAMNH+o7WTPDi1040bUqy5XjmjXLqVnpSEzmhBVG4KAdJWPPsID7eRJFItPk0bpmbkmol9PRr/ouXPnaMWKFVTBHXIqT5/G31q1lJ2Hq+LhAS30+nWU2WzTBtWOtO4wbosmRNVERARuuhEjkNR96ZIW3OEomTPDvyeTEI2MjKSuXbvSypUrKXPmzLIcw6mcPEmULRtKR2rYT+7c0EJ//RV/a9RA9yQNt0MTomrhzh3caMePE/35J6oOpU+v9KzcAxmLLgwYMICaN29OjRs3trhtbGwsRUREJHqpDtEfmtJqLcuBhwe00PPnkcNdvTrubQ23QhOiamDPHn2HlbNnUbNTQzoCAuDne/dO0mE3bdpEFy9epJkzZ1q1/cyZM8nPz++/V/78+SWdj8MkJCB4TfOHSkvx4jivDRqg1dq0aVqgmxuhCVElYSaaNQv5ZvXq4UbTyvZJj78/zrXo75OAZ8+e0eDBg2n9+vWUNm1aq/YZO3YshYeH//d6prb6q1evEn34oEXmykHGjETbtyPXe8IEoo4dJV/UaSiDJkSV4to1oly5iMaOJfruO6KdO1GfU0N6ihVD5KSEftELFy5QSEgIValShby9vcnb25uOHTtGixYtIm9vb9LpdMn2SZMmDfn6+iZ6qYrAQKRPVami9EzcE09PCNGdO4n27SPKmZPot9+UnpWGg2jlbpTg8WOiLl2IQkNh6hk/Xqt7KyceHpL7RRs1akTXrl1L9F7Pnj2pVKlSNHr0aPLy8pLsWE4jMBACNF06pWfi3jRpQlSkCNLYBgzAAu/jj5WelYadaE9uZ3P4MIonREej0snTp0Tffqv0rNyfgACYyxMSJBkuY8aMVK5cuUQvHx8fypo1K5Vz1WpSWpEF5/Dtt0R376IjTM2aEKpz52p+UhdFE6LOgplo4UJ0gfjoI6Jz56CNLltGtHIl0YoVSs/QvfH3h7/vyhWlZ6JOXrwgevJEE6Jys2IF7vdly4gaNkS07qhRRCNHov5uVJTSM9SwFdaQn+ho5i+/ZCZiHj6cOT4+8ef9+zOnSsV88qQi00sRREczp07NvGiR0jP5j/DwcCYiDg8PV3oqzFu24Pp8+VLpmbgvgYG4z/v3T/7Z5s3M6dMzf/QR8+PHzp+bht1omqjcvHiByNvNm4nWrYPZJmnnlQULkCPavr3Wm1Au0qaFv0/r6GKcwECiwoVRJEBDeoKCcH9Xr477PSmffYbfICwM7p5jx5w/Rw270ISonJw6hRvi5Ut0ePjiC+PbpU6NmpuenkQdOqDtmYb0BARoQtQUmj9UPuLikNLi4UG0dSvud2NUrAg3T4UKRI0bEy1d6tx5atiFJkTlYt8++DyKFUPFEktpA7lyIY/swgWiwYOdM8eUhr8/0fPnRGrLz1SaqCg0l9aEqDwMHYoiKtu24T43R7ZsCDgaMACvqVO1gCOVowlROdi5E5VJPv4Y3Rxy5rRuvxo1sPpcvpxo1SpZp5giEYWE1l80MefPI2pZq1QkPb/8gnt6yRLr62B7exP9+CPR9OnIKx0zRhOkKkYTolKzcSNMsm3aYOVpZTWb/+jVi6hvX6xCJaywo0FYzBQtqpl0kxIYSJQhg9boXWrOnkUjia+/xstWxo2DMP3hB6TFCILkU9RwHK3YgpT88gtR795E3boRrV6dPIDIWhYuRAm29u1h3rVkAtKwnoAATRNNSmAgtCRXLBChVoKD0aj7o4+IfvrJ/nEGD0Yjij59YHZfuVL7nVSGpolKxeLF0CL79CFas8Z+AUqEwIOtW2HC6dhRCzSSkoAAosuXiSIjlZ6JOmDWgoqkJj4e0bYJCbBGOdoL+OuvEdm/di1R164YX0M1aEJUCkRzy7Bh8H9IUcIvd27cgGfOIDBBQxr8/Yl0OkRBaqByzps3mj9USkaOxMJk61aivHmlGbNrV6ItWxB82KEDUUyMNONqOIwmRB2BmWjyZKLRo9GZYe5cafsw1qoFDXfpUpiKNRynTBkiPz/NLyoSGIhrtkYNpWfiHqxbB3fMwoVEtWtLO3a7dmjw/fffCFzUqhupAk2I2gszynVNmUI0cyZC0eVoZPzNNzDn9OuHQAUNx/D0xOJE84uCwEAEFPn5KT0T1+fiRdyvPXvifpWDpk3RfzgwEDV31djYPYWhCVF7EASigQOheS5ciBB0OfnpJ6LKlbESDQ6W91gpgYAAFMLQoh01f6hUvH5N1LYtFiRLl8qzoBZp0ADa6NWrSKN7+1a+Y2lYRBOitqLTIYBILBw/aJD8x0yTBv5RnQ4BC1pggWMEBKAh8q1bSs9EWd6+Jbp5U/OHOkpCAlGnTujMtH277Wlt9uDvj45QDx6gqEtIiPzH1DCKJkRtIT4eDv516/Dq3dt5x86TB4EKgYFEw4c777juSPXqSBNI6X5RMQ9Z00QdY+xY1LrdsoUof37nHbdyZaKjR2GdqlcPdbo1nI4mRK0lNhbpJtu342bp2tX5cwgIIFq0CObdX391/vHdBR8fokqVNL9oYCBRjhxoEK1hH5s2wa0zbx5R/frOP365ckTHj6PNX926aGen4VQ0IWoNgoACCvv2oaRfu3bKzaVvX30+6vnzys3D1fH3V0wTXbJkCZUpU4aqVaumyPH/Q/SHyum/c2euXCH66is0lnCGW8cUxYsTnTiBf3/yCVKWNJyGB7NWlNEiw4ejfdH27SjnpzQxMTDfBAVBkObIofSMXI/Nm4k6d4YpTKHzFxERQX5+fhQeHk6+vr7OPXh8PFGmTEjRGjnSucd2B96+RYemTJnQoSl9eqVnBP9orVoQqgcPEqVLp/SMUgSaJmqJRYuI5s/HXzUIUCIELmzbBhOzFmhkHym9GP3Vq8gz1IKKbEenI+rSBekl27erQ4ASoS70n38SXboE7VinU3pGKQJNiJpj+3aiIUOIRoxASouayJcPgUYnT2qahD3kz49XSg0uCgxEecnKlZWeiesxYQI0vU2biAoVUno2ialeHfPauRMV1DRDo+xoQtQUgYEIHvrsM6LZs5WejXHq1EGXh4ULES2sYRspuRj9yZPoceuMdAx3YutWFFeZPRuNs9VIq1ZovbZoEdxQGrKiCVFj3L2LC7F6daL//U+aWrhy0b8/UY8eqJRy8aLSs3Et/P3hU06JdUgDAzVTrq3cuIF7rVMn9aeZ9e2L1Jvhw5FNoCEbKpYOChESgtJaOXIQ7dih/pW6hwcKP5Qrh4opoaFKz8h1CAhAh5yUtvh49gwvLT/Uet69Q0xEkSJoc+gKEc3Tp8Oa1q0b0mA0ZEHrJ2rIhw9ELVog4OLwYaIsWZSekXWkTQv/bZUqWCX//bdjrdhSChUqIGf05EnHBIpOh7SCkBDTr9jY5PuJAWHt2iX+vby9ibJnx0JO/Gv47+zZHVvciSZsTYhahyAgUOf1a1gufHyUnpF1eHigcUVQEFHr1rjOy5RRelZuh/akFUlIQMrDzZtYtRUsqPSMbCN/fqLffydq1AhdZebNU3pG6sfbG91LrAnOevsW5rzr1/G6fRvpMSEheLgmDeBIk4YoZ0690DP24BWFqI8PUapU+vfj4jD+iRMY//375PtmzKgXriVKwBJRrhxR2bIIOjOnKQUGQqPKmdP8d9YAU6ag6PuePYiAdSVSp8YCu04dWNhOn0abRQ3J0PJEifAA7N8ftXD//BPdEVyVRYuIBg8m2rCB6PPPlZ6N+pkwgejnnyEQPTxgjbh5Uy8sxdfLl9je25uoZEmi0qVRilEUZElfGTJYNPlZnScaEwMzfUiI/q/471evIHBv3MDciYh8ffVCVRSs5crp82GrVcP8166V4AS6Obt2wYw7YwZ8jK7K8+dENWviGjh2DIswDUnQNFEiRNotX060apVrC1AiNAe/cAF1fcuUQXk7DdMUKwZh1Lo1BNHDh3jfwwPaWrlyaG0lCqQSJbC6dyZp0+pTckwhCERPnyYW/GfOIDAuLg7bZM+OlJYLF1DZRhDUHTSnNLdvw5/Yvr38nZrkJl8+or170eO0Y0ei3bsTWz807EbTRDdsgL9j4kSYbdyB6GjcLG/ewIeTLZvSM1IPISFER44QHTqElyg0CxeGxlGhAoRl6dKy+76cUrEoIQGVbK5fJ7p2DZaWCxfwWdasaKvVqBE6gRQv7hoBM84gIkLfqOD0affR3A4fhqLwxReuEyClclK2EBUvqK5d4YB3pwvqyROUJatYETV/U2qgUUQEfNyi0Lx2De+XLg3B0agR0XffYdGxYoWTp6ZA2b8ZM4hmzULk+dGjuAfOnEFwVL58+nPSsCH+nxIRBAR7HT1KdO4cFhfuxPr10LAnTULZRw3H4JTK48fMmTIxf/wxc1yc0rORh8OHmb28mEeOVHomzuXdO+bVq5kbNcL3J2LOn5+5Rw/mdeuYX7xIvP033zCXKeP0aYaHhzMRcXh4uPMO2rw58yefJH4vIoL5r7+Yhw1jrlQJ54uIuXx55pkzca+kJKZOxfffvVvpmcjH9On4jrt2KT0TlydlCtG4OOaaNZkLFmR++1bp2cjL/Pm4WTZuVHom8hIdzbxtG3O7dsxp0jB7eDA3aMC8dCnzvXvMgmB6319/xTly8rXgdCGq0zFnycI8ebL57UJDmbdsYe7cmTldOpybgACcy9BQ58xVKXbvxrUzZYrSM5EXQWBu04Y5c2bmJ0+Uno1LkzKF6OjRzN7ezKdOKT0T+REE5q5d8TC8fFnp2UhLQgLzwYPMX33F7OeHh33lysxz5zI/f279OPfuYd+//pJtqsZwuhC9eRPf88AB6/eJiID23rQptHpvb+ZmzZjXr2d+/16+uSrBnTvMvr7MrVphweHuvHnDXKAAs7+/+1rjnEDKE6L79uFBMnu20jNxHh8+wExXuDBuHFdGEJjPn4fpMU8e/JZFizJPmAAhYe+YOXIwjxsn7Vwt4HQhumoVs6cns73HCw5mXrwYD10i5vTpmbt0gfbm6g/hiAiY9EuWhDsgpXDyJBZHY8cqPROXJWUJ0ZcvmbNnZ/7005Sx0jTk0SOY8j75BBqcqxEfDxNj9ep4gOfIwTxoEPPp0+ZNtdbSpg1zvXqOj2MDTheiX33FXLGiNGM9fAi/Wpkyep/zvHn2C2glEQTm9u2ZM2ZkvnVL6dk4n1mz8Bvu36/0TFySlCNEExKYGzZkzp0bK+qUyIED0ERGj1Z6Jtbz/j3zokXQoonwG+7eDaEqJXPmQLNyokbldCFasiRzv37SjikIzBcuMHfvDlOvry/ziBHMT59Kexw5mTkT19aOHUrPRBl0OigWOXJA0dCwiZQjRKdORcDA4cNKz0RZ5szBA2PzZqVnYp6XL2FiypwZ5qbPP8fDWi5OnsR5OXdOvmMkwalCNDQU32/9evmO8fw5Fmh+fhCoXbsyX7wo3/GkYO9ePBfGj1d6JsoSHAwFo2FD17RUKUjKEKLHjkEDmzhR6ZkojyAg6jJ9euarV5WeTXKuX2fu2ZM5dWqY14YNc070YEwMonoXLpT9UIsXL+bSpUtziRIlnCdEd++GEH34UP5jRUQw//gjot9F68GePdKY3aXk/n2kuTVrpgkOZigYHh5QODSsxv2FaGgoc968zHXrSm8CdFUiI5krVEBAjlpSfI4fRwQoEX6vOXOcH+AREMD82WdOO5xTNdExY5hz5XKuIIuPh8WjWjX8rmXKMP/vf+oQWJGRyIMtVow5LEzp2aiHiROhcBw7pvRMXAb3LpzJjLqnMTFEv/2Wcqv2JMXHh2jnTqKwMFRr0umUm8vdu+iDWrcu0YsXROvWoRTfiBFEfn7OnYu/v75NmLshNuF2ZlUub2+izz5DRaTjx1GnuEcP1O89cMB580gKM1GvXrjOduwgypRJubmojYkT0fGlSxetN7GVuLcQ/fFH1Apdu5Yob16lZ6MuChcm2rSJaP9+3DjO5s0bdJspW5bo0iUsci5dQk1PZxd4FwkIQLeLp0+VOb5cxMcTnT2rXP9QDw88mHft0teh/eQTombNUPTf2cybR7R5M4rzlyvn/OOrGS8v1BOPi8OCRxCUnpH6UVoVlo2zZ5lTpWIePlzpmaib2bNhatu61TnHi4lBMYRMmRDJOWsWqg2pgeBgnIvffnPK4Zxmzj17Ft9LLcVFBAHXW9GiMB1+8w3zq1fOObYYoT5mjHOO56rs2YNrZu5cpWeietxTiL57h5SI6tWZY2OVno26EQT4AX18ENQj53G2bGEuUgTRtv37M4eEyHc8eylenHngQKccymlCdMECBE3FxMh7HFuJjUVZysyZmTNkYJ42DYVB5OLRI+asWV03V9rZjByJKOvTp5WeiapxP3MuM9E33xC9fQtzpVKmQVfBwwMdbIoUgW/y3Tvpj3H6NLqkfPYZuqdcu0a0ZAn6W6oNd/SLBgaiEXeaNErPJDGpUxMNHUp0/z7R11+jFWHJkvCLS21GjIrC9e3rS7RxI8yWGuaZPp2oShWiTp0QP6FhFPcTor/9RrRlCxpsFy6s9GxcAzHQ6PVr+CSleoC9fUvUvTtRrVpEHz4QHTwIH3Xp0tKMLwcBAURXrhBFRio9E2lgJjp5Ujl/qDVkyUI0fz7RzZtENWrgmqlZUzp/qbiwvnMHgURZskgzrruTKhUUkXfvEOinYRT3EqJhYUTDhmHl1KGD0rNxLYoUwQp9zx5pegzu2YOgjd27saC5cAF9KtWOvz+ilc+eVXom0vD0KdHLl+oWoiLFihFt3Up04gQWMZUrE/3wg+PR44sWIVhm9Wr019WwnkKF0H/2l1+I/vlH6dmoE6XtyZLSty+CVZL2i9SwHkdLoL17hxqtRMxNmtjWTUUN6HQIenJCwrlTfKK//YbfQo3+Z3NER6N8oIcH2hbeuWPfOEeOwAevBRjaj07HXKMGc9myrt9oQAbcRxM9c4bo55+Jpk0jypNH6dm4LqNHQ4vv3p3o1i3b9j14kKh8eaLffydauRLaqKulFnl6wvx88qTSM5GGwECi4sXV6X82R9q0RHPmQPt5/Roa5I8/2uZqePoUfvh69aBNadiHpyfR8uV4HixYoPRsVId7CNGEBKK+fWH+6d9f6dm4Nh4eRGvWEBUoQNSmDVF4uOV9IiNx3j/+GA/sa9eIevd2bmK/lAQEEJ065R45coGBrmHKNYW/P3zU33yDIKT69YkePLC8X0wMUfv2ROnTIydUK7TiGJUqIa978mSix48Vnoy6cA8humQJbrTly7WoOynIkAEBGMHBRN26mRcmJ05AS/j1V6LFi1GJpmBB581VDvz9iSIilCkEICWRkbgvAgKUnoljpE9PtHAh0dGjKIZRsSLR0qWmr0tmon79iK5fJ9q+nShbNqdO122ZMgVBWd9+i3OsQUTuIERfvCAaPx6aUNWqSs/GfSheHJHOf/5J9P33yT+PjUUQV716RLlz42E9YABMP65O9epYjLl6qsvZswjKcWVN1JB69YiuXsXCbsAAVD16+TL5dsuWoRrRihWwTmlIQ8aMCNL6809Un9IASjtlHaZjRxTWTknd6J3JtGkITNm1S//ey5fMtWqh08rcue6ZuF6lCnO3brIeQvbAou+/R5CUOzag//tvNCrIlYs5MFD//okTKBAwaJByc3NnBIG5eXPmfPnQ61eDPZhdWC/fuxf1N3/7DQWTNaRHEBBodPAgNJvwcCSte3jA5Fu9utIzlIfBg4n++guFAGQiIiKC/Pz8KDw8nHx9faU/QLNm+Ltnj/Rjq4HgYFybZ87AvNu0KYoDlCyJ6zVVKqVn6J48eoSa1wMGIPgrheO6trfoaKKBA4kaNybq3Fnp2bgvnp7wd+bLR9SgAbqtFCpEdP68+wpQIvgRHzzAg9oVEQQER7mLKdcYOXMSHTqEILavv4bp1tsbxVY0ASofhQsTTZiASN2rV5WejeK4rhCdMQNBBkuWuG4UqKuQNi38za9ewf956BD+ujOi8HHVVJdbt1Bpxp2FKBFKBy5dii4xISGul8rjqgwfTlSiBLIi3CGK3QFcU4jevk00ezbR2LH4ITXk4/Vrok8/RTWjfv2InjwhmjtX6VnJT758SPNx1eCiwEAER7mztUBkxQpEiY8bh4Ve1aqwlGjIR+rUyIY4dQqVoFIwridEmRGJW7Ag0ZgxSs/Gvbl8GQ+k69ehfS5dijD3SZMQoefu+Pu7riYaGIhUkAwZlJ6JvJw6BbdO//4omH7+PIqt1KmDQvYa8lG3LnqOjh4NK0AKxfWE6IYNREeOwIybNq3Ss3FfNm+GEMmaFQ+munXx/vjxRK1aEXXtSnT3rrJzlJuAANT8jYmRdNglS5ZQmTJlqFq1apKOmwi1F52XgqAgFFSoXl1fSSdvXqJjxxBo2L070rASEpSdpzvzww9wp40cqfRMlEPp8GCbePuWOUcO5s6dlZ6Je/Pjj0hr+fxz4/0dw8OZS5ZkLl2aOSLC+fNzFhcu4DycOCHL8LKluISEOLW5uCLExjIHBDDnzs0cFJT8c0Fg/ukn1M3t2FHrKywnq1bhejtyROmZKIJraaIzZiAqd/58pWfivsycSTRkCNGoUUTr16NaTFJ8fdE67flzoi+/dN/AggoV0CbO1fyi4nxdvVKROYYORcrVtm1EuXIl/9zDA2berVtRGKB9e8ktChr/0rMnWteNHJkiKxm5TkHJ4GCYcEeMcP/IUCVgJpo4EQX8J0/Gv81FPZcqBSHbujWKe48b57SpOg1vb/S3tNYvGhdHFBoK/5CxV1hY4gVHfDz+duqkT8nw80OEaY4ceCX9t4+P5Wj0wED4BfPnt/07uwK//AL//M8/o1mAOdq0IfrjD/xt2RKLPx8fJ0wyBeHpCX90o0bIrW7RQukZORXXKbYwfDiiwB49IsqcWenZuBfMOL8LFsDHYYt/Y/JkoqlTEWgkJve7ExMnooxcSIheeH34gAbS168nfhkrQZcxo14IZsmSqLZzRHw8+e3dS+FNm5JvqlT4HcLD9UL37dvk42XKhET3cuXwEv9tmNpRpw4Wmlu2SHsu1MDZs/h+X36JqFxrOXYMD/dKlfCgl6O4RUqGGc0BIiMRQ5GC0g5dQ4i+eoWm0aNGSdMwWkOPIKDyyPLlKCA/YIDt+7dujRSD8+fRWNmd+OMPfL+BA9Fa6/p1LOSY8aAoUkQv0IoWRQEAQ80xXTqTQ1usWBQfT/TmjV6ohobi2DduYB63b0P7JcLxRIG6bBkWNmPHynRSFCI4GBWJ8uWDUEyTxrb9T59GVaNixYj278eiRkM6jh2DIN25E/dMCsE1hOjQoWjP9fgxVuIa0pCQQNSrF1IBVq0i+uor+8YJD0eEZKpUSDnImFHaeToTnY7o0iWk9Bw+jMVBdDRMgHXqJNb+Spd2yDTocNm/hASUJRQ14Rs3YHoOCoKJrUoVooYNYWYLCDDu33YV4uNRnezOHURM29un9tIlFK7PnRsdh3LmlHaeKZ2GDWFBuXjRPZpRWIOSUU1W8fIlc9q0zFOmKD0T9yI2FlGLXl7SRHHevMmcIQNz+/aIjHQlgoOZly5lbtsWBduJmH18mJs2ZZ4zh7lYMeZevSQ/rCzRufPmoTHA8uWIrs6ZE98ndWrmevVQlP72bemO5ywGD0ZheSkipW/cQFRvyZLMz545Pp6GnuPHcb1t26b0TJyG+oXooEF4sGldWqQjOpq5ZUs8WHfskG7c7dtxA82cKd2YcvH+PfO6dRCUXl54QNepwzx5Mh7UhikRffognUdiZBGi7dvje4gIAoTGokXMbdow+/riN6pcGR14nj+X7thysXYt5rx4sXRj3rvHXKAAc+HCzI8eSTeuBnPjxszly7tn9yAjqFuIPn/OnCYNVs8a0qDTQQNNm5Z5717pxx8/ntnDQ56xHSUujnn3buQZp0uHB3Pt2tBCQ0NN7yc+xN+8kXQ6kgtRQUBrsDFjTG8THQ0toX173FseHswNGjCvXIk8bLVx4QKu1Z49pbdwPHnCXLQoc/Hi5n9/Dds4eRL3y5YtSs/EKahbiA4cyJw5M5L7NaRh2DA8OLdvl2d8nY65WTNYD+7fl+cYtnLvHq6lrFlxc5ctyzxjhvUayP372O+vvySdluRC9OFDzPOPP6zbPiyM+ZdfoDl4ejKnSgWT9tGj6jDJh4ZCW6xaFcJfDu7fZ86eHf1xo6LkOUZK5JNPcJ+lAG1UvUL02TOYG6dNU3om7sPChXjI/vSTvMcJC4MfsVw5ZRv3njzJ3K4dFg3ZszOPGsV85Yrt4wgCfIvjxkk6PcmF6Pr1+H3t0apevmResAAPPiIIro0bmePjpZmbrcTHMzdsiN/t6VN5j3X2LHP69FhAuGODeSU4dQrX0aZNSs9EdtQpRAWBuUUL5ixZNC1UKrZtgzAZMcI5x7t+HYFGn33mXK0mIQHftVYt3MQlSzKvWOG4ltG2LXP9+tLM8V8kF6L9++P7OoIgwBTfqBHOX8GCzPPnO7+844gR8FU7q5Tc7t3Qxr/9Vh1auDvwySco0xocrPRMZEWdQnTvXtzA2bLB3BQXp/SMXJuTJ+FX6tTJueaVbdvwO/7wg/zH+vABgSdFi+KY9erhwSjV9507F35UCa9FyYVoxYrwHUrFpUvMX3yBoCs/P+aRI5lfvJBufFNs3Ijf8Mcf5T+WIcuX47hz5zr3uO7Gu3fM06fjmiFiHjJE6RnJijqFaN+++AGaNdOvhpcskc8v4s7cuQNfYN26ypy/ceOwwv/7b3nG1+mYf/2VOV8+HKdTJ5jnpEY0T507J9mQkgrR8HB8/1WrHB8rKc+ewRTu64uFxIQJ8pnpL1/GMbp2VUYjHDsWv/PGjc4/tqsTGorAQj8/BK317w+TfIkSyrkFnID6hOjjxwhwmDUL/796lblLFzwgcuVC3p6SfjZXIjiYuUgRpGcoFXmZkMDcpAlM8w8fSjv2kSNI1SBCtOm9e9KOb0hMDB4MEmpHkgrRAwdwHm7edHwsU7x7h8jfNGlwL65cKa0P8c0bpJx89JHx7kHOQBCgfadOjQArDcu8fMk8fDhyq9OnR/CiaLE4fx7X5dq1ys5RRtQnRL/5BmbcpILy7l0kvHt744E8dSoCWDSMExnJXK0aHnaPHys7l7dvYWatWFGah+OdO8ytW+PmrF5dtlZlyahdGz5eZp4xYwZXrVqVM2TIwNmzZ+fWrVvzbRuLGEgqRKdMQSS7M8z1jx+jkAMRgsf27XN8zIQE+NCyZlU+bzM2Fj7hTJmQY6thnCdPmAcMwKLK15f5u++MB7W1aoVAQzfVRtUlRB8/hpA050N78gTpCmnTMmfMiJWxmzuubUanw4WbIQPzxYtKzwZcvYqVapcu9pvpQkMR+OHtDRP/b78518c7ahRz3rzMgsCffvopr1mzhq9fv86XL1/m5s2bc4ECBTgyMtLq4SQVop98AveHMzlzBgsLIuZPP8VvbC9jxsDadPCgdPNzhHfvUDCgQAEthzQpd+8yf/UV7sOsWZFBYU6huXgR18i6dU6bojNRlxAdOxb2dGseREFBCHTIkAE+lMGDtRJeInPm4KL980+lZ5KYLVvsD9zYsgU3rK8vTP1K+Hd37cL8jWj2ISEhTER87Ngxq4eTTIgmJOC8TJ/u2Dj2IAjIOS5WDNG0Eyfa3gD7999xXufMkWeO9vL0KaxizZtrEbvMzNeu6V1ruXOjxKS1rrWPP0bEvBuiHiEaF4dcvIEDbdvvzRvmSZNgykqVCubgBw9kmaJLcPo0VogjRyo9E+OMHo2b8MAB67YPDUWwkOj3fPVK3vmZIyQE8zBSa/jevXtMRHzt2jWrh5NMiF69ink5Kx3EGLGxuA+9vZkrVbI+H/faNVgoOndWp6D66y+c23nzlJ6Jcpw7h5KRRNDMly61fRErRurbk6etctQjRMXVqA0PoURERDDPno28JC8vBAekNH9GWBhzoULMNWqoNy3IFt/Xrl1YWGXJgmhJNTxkS5SAH8gAQRC4ZcuWXLt2bbO7xsTEcHh4+H+vZ8+eSSNEly/HNa9UMI4hFy7AT5oqFcx85vxgoq+8fHnrrE9KMWIEFgdnzig9E+dy4gTM9EQojbhmjf3Plbg4xGf07y/pFNWAeoToxx8z+/s7Pk5UFIpt58uH4gLt2uHGdncEgblDB5jDlQ7MsIQYhVmpkvEHf1gYc/fuuHlbtED0n1ro2RPzNqB///5csGBBfmbBnTBp0iQmomQvh4Vo9+7MVao4NoaUxMTANePpicpHxhazCQko/p85s3rKQ5oiNhYL00KF3D+YURCQjla3rj5wbONGaaKwv/sObgc1L5jsQB1CVKxN+r//STdmbCxy5sTk+6ZNmf/5R7rx1cbSpa7VgujKFYTDJ80HPHAAwTu+vlj5qkH7NGTlSgiHfyv4DBw4kPPly8cPrUjfkU0TLVYM3Y7UxunTqKCUJg384Ia/pdioQIrIXmfw6BEWqB06qO+alAKdDpafatXwHKlWDf+XMnDv8WP85nLkMiuIOoTo6NEIJ5ejAHR8PPOGDfqaoPXq4UHtTjfC5cv65GZXQqxMs2ABfo/Zs3GTNW4sf71Ue7l5k5mIhQMHeMCAAZwnTx6+e/euXUNJ4hN99QrncPNm+8eQk6go5qFDMccOHRCIsmMH/j9jhtKzs42tWzHvpUuVnol0JCSgvm358vhudesy798v3/OxaVMIaDdCeSEaGws/ptwraZ0ON2/Vqvr8wl27XF+Yvn+P1X7Fiq5Z0WnkSGh2DRvid/nuO3UXAdfpmDNn5j+qVWM/Pz8+evQoBwUF/feKsmEhKIkQFQWS2iPTt21DAFGJEvjris3bmfV5kZcvKz0Tx4iLg6WnRAl9itLx4/Ifd+dOHE8tqXcS4MHMTEqyZQtRp05EN24QlSkj//GYif7+m2j6dKITJ4jKlycaN46oY0ciLy/5jy81PXoQbd1KdOECUcmSSs/Gdh48IKpUiSgykmjpUqJ+/ZSekWVatKD9f/1FTYx8tGbNGurRowdRfDxRaChRSEjiV1gYrkEiioiNJb+5cyl8xAjyTZOGKGNGohw5iLJnT/zXx8f0XEaNItq4kejZM1m+qqScOkVUrx6RIBBt307UqpXSM7KdmBiiWrWIoqJwz2XIoPSMbCMmhmjNGqLZs4mePCFq0wbPv2rVnHP8hASiQoWIWrQgWr7cOceUG6WlODdsiIRtJTh2DJGiRFiRuVqx+3XrXLuk1pEjyMMrUIA5Tx6U8HOFno7Tp8Nnm5AA3+jp0/DzDBkCU3SuXPhdkr58fBDwVrAgc8GCHJ4/PzTR/PlxDjJlMr6fry9zzZrMvXuj7ODBgzDjCgJzQABSgNSOWAAkY0bc756e6A7jitronTv4Lb/8UumZWE9kJNJ0cufGue/c2bHiGI4waRLy+53dGUgmlBWid++qo5LF2bOJ86AWL1b/wzw4GJGNXbsqPRPbEQT0NPXywiLq9Wt0DEmXDpGman6wPn+OovpECIASBZ2HB9IA2rVDgfZffkGxi7NnEZRiJCLRqDk3NhZ1Ry9fRpTkhg0oLtGtG2rKpkmjP2a2bHggtm+P1DA1n7cpU3COdu/G4mPUKHyHbt3Uf68ZY80azF+uxgpS8e4dUo2yZUOaTs+eWAQoybNnuG6XL1d2HhKhrBAdORI5gGrx5RkWu8+ZU93F7rt3x7kLCVF6JrYhCPjdxRZJhnmEGzbg/YULlZtfUiIi4M8Te3UaaogNG6KDzIULdgkCu3yiCQl4CG7bhtJrRMjJJEJsQefO6J+qJh/p7t2Y39Spid/fuBELp/r1XU8rEQTMu1gx9Ty/DAkNRXyBYUcVpWtoG9KyJRaFal74WYlyQjQmBqujoUMVm4JJxGL3qVJBUE2ZolwXFGMcOYKH0sqVSs/ENnQ6BGaYE5RDh0JDVbKDRmwsgs4++ww1monwsOzTB+UHQ0IQoPbFFw4dxuHAorlzkSYUHo6I87Fjkc/o6Qmtr1495p9/Rl6uUty5A3N069bG0yVOntSbq10tB/PmTTwjJk1SeiZ6Xr5EF5X06WFyHj5cXXnWIn/+iftKjraFTkY5ISqmN8jZuslRkha7Hz1a+WL3sbFobebv79zi646SkABTkoeHeeEfH48Vfvbszk1z0engI//mG5jJiZgrVEDajbHiFYMHo2CEAzgsRNu2xblKSlgYcq4/+QQCNVUq+CM3bXJuVaOICOYyZaDBm/uO589jsVqpkutZVr77Dm3TlDaRPnrE3K8ftE4/P+ThqrlwfkICXGe9eik9E4dRTojWr4+cJFfg1Sv4cMRi94MGKWcumzEDmpor1aCMi4OZ0cuLef16y9uHhOAGq1pVflNZVBR8M2Kof8GC0OgslZ8Ui+k7sMp3SIgKAlwOY8ea3y4oCFp/9eqYb+bM8OnKrZ0IAvzDGTNat1C+dg3fp0wZdWpOpoiKQs/eRo2UMU3eucPco4e+o8r06fCDugJTp0JjdpX5mkAZIXr7Nm7oDRsUObzdJC12//XXzi1Z9uABtOIRI5x3TEeJiYEpL1Uq26opXbiA79qjhzwPp+Bg/JaGwTnHjlmv3T9/jmt461a7p+CQEH3wwPZOPffuwVyeIQO0p549ma9ft/3Y1jBjBua3Y4f1+9y5g+jlokXV5b+zxN69zn+eXb2KhanYUWX+fNcrp/fiBRbWS5YoPROHUEaIDh+OVVNMjCKHdxjDYveenoiQlethJCII6BeZP796g52SEhWFJO60adENw1bWrsXD6aefpJvTw4cw2aZNi1XwwIH2L4QKFoT/yU4cEqLiubHH3xkWhp69YnRxkybSJtrv2wez/YQJtu/76BE0u/z51V9T15COHfE8kDt24uxZfUP6ggWZly1TZ2CTtbRtC7eJC+N8ISoIuHnVWOvTVgyL3RPhgjh/Xp5jia2EbFnZK0lCAs5HunTMhw7ZP87gwTBVOfqQDwvD4i11apgNp01Dao0jdOmCQB47cUiI9u3LXKqU3cdmZvjX161DtSsipHk56tu7fx/5rs2a2e+zf/4c5vWiRZWPQbCW589huu7XT57xk+a0O9JRRU2IFYyU9ik7gPOF6PnzOGmHDzv90LIhFrsvVky/sj9xQrrxIyKw8GjZ0jVCwgWB+dtvoaXv3u3YWHFx8J/nyGGfHzouDgudrFkRrThlinRmr8WLYaa2M8/RISFaoYJ0QRk6HXqkFiyIBcu339oXlBIZiRqsxYo5Hmn76BGKVlSr5jpmyoULoYGfPi3NeIIArb5OHX2g2+bN6i6LaSsfPsAqpLaG7DbgfCE6cSJ8iu6wikpKfDweRmKxe6mKOY8cCY1O7S3ORObOxfeXKpk6OBjmverVrTddCQLSVEqUwIOtVy/pA1YuXcL3tFNLtluIvnuH7/TLL3Yd1yTR0XBT+PoiwnPOHCwQrUEQUDnJx0c618bFi/Dftmxpvi+pWoiPR9WtypUdi5w3Vuf7jz9cYwFtDy1bKle1TgKcL0QrVnTNKju2kPQmqFYNZgt7bqyXL7FSU1Mumjk2bcJ3thQ1aivnziF8/6uvLD9MQkLQMYQIfWrlimSOj8dDfuZMu3a3W4ju34/vdvu2Xce1SEgI8nm9vKBZXrpkeZ85czCn33+Xdi779mEe/fq5hhA5dsz+loQJCUj9K1dO33Hq779d43s7wqpVsFq5WnrTvzhXiD5+jItjyxanHlYxkppjypWDpmqLOWbQIPiYXCEM/Ngx+ByT9giVCrHU2rJlprfZvh05plmzwvQl9wOocWPkYNqB3UJ00iR8P7m/2+XLWPR6e8MMbsp6dOAAHoJjxsgzj9Wr8bvbuVhxOo0aYfFh7aI5Lg5WheLFndtRRS28egXLypo1Ss/ELpwrRBctgg/J0SbErsixY7g5iHCzrF5t2VT2/Dm0r6Tl0tTIjRsQ9g0aWG8CtIeBA3ENJW2w/vYthLcYIPPqlXxzMGTiRLsFmt1CtHFjmMCcQWwsomy9vGCmTJo/+/AhCiV88om8vrrJk/HbWpNnrDT//GOdVh4djfSOAgX01+25c86Zo9qoVQvf3wVxrhBt3Bg3W0rGsNh9/vxI3zAVmDJwIPzHal90vHuHtIRy5eQv3RYXB80+Vy7kmTFDE8qdG0J83Trnmr+sNa3GxCAw6sIFBJ2dOMHh+/ZBiO7bx3zmDASSpSCahASYkJ3d0PrcORRCSJ0aPm9BQFBIpUqo3CR3aUFBgCk/VSrXEDQff4zYCGPa6Pv3OIe5ckGD79LFcnEPd2fWLKScuWAzAucJ0XfvYBZy8cRaybh2jfnzz/XF7n/4IXER7mfP8MCaNk25OVqDICBHztcXBQCcwatXiFauUQM3n6cnHlrPnzvn+IYYBvmEhUELWb4cC6AGDWB18PMz2uIsnAhCNOln6dIhUrZmTRREmDcPboHnz/XBTMeOOf+7Rkfrmwd06oRX+vTOq54VG4v4giJF1O/eCAzEedq0Sf9eWBjz99/DcuHtjUXB3buKTVFV3LqF8+VoNL8COK8p96ZNRF26ED19SpQ/v1MO6RLcv080axbR2rVo8Dt4MNGgQUTjx+OcPXpE5Our9CxN8/PPRH37orl6x47OO+7x40QNGqDB8+jRaLLu7KbqQUFEhw7h94qPR2NxIsyjRAmismXRgDhHjsQvHx8iDw+KiIwkv6pVKfz8efL19kYTb8NG3s+fE928iYb10dEY28cHDaHnzCFq0gSN7D08nPu9t24l6tqVKC6OaNEiom+/dd6xHz4k+ugjfPdNm5z/3W2haVM0vj58mOinn4gWLyaKjSXq3Zto5EiiggWVnqF6YCYqWRJN21euVHo2tuE0cd2lC1rfaBjn6VPk56VNizQBLy/UOFUzV69ivn37Ove4T57AP5c6NVavP//snOMKAvOpUygAUbq0XnPMmhVm9w0boJVZWYnLap+oTgctf9cuBKxkzKj/7jlzwqKxY4fzKoAdOQLt388P392RYhr2INYtdtbvbi9ip5LUqdXdUUUtjBiB69mVGmuws8y5cXG44SZPdsrhXJpXr+BnIkJQ0bffOrebibVERqJiToUKzvVjnDyJ6NuCBWHa7N8ffrLAQPmOefMmumIUKYLfJU8e1E3etAk5rOvW4X0bKyDZFVhUuDD6sH74gPSH0aPxGxDBJ9y7NwqZyBXk8+QJzn+jRvjujRtjwbd4sTzHM0XfvljAXb3q3ONaw6NHmF/q1PrC8K5SeUlJTpzAdXzqlNIzsQnnCNGDB3FyLl50yuFcmsePIRQmTsSiQyx237s3CoirhZ494Q+7dct5xzxwAP7COnX0OWWxscwBAQgsknKVHxeHdKRq1XDt+vmhYIMxAWVPMXi2Q4i+fGk6RezGDbTlKlwY2+TLB3+xlLVco6KQ+1ywoL6iUXw8hDqRc/33UVFYPJQqpZ6KRrdvM3/5JRYV2bIh+Evs/btundKzUz8JCThvUueYy4xzhOigQYhEdfekYSno0wcXklhkPiICQUc5c8KE9vnn8he7t4Soef3vf8475u7d0MybNk2u+QYFQTv093c8vSY8HIE8+fPrizVs327eVCoIiLS08ea3WYhu3Yo5iVHJpuZy6hQEvmhGHDQIkb+OIAgQEGnTIsI46Wfff68vsuGs+/zWLSzkevRwzvFMceUKGrh7eOA6XLAgsWBv1QqlEN2pXJ9c9OiBKHAXQn4hKghYuQ4YIPuhXJ63b/GQmj49+WdRUUiHER/ucha7N0dQELQyZ1ad2rIFZrG2bU0Ls1OnIDTs9c9GRMBk6+sLzb97dxQbsJZ27Wzuj2uzEB02DDmF1hIUhO+UJYs+lcJeYbp4Ma67tWtNbzN/PrYZNMh5fq1ffsEx9+xxzvEMOXMGApKIuVAhFAExdn2ePo1t7OlklNLYvh3nSk1WNwvIL0SvXMFJ2b9f9kO5PAsXQliYKxSQtNi9s6ubfP45NGVHO6BYy9q1eg3cUv3UlStxTlautH78hATmFSug6Yu9Wu1JlZk3D/vbUBPaZiFasyYEoa18+IDUsty5sdAYOdK2fN4TJ3BdWtN5aflyaGS9eztH8xIE+GcLF3aOb14QmI8ehYWCiLlkSeZffzX/uwsCKj/ZWdkqRREZifto3jylZ2I18gvRqVMRTShnFRt3QBBgxujY0brtxWL3Yp1NqYrdm0P0bTurPNdvv9n+QO7TB4LCmk4a+/frz1/XrgiasZdTpzDOmTNW72KTEI2Ohoa8aJH9c4yMRPm+9OkR7LJokeWFyfPnWGDUrWv9AkFc+Hz9tXNMu3fu4Df/7jv5jiEIaL5du7Z9HVWWLcM5sacTUUqjRQubrTpKIr8QrVbNesGQkhEj0w4csG0/nQ7F7cUAGEeK3ZsjOhqFA+rWdc6D8cgRvVnVlu8SE4MSYnnywJxpjLdvmb/4AuerTh1UkXKU2Fj4bOfPt3oXm4SoWEouqT/SHl6+hM/UwwOBQjduGN8uJgYFLfLls72Moljn2FnBRpMm4Xq5eVPacXU6mBirVHGso0p4OPzTWoaCZVaswIJD7upnEiGvEP3wAZFqUrXEcme++AJNiO0VfoIAzapuXfuL3ZtjyhSY9JwR1HT9OvyujRvbZ8F48QKmy9q1k++/Zw8ErJ8fAqOkXBDUrs3cvr3Vm9skRGfPxkNYypZgZ84g3zV1aoyf9Fr5+mssDOxdZEyd6rwAtOhouDjq1ZPmN42PR96v2Nawfn0scB0Z++uvsSBxhbZuSnLnjku5AOUVoseP42RY00opJfP6NR5Ws2dLM96JE2gMToQHy6pVjpnT793D/OTq0mHIixcInqpQwbGawSdPQjMRA9oiImAWFv3IcpjVRo+G8BYElKU7eRIFAUaMgEbdpAmKRBQqxJw/P4fnzQshmjcvFj0NGzJ37ozc4PnzkQf64gXGa90an0tNdDTm5+EBDV4sQ/fzzzhXq1fbP7Yg4Jx7e+O7yM3ff2POv/5q/xixsfjOYsxBkya4n6Tg/HmM+ccf0oznrggCUvtcofEGyy1E58yB/0VbeZln/nw88KVOyD53DhGt1hS7N4UgoGlAwYKwLMhJeDgCMPLlk6YO7vLl+hZaZcqgcPuKFdKbowUBwqdfP30xBrGakZcXLAwBAfgt+vRBJaoJEzh81CgI0ZEjUWv3s89Qb7dsWeTDimOIucL168PXK8f99M8/EBy+vrhvU6VCIQtHiY9HWlLGjLZFO9tLly4IfLO1IH5UFCKQ5Y5+r1KFuXlz6cd1Nz79lLlZM6VnYRXyCtEOHVzKQawIgoAIv06d5DvG9esInBGL3c+enbjYvTnE0mVyr54FAd1tfH2l7WghauQFC0pbGEKng6Wlb1/9g9fLC39btEDLrsuXoemZwKw5NyGB+f59+LfFYgaiYPX1xUN+yxZpI1LDw7FgIkIqjVRlBN+/h/AoUED+bi9BQRDYw4ZZP7c5c/QdVT7/XN6OKitWQOt//Fi+Y7gDzuqZKwHyCtH8+RFOr2Gao0fx0HJG/dF792BeS5UK2s3kyeYfaoKAh1/t2vJfzIsW4Tzs2iXNeIKAij0eHiiHlzu3ND1Gb91iHjVKLzgLFICQ+/NPLExKlrRag7PaJ/q//+FYwcGIAp42TR9IljEjiiAcOuT4bxQbC5NuhgwYu0MHfdEPR3nyBNdcmzbyX0sTJyJNwlwFK8OOKqlSIdDKGbmJ79/jNxs/Xv5juTJ79rhMvqh8QlQsUbZ1q2yHcAu6dEHUqzNXXE+fIucvbVo8MEeNMi5gdu1yjoC/cAHBLdbkIVpDQgIeikQwnT55YnuahiFibmCLFhgzSxZooCdOJA8E69kTJmkrsFqIfvON8Soud+8i4KtECcyrYkWkl9jr/+7XDwLl1CkUtM+QAT5cscSfo4jXkyNpOtYQFobAscGDk38WEoJrwtcX1//AgY6lNtlD375Y1NlzLaYUXr92mSbs8gnRHTtwErS8KNOEhUF4zJmjzPFfvUIwTMaMeKAYFrsXBHTdkTulJSICvrjKlaUxH8bFYWHi6Zk4wMSWggEiggAzdtWquJbLlkWFHHPzXLUKx7YiKMpqIVquHCI7zc3z0CH4HkWf7IIFtp1PsfKPYWeUy5dRbL5sWdPpQrYyeDCueSlSdcwxZQqC4cQSiS9eMA8dihgNHx9YyKT6TrYi9oR1wd6ZTqV4cTyTVI58QnT0aNzMGqbZsEEdC423b/HQyZJFX+x+6VLM7ehR+Y4rCPBBZcwojdkmJga+Qm9v40XalyyxPnrz4kUE+RAhbWLvXusWE2JzYSuiUa0SomFhMElbW+Di+nXUH/X0RNeZ33+3PO8zZyDYevdO/tmtW7iPixWTRmOLiYGLoFgxx6KvLfHuHcz4X36pL8Dh5wdTr7OqbZlCECAgjJ1vDT3dusFtoXLkE6L16+OBpmGaTp2ggamFiAhoxTly6HtVyhlkIRay/+03x8eKjoYmliaN6RW+IMDcmiaN6cjL0FAIIQ8PdAj580/bNHGdDouRSZMsbmqVEN27F+dITD2xlhs3EN1IhML8piJjg4MRDV2jhmnN9cEDpOUUKIB/O8r9+1g4yVk4/tYtfUvBLFkQoS2n0LYVF+2d6VQWL8ai3kxwnhqQR4gmJMCfMmuWLMO7BbGx8MtMmaL0TJLz2294+IjCtE0bpMtIyZs3SEWQIipZp0NVrLRpLVd8io7G6jZ//uQpRTt24DtnyQKt1V6fVfPmqK1qAauE6IQJOE/2mtQPHIA52NsbeXeG3ykuDlp2zpyWU4qePYP2WKyYvg2dI4h1jo8ccXwsQy5fxrXg4QG/Y/r08CmrDTGHXs4+uK7OuXMucY7kEaJXr8pzg7gTYmK42gpRCAIKHTRqlDzxXMpi9998g0WEFD1Ahw3DQ3PHDuu2f/YMwrJePQgSwzKArVo57iubMQOLyEuXsCAZNw6VjGrXRhBQpkzM6dJxeNq0EKIZMqCAevXqCF4aPBhC5tQpzNHRwuWxsagr6+UFU6pYdWrQIAhXa3/TBw9w3mrWdDxnWKeDhlyqlDR1tU+fZm7ZEr9h4cLw7cbE4LdInVqavGMpEXtnOqOAiasSG4uF8YIFSs/ELPII0ZUr4ZORKjzeHRk4EOYxteVBiStkQ40uIYF540Z9sfY6dZj37bN/7oGBGGfxYsfnu3AhxvrpJ9v2O3YMAuTzz2GqdLQMoCCgY9HcuTCNioUSiJjz5kUJw27dmIcP55OtW/O0nDl5fPbsEKLTp6MPZ69e0GJLlMD9I+5foAAE8eHDjkV0nj2LMn/p0ukLQ9j6G5w9C+2uTRvHS0pevQrBbqz1nzUIAhbqjRvju5Qqhehkw2IUaq5Z26MHfg8N0/j7y5tDLwHyCNHevaHNaBhHEGBOHDhQ6Zkkp2tXaJ7GfDVJi91XrQrtzxa/Tnw8ro2qVR1/CO/YAQ10xAj79u/ZU1+Iwd7k9/v3YSYtVUpfEKFhQwjBoUOh5ZrArDk3KkpvVm/YEFoLEaJlBwxASUF7BH5UFAS1WF/ZnupHf/6J7ydFStKIEdA2bPG1CgLyCAMC9Kk9W7aYvp7UWrNW7J1pq787JTFsGBa5KkYeIWopJD+lI4a4O6OeqC2INXx/+MH8dkmL3Zcti0hjax5S8+bhAexoSTUxOKVDB9uDMwQB5k3R9JcmDaJxbdn/wAGYt4lguu3WDQ92MTinenVouWaw6BM1DKzQ6ZAWMmIENFsi5I6uXm1bKktoKDTb/PmhBX7yiX0BN2L09qZNtu9ryPv3mEvTppYXBTod87ZtCMYjgsa/e7fl/dRas1bsnTl3rtIzUS+bN+O3k6JQikxIL0QjIqAdrFol+dBuw+TJ8AeqrcfqvHnwH9kSOGJLsXsxCd7RmqximkTRorYLAJ0OKQ9EKH/44QPGKljQclEBQYD2ULEi9q9UCeZDY/7BoUMtrqAtCtEuXSAojH2HQ4dQlN7DA4FBP/xgOYoxPh5abfbsyAc+eBC/R9WqtpfjEwQUy8+YEQsaR9i2zXxRD7GjSpky9ndUUWvN2hYt4B7RMM6TJ9JWMpMB6YXo4cP40nKmRrg6lSurz84v1vDt3Nm+/c+fZ27Xznyx+4kTYe50NHBnyBBoaLZqs/Hx0Bg9PRPnXT55AsHSsKFpbfrsWTzsiBB0dfCg+Yf4779jWzHZ3wgWhWjBghDG5rh9G1Yfb29sv3Gj6XkNHw7t0zDg7+JFmIrLl7d9tR8ejoVMlSqOFcoQBAjypOUlY2MRX1G0KM5l06YolG8PYo9KZ1cnsoQ4L6mqQrkbgoC6xmPHKj0Tk0gvRGfNgnlLqj6W7sazZ3ggbNig9EwSI9bwdTSiOmmx+x9+gHXi7Vto39YWBjfFH39gnj/+aNt+8fFIffD2Nm6CPHIEAmb48MTvR0RASIk+xH37rDveixfYx1jRh38xK0TF/a0tm3n7NjRTIgijpH420b9qLNLxxg2kg5QsafsC5/x5LGiGDLFtv6SIjQ7+/huLr59+0tcnbtfOcfO/WLN2wgTHxpGaoCBYE5zRc9VVad0awWMqRXoh2rMn/EEaxvn5ZzyszQScKEKXLogKlSpa+P59CB+x2H3dutBCHfFthIVBMLdoYds8BQH1Sr28EBhlih9/TLzAOXwY2p2PD/JGbV0YFiwI4RIUBPPjkiUowtCvH3PXrhz+2WcQot26YaU9fz7md/++3hdkawrQ4cPQ3NKlQ41anQ65k+nSYXFj6rzdv4/KRJUrW9/hR2TBAszVXi2RWd/soGBBpNGIHVWkbALfrx++o9oCjGrUQBs8DeOMGqXq4CLphWj9+toFYY4vv1RXlSJmCPTUqeUJcHj6FDmhRBCopordW8OAAbBy2JrzN3OmdQ2mBQH5omnT6ht416vH/PChbcd7+JB52TJoUt7e+lQVb288xCtVYq5bl8Nr14YQFRt1G/YQ9fbGPKZPR1k+W4KnIiNxrkTTc4ECOKal3M4rV6CtNWliWypNQgIWzuXK2ZeCI5adFLvHNGkiT/eOy5fVWbN2+HD8RhrGWbYMC2C1LX7+RXohWrAg6uZqGKdUKayI1YRYw1euhPTvvoOA+PZb48XureHsWZi9bE28Fr/bxInWbR8UBLMzEdpVWSu8QkORs1q9ur63aJEi0Kg2boR5NYkmm8ycKwgw4+7fDwGcJw/Ol5iGM3asbT1R9+yBMPb0xJjWcPAg9vnqK9u0/YsXcRxLkd2GBAej2IB4TQwciAVmzZry5U+XLIl8XDUh+s+lKDzijuzbh/Nj62LWSUgrROPj8fBYvlzSYd2GsDBcDGrzf3z2GQI75CA6Gj0bRZ9Z0mL31vRxTEjAw7VSJdtWo1ev4uHcvbt1D+UnT/CQzZgRr0aNLB/v7l0sitKlw/dp1Qqm2IgIveZz7JjRXU36RKOiIMgWL4Zmd+QItPnMmTFes2Yw21r6TmPGYOFRqBC0PGsF6a+/Ju/oYg1DhqAQg6Wc22fPUJUpXTrMa+RIvXVi924c++xZ245tLSNHwlysppiNp0/xna2tuJXSuHPHOS0Z7URaIfrwIb6stTdrSkMs9WeLNiE3sbEQGFOnyjO+qAnevp34fbHYfa5c0GC6dDEd0b1sGYTB6dPWHzcyEtVgypdPHiVsjHv3YFIrVAg37aFDmJeppvIvX8Lk6+mJh/L33yePsExIwLmdMcPoECaFqFg1KmlJyJgYCLgKFfSmZlM1jUXtZs4cBNU0awaTvbWpAn36YAFy9ap12zPjN82bF7m7xnjwAIuB1KlR+nDSpOSpNQkJ+B3k0hb/+Ud99VgFAYFdWglA48TEqDptUloheugQLtA7dyQd1m34/nvk5ampc8P+/fjNTHX5cJS6deEnN0VUFDSuAgUwj9atE2sh0dEwa37xhW3H7dkTWtHNm5a3vXkTD7ESJRK3pZs3D3PavFn/Xnw8ItB9fKBhL1xoPj/z449N5ieaFKIzZ5qPcBcE5r/+QpELIvjZDYXRtWuYX6dOem01Nhb1e729E38fU0RFYQFSujQWJNYi9iU1XADcvInUIi8vpBLNmmU+v3fqVPx2795Zf1xrSUjAHNTmcmrb1vx9ktLJmxduIRUirRBdtQorBimaK7sjLVpY1d3DqQwYAH+bHD6omzfxQN240fK2YrH74sWxzyefQCNbtAjani0Ls61bMcYvv1je9vlzlIQrWzZ5wJPY7zR9emhkt28jklIs6RcWZnn8SZNgutbpYC4+cgRCbOlSDp8zB0J0xQosZm7ehPm2ZUvrQvrj42FyzZQJi4A//8ScihWDAEwq/OLjsRjx9rauWtatW/juthTHiI9HdHCbNhCkYkeVvHmx4LCmcP2LFxC4S5ZYf1xb6NkTsQlqQlyYqcnMrCZq17ZYAUwppBWi332Hm0UjOYKApPbx45WeiR6xhq9c3eOHDsV3tmVRJRa7L18egjB1aggUa4V8eDg011atLO8THg7TaP78poOqPnxAhaIcOeDDK17cOlOgTof6tt2743v4+CQuSu/lxeGpUkGIGr6fKhUESI0ayIm1xhT9/DkKEYhlDDNlMl1FKD4e22bMaJ31YeFCCMEzZyxvKzJhgv77GHZUsYW2bXENyLG427lTfRYzMU/7yhWlZ6JOunVjrlVL6VkYRVoh2rUrVgwayXnwADfJn38qPRM9Fy8m79giFdHR0MBM+RQtodPp00yIkEO4fbtlU/jgwdYFt+h0SKXw8zOfi5iQoC8TmDu35RzKR49gKhQLBfj54W/79vjt79yBmVIQ9ObcoCDsd+SIXgCJvVwzZoS59p9/zAsUQYD2TsT80Ufmy/i9f49ArTx5LBdXiI+3LqhL7KjSqJF+MfDRR/anJYgRmXL4Lj98gL93zhzpx7aX9+9h4VixQumZqJOJExE/oUKkFaL+/lgxaCRHrBgjRUNjqZg6Vb4avmKUpTU+SWMkJEAQde8O02O9epaL3duSZjF9OjQsc2bN+Hh99aVvvsH2poI/rl9HlLOnJ6Jo+/aFOTohAZrsV18l28WoT/SXX3CcsDCYjydP1vdzNVdwfccObNOrF3y15cqZF5BBQXgoNWpk2YRoLr1I7Kji76+vJ/z77yir6EgQnU4HIW+p7KG9tGyJa0pNVKyovvQbtSBeT9ZYZpyMtEI0d271ldVSC4MHI29QTTRpAtOeHPTujUAdexGFsGH06T//6M2WxYqhrqrhAqBxYwTCWEr4P3ECws6cad1YIM6cOclL+YWGQlP19ITpcunS5H6//v2N+uCMCtHevSEADdHpcD7E2r0NGyaOmr11Cxpr+/YQaqYCpZJy8CCE47RpprcR6dsXZmIx2Eeng+9Z7KhSsyY0bVHAx8TAlO+IEOzbF/eMHCbduXNhsVBTAv8332CRqJEc0dxt76JcRqQTolFR1gdzpERq1UIah1oQBJhb5WhWrNOhPJ+9fT6ZEYRlqrLThQv6Yvf58iH46OBB/P/3382P+/49IoHr1DH9ABUEWFRSp07cPkvsXOLjAyG2cyfMrpkyQUszpdGvX4+5JUmBMSpES5eGUDY1r927IZBTpUK09+vXyG0tUyaxqfn+fXzPMmXMl5gcPx4LAFOpMiIvXsAEOmkS87p1+o4qDRqYLsY/ciS0cnu1hz175GtmceKE8TQiJVm9Goua9++Vnon6EHNp//pL6ZkkQzoheuuWNAXM3ZWsWfHQUwt37+L3sragui2cOoWxT5ywb/+nT/Fgt5Tsf+MGok29vCBUcue2nBYxfDgChMxVPxF7jRqLKo6MRMCL6Ots2dJypRkxN7FZM6T8FCnC7OvL4WnTQojmzYtKR507Y7vvvzfv+42JYR43DucoSxZoocaCZG7fxuf16pkO7ImPhxmxcmXzZt3YWJh+PT313+XkSfPf+949bLt2rfntTBETg1Sf6dPt298cHz7gulm2TPqx7eXkSZwvW3JzUwoJCbjHFy9WeibJkE6I7t2LC8BSQEdKJCIC52b9eqVnomftWsxJjkL4Y8di0WBvuP7EiXh4WlsIfdMmfam9zJmNJ/EzI/LRy8tk8QNm1hd+nz3b+OdhYTBdEkEbM6XNvn4Nk+FHH+mDo7Jmhd909GjmOXM4fPZsCNGRI5l79ID5Vdw2b14I/KRFKgzp1QvbZs9u+sH7zz9oOm5Ku2XGosfDAxp9UqKi8L4YKCX6h62lUSPmgADrt09Khw7yNbSoXBnnXS28fIlzrOLemYpSrFjyLksqQDohunQp/Edq8jGohatXcXNYWrk7kwEDHPNZmqNMGUSU2oMg4GYxEohjkg4dcMynTxOXkxs1Sh9cIwjQyEqXNm12ffAAgVaGRQoMef0aD97MmSEgPT2TJ4CHhOgjhNOkwdy2bEHKzb/NlxcvXsylS5fmEiVKJDbnjhsHgXjoEOrIZs+u13aTCsnduyH4hg9HME/mzKZL5f38c/KiEUnp0wff/fVr/D8iAgsJsaNK167Q/AcMwHvWBqOJAXW21Ek2RFzsmenLajf9+qkrX1QQYDK3tc1fSuHjj5H6pDKkE6IjRyKwQiM5u3apr8B01aqIfJUasQ+mNVVxjCG6BazttBEUhMXbwoX694wVNhcf5oY+TkMSEhD9WqSI8Wo64eEQVtmz63P5Zs3CmFu3wvy6eDHMvL6+0IYNI7Hnz8dcDIRPMp9o/fooUiASE4MYg+LFIcj69YNwu3MHx2jVCscNC4PP3dfXeO6nIGBh4OtrOtAoOBiCf+hQ+MkzZ4b5rHfvxDmn164lD64yx7t3+jrA9vD6NY63bp19+5tDrBFsTdEMZ1GqFBZhGsn55hvcgypDOiHaoQOiBjWS8+OP0ErUUu4vOhoPSDkqwmzb5lhHmFmz8DC3NhhlxgwIJ2Nm6bdvkcYjFm7Pls10gv2SJdjGWE/MuDjkYPr5JdYIBQEVedKnhwAmgkZnLI3p9Gl8Ltb/jYvj8Bcv9EI0Lg7jGMtdjIvDNeTjo6/vW7JkYmEfEQHTcZ48xrW+d++Q0tK+vfHvHxysN1OnScM8aJBp7TEgAGZaa2nc2LFKXSVKYCEkNbdvq6/Wd9OmWBxpJGfmTNyDKkM6IVqnju31TVMKgwfjoacWxMCf8+elH3vUKMeqVvn7J9bGzKHTwfphyXQsRnlmymS82P2rV7g5e/c2vv/w4dCmjHWR+OcffOblZboLR2wsPvPyggD8V6iHE0GIpk+vL3f43XemCyU8eIBAISI8UJLy8iVKOFarZtzcunFj8oIfz55BYKZLByGdKpW+444pRBOrqapISfnpJ4xrby3cbt3k6TKk0+GaUFPAX//+CFzTSM7q1bjuVFYa0ZOk4sMHIh8fyYZzKx49IipcWOlZ6Ll9G3/LlJF+7DNniGrUsG/fkBCiU6eIWrWybvuLF3Fuu3c3v92aNUSlSxO9fEn0009EJ08SlS9P1KYN0blzRNOmEXl6Es2alXzfPXuI5s0jmj2bqGHDxJ+dOkXUvDnG9vEhWrWKSBD0n794QTRyJFGuXERt2xJ5exPpdEQjRhD98gteRETffUeUIweRhwfR9OnYvlMnovPnEx9v82ait2+JGjUiGjsW38WQ3LmJfv+d6PJlfJ6UTp3wHUaPJrp/n6hPH6IiRYjWrcN7T5/ivY0bieLjTZ/PDh2I0qcn2rbN9DaGtGqF8fbts277pNSsie8UHW3f/qbw9MRvd/eutOM6QuHCuKaZlZ6J+hDly4cPys4jKZKJ45IlmYcNk2w4t6J8eXU14p40CekgUpOQAG3GVGSrJcTIWEul6EQmTIBWZ664wqtX0BQNgzXi4vS+RiIE6BjTQsPDcZ6aNUseaHT1KnyMdepgu717Mc6ECfBlTpoE7c7PD/my164hKjdXrv/GSuQT7dQJfs2gIOScilWKOnWChimOP3489h8+3HRetlgU4tSp5J+JvmEPDwQIzZqVOApa9Hlayrdt0wZWA2spU8Z8hLA5zp+XLzDv88/VVapUbJ6QtK2ehr4Ai5piS1hKc26+fFq1ImMIAgJcrClF5yy6d7ftAWgtV67gIjfRhNoiw4bB3GktFSogatQcs2bBx2fMRJqQgKpNXl6Yd+3ayJsVBeagQVgUPHmSeL+QEFzvFSsm9kvOmIFxChaE+XLMmMSfiw+BBw+YOYkQzZ8/cXGKhAQIyJw5IYh9fCDMRb+6ICDQwtsb1VySfq8qVTA/MVr+0iXELXh44HzkymW6xZm/v2UfplieMDjY/HYiX31lf1BIXBz83vPn27e/OdTWNOPCBVwjcjUld2UOH8a5uXdP6ZkkQjPnys3bt0Tv3xMVKqT0TPTIZV4+dw4msipV7NvfFlPwkydEV68StW5tfrutW2FOzJIl+WcxMUSBgUSjRhHt2kUUF0fUpAlRtWpEy5YRLVlCNGECUYEC+n0EgejLL4liY4n++ovI11f/mb8/UapURM+ewfQ6c2biz2vWxN9584hmzCCaMgX/X7gQ+1Svrt/Wy4uoZ0+is2dh2vvwgahWLZxfIph+lywhqlOHqEsXmMIN9122jOjKFaLJk4latCD66COYv3/+mWjnTqJXr/B/Y3TvTnT4MK5dU7Rogb9//ml6G0Nq1MDvZY8pLlUqosqVcX1JTeHCMPPHxko/tj2I9+WjR8rOQ424vTk3dWoEEGgk5ty55DVglSZfPnka3I4Zg+hRe7BV27AmPeH5c/NFLkRt6tEj/F8Q0NFGLHbv7Y1tDHOfV63CZ3v3Jh7r+HGYb+vXh2ujRAl9IM3bt6i6I5po/y28EF6gADRR8b3MmWH2FwssiKkp6dMzf/218SIQL18i6tiwpKQgIAhKzDMtVQopIuL30OlgyjalxYtpSpbSSqpUsT4f2FErRY8e8hRdOHRIfW3R/PxgQdFIzPXrpiPoFUQaIRofr9XNNcWff8qXLG4PMTEQHKtWST925872d8aw1e9loqh7IpYvh6nWVFWmevWQupKUoCDsV6oU5lS0KIrdBwUhOjZpfu29e4jybNAA6UN37+JB2KIF8iMzZcIC4csvYSYtU4aZDcy5vXvDpDhmDMysXl5I6fj++8T+yfHj8f9NmxIfX1xQHDyI2qK1auH/JUsa354ZwjhNGtMm3erVYf41x4AB1kedO+ovnzIFiwKpefhQvvKX9lKypCor8yjOo0fqS0liqYRoeLhjCfbujBgsY294v9SI9UyNpWs4So0a9pdR++UXzCtpBxRTVK5sWQtq3/6/KkHJeP0awspYfd7p06FVhoXBR9W+PRYeGTPC4mJYdzc2FnMpViyxViwGiBAhaOnVK7wvFhl/904vRD/6SN9CMCYG1ZDSpsW+X3+tH1MQoHFmyJDYL5SQAM0yfXrsU6uWvqNK7drGczrv38e227cbPz9iwQVzHVRsLR3p729/Gpx4LKmLs8fHq6+GbuXK6GCjkZiQEPPXrEJI4xMVbdTp00synFshnhu1+ItFX4scPlFHfK2PHiFFw5prKDoa/jVL/tPTp+FHNMaePUg3EX17IsxEv/5K1LEjUaZM8MVt3QrfZEwMUjVq1kTKS0QE0fz58D1u3oztiYiioogWLCBKnRr/b9mSKGdO/NvfH8fYsIHo2DG8d+UK3iciSpOGqF07/PXxga9WTEny8IBPM1s2on79MJf165Guc+8ejjt3LlJ4mjfH9t98Q3ToEHyuhhQtihSnXbuMn59atYjCwjCuKUQf79mzprcxpEgR+3194nX1+LF9+5vC25sof351+SB9fNTn91MDGTLgr0znJiwsjIYMGUIDBw6kJk2a0C+//EIxMTH07bff0sCBA6lr16508+bN5DtKIopF7ebwYUmGcyt++gnai1oQTX+munrYy4cPjnXs+OIL6yOGL10yncIhIvpDt20z/nnPnoheTYqpsoM//4xCDSdOICo2dWqkuKROnVhrEARorj4+zIGBSAXx9YWf859/8P9/NdT/ii0QIQ1q7Vqkm1SqhCISd++iv2Thwom1ve3bMUauXIk7qpQujQL3hpgruzdsGCKJjREWhrF//dX45+J39fFhnjfP9DaGTJiAikr2IP6epso2OoK/v/21nuWgSRO0+tNIjCDAirN8ueRDx8bGcufOnfnFv263x48fs4eHB7dq1YofPnzI+/fvZ29vbx4wYECyfaXVRNWibakJtUUtf/iA1XeaNNKOK2oI9kYhP35svRYrag1FipjeRixUYBjxasiZM8a11D//JEqXDgUNDFm/nujTT4lq14Y2+OABEvXj4ojWrkWE76tXKOywbRsKGNSqBa02Vy5ozbVro8hB6dJEVauigAARfou8eREVW7gwNM+dO4mKF0cE8Lt3RAMHQtNctIjo22+xX3w8Imz/+guabK9e2O/9e/28/fyI6tc3HkVbsyainF+9Sv5ZpkxEJUuaj4j18MDvba0WJ0bCxsRYt70huXPjPMmhMapN8/PxIYqMVHoW6sPDQ7bfavny5dSzZ0/KkycPERGlTZuWmJkKFSpEhQsXJp1OR8WLF6cuXbok21caIRoVhb9qEhZqISpKXeflwwd5zO5iikXu3Pbtb4sp+NEjfIfs2U1vc+8eznvevMk/i4ggunXLuDk4MBDCJV06/Xtv3sBE2q6d/r3cuWEi/fJLosGDiZYvh0AZMADVidq2xXapUyO9JjycqGJFokuXICxv34YZkQgCcO9eCMc3b3Bs8ZopWBDpML/9RpQvH9GwYag6NH8+thVNxkQ4Zlwc0d9/J/5O9eqhupJhNSUivTn2zBnj57BkSSwWzFG4MNHDh+a3MdyWCILbVjw9UdXp9Wvb97WEGoWomuajJmQ6N5kzZ6ZPPvnkv/+f/3cR3qRJEyIiatq0Kd28eZMCAgKS7atponIjl9CyF7k0Y0euAUGAhpIvn3Xbi1qrh4fpbUShbGybe/dgUC1XLvlnZ88m115PnMAcP/1U/97p05hznz4Qck+fQnuNiSH64w9ohffuQehdvoyc0StX8NffH5rGX39hrBo1iI4eJVq6lKhvX/g827SBhjh5Mkr4eXgQZcyIEnVr10JY+/rCXytSpAi016NHE8+/Zk0I8aTl7fLlw5h37hg/h2IJOnNYs42IuGh48cK67ZMil3DJkEFdQksToqaRSUvv1q1bov8fOXKEvLy8qHbt2hb39ZZkBpoQNY0azblqE6JRURBqhoUJzPHihWWBa848bCq46v17jF2+fOL3z56F5ikKASJoe1mz6rXZTJmIgoJQ/KFOHQT4rFmD7zV2LNGYMdASJ07E50T6mr+//w5Ntl491MM9cYKocWNooV5eRF9/DZPw+PEQekTQQD/9lOjAAaJJk/TzqlkzuWYpLhZu3yYqVUr/voeHeSEofiYI+iIPScmf33qhKM7d3oegXGZOtZlP1SbU1YSTFhiHDx+mKlWqUEbxmjWDponKjSZEpd83MlL/QDZFcDAEjzGePMGxklYxMuVrvXMnudZ67hwEqChcgoNhqv3sM6LhwzFWtWrwP8+cCeFarRqsEv/8Q5QnD1GJEtj3xQv4Pbt1w77Nm0N4pkkDU+nChURffQVhdvCgfg61ahFduIAoY5GyZTFfwwLmOXPCRGxMWBYubDriNVcuVPIx9LEmRXzgW1Mw3dGKM3I9QNWm+altPmrCCecmLCyMrly5QvXr10/0/qpVq4xuL50Q9fBI7EfSAClJiKZJg4e/PfsSWT8va77Dhw/6kPikvHsHAZrU1BsUhL9J/ahPniQPmLp3DwFCIhcu4K8YrJQ6Ncyno0dDI719m6hZMwQDicLp+XNsq9NBq+3ZE6baMWP0AULiNjlzQriLxyGCzzImJrEmWKgQfL7v3unf8/DAd3r5Mvm5yJwZpl5jWJNS4OOD+cfFmd5GRHw+aELUPGqbj5qQ4dyEhoZS9erVacq/ZTj37dtHgiBQdQO3TmhoKAUGBhrdXxpzblQUVtjmfFQpFbUJUbkCnRz5nrYGplkrRE1tY+oz8eZMKnxFoWvIq1eJg6iePsUCQqyz++wZ9vP3h/D8/HNEyvr56YW1SHy8vnbr0KFEAQF6ofTLL3ohxwzz7aFD+L+oQW7frteURb/nH3/oFwMeHqiDe+EC9jXUGp89g+YqargeHvrPr13D3717YVo2xq1b+Lt7N0zazImfA4b/F/+ePg1hn1R7FT83fN9w/9BQXCsHDiT/zNxxDd9LCjMWSaGhRPv3G9/X2DFMzdHY+ObmZWychw+xgNq3z7bvYe//Tf3b0meOvGfv/MLDJTe9Hzt2jM6dO0fNmjWj6Oho2rx5M+XJk4ci/z3Ohw8faNCgQTR79myj+1slRJmZ3psz6QQH4wFy/Ljt38DdCQ5GoIhazs3z57gppZ7P7dv2jysWE7hxwzqN5s0bRGmaO9b79xB0xrZ5+JAoISH5Z2JB9vPnidKm1b//7h1+R8Pto6JwLsX3rl6FJn7yJP5//z7+PnuGbd69I4qJoehhw8jj9WtKs2IFiXdUBJH+wWDo3yRCIfllyxK/17hx4v8PHZr8O/bokfy9I0fwMsbHHxt/n4iod2/Tn4l07Gh5G5HVq/GyF4MoSkn5NxJTNTRtqvQM1EtEhNmPM2bMSB5WKnWffvop9erVi0JCQqhPnz40c+ZMioiIoHHjxtGxY8coLi6Oxo4dSwUMG1EY4MFs2ZkRERFBfn5+Vk1IQ0NDQ0NDScLDw8nX2kBFB7FKiFrURH/+mWjUKPoif35av369lPOjL7p1o/Xr1kkyVuSHD9S0WTPau2cPZZDQpGl2juPHw1Q3Z44049lBovGmT4dvbMkSu8czeh63bsV1IJrabOHFC5g758+3ro1a//6ICh071vQ2X3xBe969o2bGigysWgXT3e+/J37/9Gn4MLduTZyD2qsXUZky9MXly/rz2KkTUYMGSEkhggY6bpx+34QEBAh164aI2HXriC5fJvb0JM6WjTwiIig8Lo4KCgI9TZuW/NKlQ5m9woWRFlOyJDSu/fv1+ZxVqiCN5ocf8P9Dh5C7evnyf5HGo0qVoh+CghBEZGiCrlwZms306cnP5Z07ehOxIX//DQ3z1i0EQhlj82aUFgwOTqy9G4OZKFMmmp4jB31nrpygKTp0QFu0jRv/eysiIoLy589Pz549s/+hOWMGfNGiRYSIqlWrRuckbL1m03g//YSykqI/XIoxLSDJeTSC1OfxdO7cVPPjj/F7mcEWTdRRrDLnenh4mD+x/95gz9KnJ9+6dSWZmMh9Hx/pxoyIoKtElCEgQNILxewcCxaEz8WG7yDpd0463qZNKIzgyPjGzuO9ezDFBgTYHlwkVswpVsy6eeXOjehcc9tmz04e798bP4+nT6NmbNLPMmaEEM2fP3GuaJkyRPHxic9juXKo4Sv+v3BhCFEPD7wnCOjhuX49tqtSBWbZoUPhfytYkARvb6L798mXiHzDwvAwP3cORRcyZIDftmZNpP6EhaHwwXff6VOBXryAUClTBn+JqIhOR74+PrjuDP13wcHw1ya97qOjIWyN3Q/i+jpXLtPpRzodIpSzZ7ccE/Gv71vw9rbv/ouLw3GM7Ovr62v/PZ2QgN/eYH8vLy9JnxE2jafT4fe3sL3UcyRy8DwaQeo5ZmMm30yZrE+HcwLSROf+q40M7NlTkuEMGTBggORjSo3ZOdoRTSb1d040npwRjkT6ICF79rV2XhkyWA4uyJaNqpnwYVCBAskjWIn0EbhJU0GKFye6eTPxeaxaFfmYoqDJnx8a586d0M4++ghVgqKjIZjF1XhMDMZ7/Ji8/vWbcrp08KdevoySgfv2Yb8PH6CRzplDtGULjmXouzxzhqhSpf8EKBFRi6JFMb6hQHvzBufLWN7s48eJm44bEhJCOi8v8+lEYpCWNav+f3/fBkmL/luLnJHlScaV9R60Yz4Oj6kQUs+xQJYs6grUJImFaK/OnSUZzhCXv1B8fGwWLC4tRO0ZW6zoZG3UXe7clhP8CxemIqYKBJjqCJI5MzSdGzcSv1+9OtGTJzSgQwf9e40aIcr2yhX8Pz4eKS/r1xN17oyUlKNHYQ5dvhzvDxqEGrj/mqKi/w2pj+vdG11dtmwhmjYNLoB8+ZB32rw5NNwBAzBv8VzpdDD1NmyYaKplIiKSV1wSO0+IeamGmCu3+PAheRUubN6y8OKF6XzcpPz7+zazJQjJEE2I2j+mQkg9x2xp07qpEBVvbJXnNqVJk4YmTZpEaaQuvm6O9OnVdV7sEOpJMXoeHRGiXl4QOhZ8QP8hVtIx5843V4mneHH8NdbWqHr15K29xApDhoUO6tZFbuf//odyfcWLE+3Yge/y2WfwJ9arR7RyJVJNvvwSQmz+fPhP06Ylz169iIgo1eXLRO3bw7c4cSLmtWMHtMxVq2AG1ulwfgoWhC94yxaYhQ3r+b58iQVAUjP1mTP4fcqWTfx+cDDMxOL5SIo19YwfPTLfCMAQceFjb31lI7m/ktzTaktDU2A+ijwb7UFtvxVJrImqSlgYIU2aNDR58mTnXihqS5yWYD5Gz6MYiCMWorcVW2qwFiqEFJawMNPbFC2KnDJj88mSBf5XY4XXa9WCGdYw1SZ3blQb2rFD/15cHITSokXQLmvVQprLzJkILhJ7haZNi4VU6tQQcC1bwtxboQKl+bd4vNfp0yjt97//oSCBt7e+N+nr16if26QJfOt9+xItXkz0xRdInTIM+Nm1C0I8aWrE8eNYHCTVKMXvb6ov6717OI/msLVxAJF9nX6YcS6S5OtKck9HRpouzKEEkZGKCFGnPxvtwVwRFYWQVog6qOG4JWoUovHx1uVj2oIpf6K1mCs/Z2xbIvPdQ8QoX1MNo2vUgLBMSrNm8JeeOJH4/c6dUcDg7l2i77+HRhgYCF/gt98iYrR8eQQO1a2LSNK7d2HCvXoVpt1Nm+CfPHsWrxw5MPb792hVNnkyImWzZYOG+fq1vjPLqlUQ5j/8ACEsCLjfihaFYH30CLmXn36aWNB8+AANulmz5N/19OnkNYFFIiOhEZuLlma2TYg+fIjvbI+ACAnB95WjmbzatBu1zUctMKvy3KQoTVQRfHyQ4mJY31RJxAe3sR6SjpAxI8ybjghRa/ctWRLCS/RHGqNQIWjHptp81auHCj5v3iR+v1IlCJVt2xK/36wZBFf58oii/fxzRMuOGEG0YgW0NiJoe2KaS7Vq+GzZMkTZduoEMywRzLaGRRSmTYOZNn9+aLx37iDq9tKlxNWH4uJQFrBaNZhHp0xBxaJixfB9mjdPPO99+xDM1Lp18nPw998Q+MaCgs6fx/cV02uM8fQpFhyG5Q/NYYvANbYvkTxCVCyIohZUKChUQWwsrkmVnRtNiMqN2s6N+BCSo7lxoULWa5NJKVwYPj+x/J05MmaEKdWUgCSCYKhRA9qWMVq0wA25Z0/y/bp1Q6DPhw+Y0+DByLP09MTnV68il69AAQjDvHmJunTRN5vOmhVdXN6/xz5Pn+qtNGJFo0GDIIiJ8F1ErZgZeZmpU8Pn2bt3YkE2ahTyGVesgMl3zBic8+LFYQYeMAA+WbHh98qVOA9J/Z7Pn0Potmpl/PycPg2zmTkBKZ5bc4LWEDUKUZ0OZnI5hLO9qNBkqQpU2ugkRQUWGSM2NpYqVapEHh4edFl88EiJ2s6NaHa1V9gZ8PjxY+rVqxcVLlyY0qVLR3tu36YHhw5RnD2m4rJlIUDEeq2WqFHDvBAlQgTtiRPGo35z58YY27cn/+zrryEAmzZF0My6dRBely5B0zQsVOHjA+3yxg0ED+l0iNrt3RvjjxoFzbVwYaIJE6BVliwJQWv4XY4dQ4BSxYowHdevr/d/iqX6li1DR5d586Axixw8CM11wwYI1/PnkWLTsCG0zT59kn/HnTshdE2Vltu7F9q6ucjcM2fwvcw1RxcRBFgOkgY3WYtYPOLf/MCZM2dStWrVKGPGjJQjRw5q06YN3THVF9UcL1/CvaE2IaqAoJg5cyZ5eHjQkCFDnH5sq1CpECWWilSpmBcvlmw4ZzFo0CBu2rQpExFfunRJ+gOcOsVMxCzH2PaSOzfzpEkOD7N3717u0aMH79+/nx88eMD32rblR56ePHz4cNsHi4627RpasYLZ05P5/XvT29y/j3O/bZvxzxctYvb2Zg4K0r934wbzF19gP09P5unTmSMi9J//8APeP3068Vjbt+P9Tp2Ya9bEORbHffCAuV8/Zj8/jOvtzVyxIofXqsVExOEZM+rfb9WK+dgx7Bcfz9yoEXO2bMzff49tBg1iFgT9ccPCmAsUYP70U/378fHMa9cy+/pin/r1mY8c0X8uCMwVKzK3bm38vLx+je+yYoXpc8vMXKsWc5cu5rcRuXULczlwwLrtk9K7N3OVKv/999NPP+U1a9bw9evX+fLly9y8eXMuUKAAR0ZG2jbusWOY140b9s1LDrJmZZ42zamHPHv2LBcqVIgrVKjAgwcPduqxrebGDfxWJ04oPZNESCdEM2Vinj1bsuGcwZ49e7hUqVJ848YN+YToq1f44bdvl35se6lVi7l7d+nHXbaMEzw9uWTBgvbtX60ac7du1m17+zbO6+7d5rcrW9b0dw0LY06XjnnGDOYLF5jbtcOY+fMzjx6Nfy9fnnifuDjmGjWYCxVifvs28We//w7h4+HBvGtX8uOFhGDMDh2Y+/fn8E6dIET79MH7Gzcm3+fpU+YMGfD5wIGJBaggYCw/P+ZHjxLvd+sW5tKzJwQmEbO/P/OePfqF3Z49xs/Lr7/ic8PFhbFz5+3NvGSJ6W0M+d//cF7evbNu+6TUq8fcsaPJj0NCQpiI+Ji4ALGW//0P3/XDB/vmJTUREZjPhg1OO+T79++5ePHifODAAa5Xr556hejZszg3Fy8qPZNESGPOJVJfFKoFgoOD6euvv6Z169ZRetHkKgc5csCkK4cP0l5sCeKxhSpVyEsQqKalGqqmsMZEK1KiBPx8u3aZ3659ewTqGDPpZsoEk+eUKYhCvXIFUbD37xPNmgXf6LhxiJIVSZUKUbbh4Rjb0HQdHg6zpZ8fUffuMPvGx+s/F90F33+Pz1aswP9nz0aebFJT9oEDqJ8bGwvTa9L61VOmIIjpl18Sp40wwzdaqBBMxJcuoVWZICBAqmlTXJdJu8GIrF+P8o3miijs24dyeS1bmt7GkNOnUdHJnkYWOh38t2YihcP/bReXJWnLOks8eoRzL+czwBbkDKAywYABA6h58+bU2NT1oBZUas5NkUKUmalHjx7Ut29fqlq1qrwH8/BwLOBGDmxJJ7GBBxkyUAwR9TH019lCjRpIC3n71vK2Hh6IOBWFgym++goCdNMm/XtiX8769Yn++gtC6ssvEbDTqxeCeogQHCQIyVuNFSoE4X3yJAKK4uIg/Pv3R8GEhw+R4vLtt/B/LlyIIKGTJ+HXK1ky+Xfx98fnMTEQjHXrQoBmzQrhu2YN0a+/6v2xP/4IITpjRuKCC0TY7vBhbJs2LcZv0QIpOStXotxhSAj8phs2QBiKPHiAc/P11+bP/65dCLYylh5jjDNnrA9ASsrNm/gNTeSzMjMNGzaMateuTeXEvqrW4kiwkxw4WYhu2rSJLl68SDNnznTK8RzCVL9fpZFMp/3oI+a+fSUbzh4mTZrERGT2de7cOV64cCH7+/tzQkICMzM/evRIPnMuM3Pz5swtWsgztj2sWgXTWmys0Y+tPY+GvHjxgosVK8b3cuRg7tzZvnk9fWrarGmM48ex/alT5rdr1oy5alVmnY55506YjYnw3o4dMCHnzs1szJ8mmjbXrk3+2R9/MKdOzdygAfavVYs5Jkb/+eXL8Bl6eeHl58dctChMiH//zeH798Ocu2YN8yefwPwq+kf9/TFXQ/Pt4MEYp0cPbDNqVOLPmWHm9vExbcJu2xam6GPHcF6IMKeVK3E9jB4N10xUlOnzGReH7zJ5sultDHn1CtfbmjXWbZ+UlSvN+r/79+/PBQsW5GfPntk+dp061vt1ncGCBXAxJP1dZeDp06ecI0cOvnz58n/vqdqcu3kzrld7XQIyIZ0QDQiw3p8lE6GhoXzr1i2zr+joaG7dujV7enqyl5fXfy8iYi8vL+4uh69wwAD45tSCGExx9arRj609jyIvXrzgEiVKcLdu3VgYPJi5cGH75/bRR9Y/1BISILz69ze/3Y4d+L6FC+NvnTrM+/bpH1QPHzKnTQuhZIzu3SGYjPli9u7VC8lDh4zvHxLCvHQpfIiZMmEORBz+74IknEgfBNS/P/yZxnj1CkFGRMxjxiR/0L57x1yuHHPJksYFzr592HfdOv17hr7gvHnxPQcMMH58ke3bsf21a+a3E1m1CkIwNNS67ZPSuzdzhQpGPxo4cCDny5ePHz58aPu4gsCcPTvzhAn2zUsOBg1iLl3aKYfasWPHf889w+egh4cHe3l5/adkqIZffsF1Fx+v9EwSIZ0QbdsWq2kX4MmTJ3zt2rX/Xvv/1Qi2bt1q32rWEnPnMqdP75TVpVW8f4+H2sqVDg/1/PlzLl68OHfu3Bk33caNuNBDQuwbcPJkaDlxcdZt/913EEDGtMi4ONx4xYphTpkz6yNfkzJtGoScsYVFZCQiQ3PnZn7yJPFnAwdivyJFEF38/feINE7KpUuYw/HjCGJ5+JDDz56FEH38GPukTs28cGHyfQWBeetWHN/PD9Gb1asn1npjYxHJmykT8/XryceIioLG2aCB8evwxg19AFK2bIhCNoxKNuSTTxCBbC0tW2LhYi/lyjF/802itwRB4AEDBnCePHn47t279o378KF1wWnOpGVLWAicQERERKLn4LVr17hq1ar8xRdf8DVrF0jO5IcfEGSnMqQTokOHMhcvLtlwzkR2c+62bbhZg4PlGd8eKlTACt8BRBNuw4YN+fnz5xwUFMQhZ87guxqLTrWGixex/8GD1m3/6BFMhatX69+LikKqTIECGKtNG0SOEzEfPWp8nJgYaACVKhkXgq9eQZMtWpT58WO8J66Mly/H/qNHQ6AWKIAFiuE4S5bgM4Mo0PDwcAjR8HC8UasW82ef6fcRBOb9+5lr18ZxWrRgfvECUYpp0jD36oVtoqORGpM6tenvN3AgPjel5b57h0XGF1/gukiVCv+fMiVxFPKDB5iLtabZyEho+XPmWLd9UsLDk/++zNyvXz/28/Pjo0ePclBQ0H+vKHNm6KQ4uuCTg3LlLFsCZETV5tx+/ZjLl1d6FsmQToguWoSbVKeTbEhnIbsQFQXDmTPyjG8PX3/t8AW5Zs0ao/7SB0T2+8cFASkmAwdav0+TJvBzhofjYZ0rFzTtzz/XmxwFAabi6tVNX6OXLkE4mTr2o0fQOPPnh38mTRoIHEPN7s4d5vbt8eDPnh1+zMBA+Ilr1Eg0XDIhOmIETKp37jDPnMlcpgyum8qVIUwNWbMGn82fz/zxxxBUe/can7e4iDOXjjJmDL7P8+f4/9OnMC2mTQs/7ZgxWASOGQNt2NqUENGUbq+2+Pvv2D+JudaUr36NLX7XwYPxe6oFQYA5fe5cxaagaiHapAkWiypDOiG6ezcudjnMoa5OWBjOzaZNSs9EjxhcZMpk5whDhjDnyWP/gmr4cOYsWYxrhMYQAw4yZoQG1asX8717ybcTA5GS5n4asmSJ6UAiZgiZ0qVx7ooXT2xSNeTuXZyHnDkxnocHNNRhw/hYhw48LE8e7pM7N4ToggXwy9Ws+Z+/lNOlQ+GGQ4dMuwE+/xzbZsiAYgrGuH4dQq99e9PjXL8OLXnKlOSfvXoFX3GGDBCoqVPbZsFo1Qravb106yaf9lGzJs6hWhDziE0VB0nplCyJhY/KkE6IXr+uymoSqiFTJmgXauHaNfxehw9LP/bhwxg7SQSv1dy5g/3Xrze/nagZZcgAIZU7t97UaoqePfFbvHpl/HNBgBD29jZeXScuDubVNGkwx759zVdNio9HlC0RtOWiRf/b97/AInHudepgu9GjzY+p00HYp08PYZs9O8y8SXnxAlpzhQqmIxoFAcctUcL0goCZ+c0b5rp19ZWVvvkGpl1zPHsGi8CyZea3M0V8PBZT48fbt785YmLwOxjzQSvF6dOqLCagCnQ6/F4//qj0TJIhnRB9/978Cj6lU7WqPFWC7CUhAZqbHII9Lg7+NEeiHhs0MB2M8uwZTI3p0kGAjhoFgUvE/Pff5scNDUVgTqtWpjWzuDiU0cuYEaZYQwYNghA5flwvyAoWZN6yxfR4W7Zgbi9f4v+CwBwezuH/uhHCw8L02xYtyvztt6bnf+ECqvcQMffpA407Tx74Uw1TlkJCIDzz5dObaI2xcKF1Pui3bxHANXAg86xZzDlyICK5WzfmmzeN7zN5MsyTornaVo4ccWwxZg7Rd68mF8vatapM4VAFL144FmshI9IJUWasiI2ZhDTgFC9VSulZJKZhQ9P1Ux2la1dEe9qLGPRh+IC+fx++XDHoZfJkaEjMEEw1apj3eYrs2oWxzWkhEREQ4j4+elOpmDNqWN/3/n0E/BDBPLhrV/LjDxmC3MwkJPOJMkMoVa2afD4XLyL1hwjmZMPFwqlTMLOKfujnz3Gt5cplvibs+fPYzxoT2dixWDCIwXEfPuD85c0LK0CHDonrQ8fHQ4B//bXlsU0xdKhjbgFziDEc5rRvZzNwoMsGZ8rOP/+YTctTEmmFaPXqSATXSI5Yo9NQ61Ca776DViZHPpiofSWt6WotMTFItRg4EK6Crl1hGsyZ03T6xdGjOKY1qTtDhkAYm9NyIiP1QTvTp8Oc1LOncY3z4EEUSCDCg/D77/XmzurVMf8kGBWiy5dDw4uMhPa3apVe8yxYEJ8by5NbsQLbTJ4MgZ0/v/lgnvBwaL1VqlgWJLdu4VxNnJj8s5gYHLtIERy/eXMIdTGX9Px582ObQhAwplwFXDp2tC1NxxnYUjs6pbFuHa4nc24OhZBWiHbqhI4RGskRu1hYMjc6k8BAzOmff6QfOyICWpy1VW2M0bcvBCcRhMJPP5mvpMMMk3mWLJbTFmJjIdyM5X4aEh2NoBwiaHaWuoQEBkJg+vhgn1KloKn16IEKRqZSXOLjYZr98UfsV7YszMaensyNGyMozVKSeePGeiFuzjccG4tt/fygSZtDEGBaL1rUfKBXfDwedKVK6YOdKla0Pzda1DysTXWyhdhYmOqnTpV+bHuxtYtRSmPqVCyqVYi0QnTMGKyWNZKj0+Gh9f33Ss9ET0ICfFumKvU4ytdfw6Rna4WRf/5BODsRhEjDhiZLFCYjOBimXmssIkFB0NrKljVtIYiPhyaYIQPm4u9vXbpGZCS0cdHUa/jKkIG5YEEOL1QIQjRz5sSfe3ggX3DpUr0f1RwRETjXRLAs5M5tPnCqe3eYMk1F9BoiagBJU2xModMxjxyp/y61ajH/+aftwvSLLyC45TDl7t+PuV25Iv3Y9iIuaOXw/7oDPXtCU1ch0grR5cvxoLG22kxK45NPYO5SE199hdBxOTh/Hg+GP/6wvK0gQEsXTZflyjH/9htWoKlTmw+OSYpo2rSmEs2tWxC6deoYNxEPHQqN8OhRRJ4XLYqAprlzrRPsM2ZA63nzhvnkSQSPzJvHPG4chw8dCiE6aRKKCfz9N4Tmxx9bV7VGEJj/+gsLVx8f3H/Pn0Njrls3+X0oCMhFJcK5tcSzZxDKnTpZ3lZEp4MG2rAhhKeYtvPRR6i6ZI1QfP0apnO5Wiv2749zppYKYsyomZs2rfWLxZRG/fqJC5GoCGmFqLjCsxT6nlKZMAEmCTXdvGKQze3b8oxfubL5hUPSovDVquH/4sM2PBzmWVuquAgCyqdlyWJd3nJgICJPa9ZMrJGKEb+LFunfi4xE9KynJ8oJbttm/vds3hxC0QhGfaLMMIFnzmxe4Fy+rDffNmyYuBjBP/9A8BtG+ep08C9bCqgSiY/HwiJfPgg1a9m6NXGqmyAg17VhQ7xfpgzOqznrxPz5MG3KUeFLEPCdBg2SfmxH6NQJVg4N4xQsiNQvFSKtEL17Vz4/hjvw55/qW2R8+ADNyt6ybJb4+WcInKR+x4QEaEPly+Oc1K2LRZgxgTRrFh6qplIpjPH6NR6WtWtbZ04+dw5Ct1IlmEIvXoRm8OWXxud07Zre5Fy2LEoAJg3Q0ekwpgm/sEkheuAAxk0aWSsIuLfE45YogUWQsfktXYptfv0VGulXX8FM/PPPls8FM3Izvbxsy/uOicGcTCwaODAQiwoiaPQrViQ/Z4IAy4gt2q8tXLiA45tqFqAUhQrB6qGRnLg4PEPMFUlREGmFaEwMbtRVqyQd1m0QK5JYY0pzJq1aOVYg3BwREfABignzsbG4PsSi8E2aIOfSHDExCJapV882Lf7ECQiCkSOt2/7qVZhC8+TB38qVLQcyHTsGrVf0R/brh+PqdBD6RMaLNrAZIRoRkbhBwL17SB0rWRLjVagAs7A5t4kgwI+UJg1SZry9E3dvMceePbiPp0+3bnuRKVNwHHNpNcxYoIilEfPlg2YsBlyJEdZyFAFhRoRxpkzqcjm9eoXvvHmz0jNRJ2K9Zmv98k5GWiHKjJviu+8kH9ZtKFJEfaWrHG1VZYlvv4V58ocfEGVLhK4/tqQ/iNrZr7/aduz587GftVVzHj3StyVbsMD649y6hQAt8ftly4b0EQ8PCAQjoflGhWhcHFJ6ChaE0CxeXB+M1K0bNFFrFxKnT8Of7OlpfZL6hQvwr7ZoYVtQz927ENhjx1q/z82bCCDy8kKA26xZ6ERTtqx8Lo9KldRV6o9Z71KxVG0rpXLwoGP1l2VGeiFau7b6LlI10aWL+vLTXr1yrDybOSIimMeN00eddu1qvFWXNXTpAuEkFliwBkGA/8vT07oApxEjsO2nn2LOXbrYdjydDpro+PEoPmIYdZs/P9JqWrRg7tqVwz/7DEK0RQv04y1eHGZrcftUqRAEs22b9QXfmWG+njEDArRcOZiUGza0bNZ+9Ah5uNWqWU7lMUQQYMItVMi2eYo8eIAygl5e+N4dO9p2zq1FdDepTeMbOxaWDzXFSqiJlSvx7FBTYQwDpBei3bohrF3DOGKlFLUlDbdsiQhKqW7kN2/0ATKpUkG78PV1rNhEUBDG+PJL2/ZLSIDmmz69+ZzY337DQ/bHH3Ee1q+H6S9XLtO+R3OUKIHUk/PnUWxj3DgUb2/VirlePQ4PCIAQbdQIaSfDh6OU4LFjWNAQ2W4duHEDgtrTE5pxdDRMpF5ezMOGmd4vOBg5nkWK2B7QI5ar+/NP2/ZLSp06MImnTQvNe/RoaYOLRoywrbGBs6hXT77KYe7AuHGwcKoU6YXo99/jwamtqoxz7x4eONu3Kz2TxIhBT2fPOjbOq1d4+IldP779Fm21Xr7E/x0pvsCs7+G5YYNt+0VF4WHl42M8qOTyZQRYdeuW+Np98QLpJkSIhrU2tzA01GIRfZM+UWZ9w2hrtGdm+Nv794ewLF48ec1fsYiDMX/88+cwHefKZbvJ7O5d/NZGKjLZxIkTmN/WrRCc4jWULh0sCY52hxIrYKkteOfNG/xmcliB3IUOHXDvqhTphei+fbgZ7tyRfGi3oUwZ9ZVHTEhAq65evezbXywKb6hFJE34HzIEBScc0UYFAQ/sDBlsf+B/+IBc3bRpETwj8vo1Gm5/9JHxQCJBQNpN8eL66kOWIoX/+MNi2UOzQlQQUDTBUlh/aCgCejJmxLmdM8e4piUIWCCkS4cFg8jDh/juBQoYbx9njpgYnLPixR1vqdewIQKmDP2wSa0ZX39tf2S7aGUw1ZRcKcRiFrbkQac0ChSAlUalSC9E377VurlYYswYrIrlqFnrCFOnwuRpS9eN+/dhojRWFD4pQUF4iI8b59g8IyLw4P7oI9v9JDExMJ2lSqXPV2zcGGZES4EdcXEwx+fOra8Te+iQ8QCc0aMR5WvGImNWiDIjgrVuXeOf3b4NzTNdOiwKBg2ybPqNisI5K1wYC4crV2AmK1rUvqCWb7+Fa8LR1l2HDuF87thh/POICBReEDvHfPGF5QjgpNSrp05tpmNH4w0HNMDLl7g2fv9d6ZmYRHohygxfkC3J8SkNscSX2nqvPn+Oh9TSpZa3vX4dAWSWisInZfx4PHgdtVRcvIhx+vWz3XUQFwdtUizu4OFhW95gTAzzmjUI2hELw48ZgxQZcS61a8MMZQaLQnTevMRVbF6+RMSwWJhC7JpkqU6wIY8fY8FQoQIWTJUqGe9FagmxqIKjtV5jY9GVplYty79jVBQWMfny4Tdr3946AS7WrVZballMDCwqaioFqjZ27MBv9/Sp0jMxiTxC1FQ7Jw2g02FVbW3+ojNp3dp84fDz5xGkY0tReEOiohC80qiR437zn3/GPObNs31fsYasWLTAnmhQQUAQUJ8+0MKJYHrq3h35kuPHm00TsShE9+7FmJ9/jt9EjNht3RoRpracd5GEhMQt1eyJpg0MhHDv1Mnx33DGDCzcbKljGxuLiM2iRfE9mjVL7gM2ZMgQWH7UFt0p/r4qbO+lGsaMsWjRURp5hOiSJbjZ1RYFpyZ69cLDW23s2YMb+/TpxO8fP65P+yhWDLVe7a3zKT48bA0OMsbo0falLVy9Ck2sQQNEbObPb7IoglXExqKO7ZAh+oc7EQKZqldHRPGYMchbXb+eeft2Dl+/HkL0l1+g0U2ciM41DRtikSWO4ecHwfzrr3CX2MuDBzBpenrqC+Nv2WLbGHfuQJOtU8fx+/vBAwjjESPs2z8+HueyTBl8lwYNYFEwfOBGR2OBo8YFa9++SAtSsYBQnPr1sWhXMfIIUbHwuLnVYUpH7pq19pKQACHQti1u7v374ZcTi8Jv3CiNL7djR5iBHe2vqtNBU0ud2nLlI5E3b6ANV6yIfMjHj/W1Xfv1czz9aN48FB7YtQuBPl9+iYbhhQvrW6QRcTgRhKioYebJgzm1b888aRL8QFWqWDQLW0QQEP3p44OH9pEjeK9zZ7x37Zp14wQH47yVKuV4HqcgMDdtisWLo+dbp0MubeXKyTvHLF6MRYOtQVNyIwj4vdVWeEVNJCTA3D1rltIzMYs8QjQuDitMWyq+pDTEmrU//KD0TJKzerW+JqyxovBS8Pw5bpB+/RwfKyYGK9bMmS37yBISoFFnyZK4aLtOhwdu+vQQFPbkhYq0a2c6IIgZgvvNGw5/9AhC9OlT08eyIkDJLNeuwXROhIIGhn7ryEj4RosWtazhhoXhOsiVy/5G64aIPlVTwUT2IAiwpIjN0StWxDXhaPqNHJw7J295Q3fg6lWco6NHlZ6JWeQRosy4kOUqIu0utGqFSjVqQSwKLwrPrFnRnksuc9PChdLVxAwLgx8+U6bkpmhDxoyBZmLKdHvvnr47Sv36KINnC4IAQWNF+TuLPlFm+0vCBQUhJcTTE0Jy3z7j2z14AEHTtKlpC0NICAKQsmRxPBJXnFuOHCjwIce1JQgQTiVK4NwVKYJUElv72srJ+PHqq+GrNlaswPWrtsI0SZBPiA4dCtORhmlWr0aUoS3RlXKQtCh806aof0xkuxCxBZ0OWmGOHHiwOsq7d1iUZMhgfPW6ZQu+kyXtX+zTWbq0vlShtcEfYrFsK6r3WCVEbW1aEByMlnsZMkA4Llhg2Xe9fz8eVsZqXr98CZ9jzpzSBMAkJEAzzpXLdONwKYiKwjGaNdP7f4sUMd45RgkqVFCnhqwmevWCNUHlyCdEN23ChStHT0B3ITgYDy9rUkrkICoK0bVi0fR27fRF4ePjkYvZsqW8cwgOxsOuYUNpfK2RkXhIp0uXWPu6dg3+v86drdd+4uPhSxTPz6efQoM1t79YAs8Kn6FVQpQZGtXAgea3uXULmmeaNDBJDxtmm99y5szklbSePMHCKl8+6YqnTJuGhYnc7RIXLEDU7/37+P+lS/DDG+sc42zu3LEvqCulUbYsXBAqRz4h+uiRbWXLUiqtWsFU5swIvYgIaGM5c0KImyoKL1ZTOXdO3vkcPIiHm1T5ctHRKITg7Y0Fyps3MGlWqGBbYXWRuDhEEleqpE8NmTYtsU9VpG9ffG4FVgvRHj1QJCEpYWGwIIhBUblzI2XE3nSdDh2gwd68iaDAXLkQDCWFD5QZgV+entCU5eTDB1zbPXsm/+zmTUQ6G3aOsaW4iBQMH67OGr5qIjwcz4RfflF6JhaRT4gKAi5krS2aef76Cw/AM2fkP1bSMmq9e5uPWkxIQE3V5s3ln9uECXjAWhtha4m4OFTUEfNZM2VyvBm6ICCFomtXfZRtrVqIxr18GebpChVwXq3AaiFq6Bt6+hSFHtq2RUSyhwc0719/ddxM+f49Vv85c+L6qF1bOpNraChz3rwIuJLbNzl3LhZQ5n5vsXNM6tS4HyZNkqdzTFKioxFroLYavmpDrGJla2UqBZBPiDJDy2rcWNZDuDxizdqvvpLvGIZF4cWC3tZWANmwARezVMLNFPHxyD3MnRtmRKkQ/WFlykjjdxWJjISfsnlzRKKLPUSJIGSPHLFYhs+iEI2MhBVg2jSMmzevvqVctWrIObWn2pApYmORLiRWYZJKU4qN1ZdWlLtGbFgYKjlZuZDh58+R25suHe6PUaPk9dWK95PaaviqjenT0bFJyowAmZBXiLrQiVCU77+HH+vdO2nHffoU2ljatChQPmaM7T5qnQ45juXKyR9J+OoVgtHKlHGsqIDItm363M/cufH66y/Hx01KdDSiQTt31lcVEgsl5MwJ7euzz+DXnDoVxUh+/pnDf/wRQnTBApgVhw1DXdhPPkEQjIeHfhxPTxRt2LoVdW+l5s4d9LlNlQrz9PBwvOMOs74yVOrUWFjITf/+EIa2CuvgYERUZ8yYuPuQ1NStq84avmrDhRQweYWo2JHcUseLlM6LF/DROFqHVOTePX1R+CxZUF/VEaF08SIe4s7Iab11C+a1evUcM0/euIGHaYcOeJC/fMncpAmux6++kn7BwgyTdNas0Lxu3EDgyMSJKLPXqBFz+fIQql5ezB4eiYstZM4M03mdOgjwGjYM/qCzZ2FmbdoU85canQ5t0tKlQxDRqVN4f+pUaWIaJkxwXt3aM2cg/H/80f4x3r7F/ZIli97lIQYnOcrNmzgXGzdKM567IgjwV48fr/RMrEJeISo6h9eskfUwbkGbNnjIOhJglLQo/Jw5jreoEhkyBNqyPd0+bOXECUSZdulinxXj3TtEFpctmzjHTBBQczVjRsfL/BmjUSObopmt9okyw1rh5yetVUcsA0gEE79htKpOhxq9vr72R+auWIGxZ8+WYrbmiY9H8FWlStL4XJMG39nTOSYpaq3hqzbEoNTdu5WeiVXIK0SZEWjRvbvsh3F5xHqyoiZgC+fOQQiLQTSLF9tXnNwcERGonNO6tbTjmuL337EAs9RPMyk6HfygmTKZDpoyLPPXubM00afx8Qg2sqFEmU1CVAy0MBZFbSvv3yOQJn16fRlA4xOEdly6tO2Lsb/+gsY9YIBzIs9//BHXi7lCG/YgpoGJnWPatbMvdzoqSr01fNWGmCamdP68lcgvRMeNg4lLbb0z1YZOhweaLc26DYvCFy8O85+9ReGt4fffcaxdu+Q7hiHz5+N4kyZZ/yCeOBEPO8Om28bQ6VDsInduaL2jRjlm4r140eYALJuE6Pv3EEo//2z/HBMSkBJj+J0tCcdbt6C5t21rvRa8fz/Mw61bO+e+l7KEpCnEgiRic4GmTZlPnrR+f1EwqK2Grxrp2BGBcy6C/EL09GnnRHe6A9On4+Fjrii7IKCIQJ06OK/ly0tXFN4SYtHwAgXsy7e0B7EIwIgRlgXpzp3Ydvp068d//x6CN106mNoWLLCvzNhPP8GHZoMFwCYhygxzpS2LLBGdDuemfHmcny5dbNO+bTmvO3ciiKhFC+flQXboAB+ao80MrCE+HhG2YmnMBg0Q+2Hp2gwIgLlfwzxij9Vp05SeidXIL0R1OiRt29vuKCURFIT8toULk3+m06GaTJUq+qLwu3Y5P/JZbF81ZIjzjinW2O3f3/T3FTWm9u3tMx8+f47kfC8vmN3GjUMwkrV06YLoVhuwWYgOHAiLg7VERaHiklhDtm5d+/ORrdHwN27E+evYUV6LiCF//CFdWz1bSHo/1qgBH56xa+/aNa1CkbW4YI9V+YUoM8qRFS+u9c2zhi5d4NcUgw+Srnzr1ZO3KLw1LFhgdX1YyVi5Eg/xHj2Sa93v3sF3V6aM44FUT54gMjZDBmhUPXsiQtbS+S5QAPvZgM1C9LffrPMVPX4MoZctG4Ji2rd3vC2hToecWFO+ZrEOdPfuziv0/uwZXEUtWih3PwgCHvwBAfrOMVu2JL5Gv/gC8QTOWli4Mv36oUqWC8kK5wjR3bu1BGNruXkTD6NFiyA4DH0w//yj9OyAIODBlTUrHmTOYv16aDodOugjSXU6RMT6+UlX35UZpsEfftAXOCheHL7Zu3eTb/vsGbbZts2mQ9gsRB8/Nu2TDg2F1lm7NrZJnx6aq1TpGcw4J8WKIWdYNHkLAioEEaHkobMsI2Jxjrx5LRa1cAqCgAAtsQNQyZKoInX1KhYyS5YoPUP1Iwj4PZ1p5ZIA5wjRqCjc1M4IdXd1oqJgIvLySl4UXk28fo2IxTp1nNtiascOXEuVKkGoTJ6MRYdcWnF8PDT/Hj1gLhZN6VOmIBUnLo5582a8b4v5l+0QomIj51Gj8O8bN7DYatYMbgAvL+SSrl0rXWpTUq5fRxTyZ59hIdOtG7776NHO1R6kLhMpJadPY2FHhHOVObPz6/O6Ii7aY9U5QpQZKRj+/k47nMsREYFFRo4cEAoeHuqvO+ysguJJuXwZkcy+vrjppCpcb4moKEQot28Ps6b4kCxYEMn5R4/aVE3IJiH64QMWU1WqoKxdrlz66kh16yKwSc5ydYaIDbXz5oV/3Nn+SLFhgdqDT8SKWURY/CxYoFznGFdgwgQsONTU99UKPJiZyRmsWUPUqxfRq1dEOXI45ZAuwdu3RD/9RLRwIVFkJFGPHkSjRxNNnkx0+DDRgwdEadMqPUvTTJtGNHEi0cGDRA0bOu+4p08T1a5NJAg4dwMHEnl4OO/4Oh3RxYv4jWbMIPrwAe8REeXKRVS+PFG5ckQFC+J6N3z5+BB5eFBERAT55clD4S9fkm/q1EShoUQhIfq/L14Q3bhBdP06rgNm/XccPpzo44+JAgIwnjM5cYKoSROiqCiiJUuI+vd33rGDg4kqVcK53bePyMvLece2lU6dcJ3u3k00bx7Rhg1EWbIQDRuGc+brq/QM1UXFikQVKhCtW6f0TGzDaeI6JASrx9WrnXZIVfPqFcxyYlH4wYMT+xfv3IGWt2iRYlO0CsMmy87qHRsezlyqFPxO/ftjpf/ll8qYzCIj9SUbb9yAaXfCBORVFi+uL05v5JWo7F/Sl5cXNL2PP0bHj9WrEVl75Ij9RTkcRSwR6O2NALcGDaCBG2sJJ9fxP/lEuibucnLtGp53K1bo33v4kLlPHwSsZcqE60SOOsiuiFilyAUjmJ0nRJkRweasijdq5ckTBHxYUxS+e3ckxktdfUhqgoLwYKtTR/7cQJ0OrgFfX+bbt/He2rVYjBQogMo+zkQUaqZC8gUBQTgPHsBXtmsX7+vRg4flycMDcuWCEF29GibS48fxnd68MR2gExeHRdfcubJ9JaM8fMhcvz6+65AhmMebNyiUX7Gic8yUY8dCMEldrlEOOnSAy8FY0wbDzjE+PqhipPZFgdwsXIjFhVy+fBlxrhD94QdcOGoXCnJw7x5zr176ovBTp1ouCn/vHjQSRwpqO4vAQCwMPvtM3gjN7783Xhj90SP9Q37AAOcVgxA7FdlR7MLmwCKRunXhl3UGgsC8fDkWKQULJl+kXLmCQK/PP5c3sGj5cvy2c+bIdwypuHwZc121yvx2STvHDBwobRtAV6JhQ3kaLDgB5wrR27el6QzhSly7htxPT0+YPOfOta0izpdfYj9XWHhs3w5NYfhwecb/80+MP2WK8c91Opi/06VDatCJE/LMw5BmzVB60Q7sFqJjxuCakDsa9ulTmE+JkOttap4bN2KbBQvkmcfu3bh/vv3WNfIH27aFhm5t68C3b7GoFjvH9OqVssoDvn0LZWHpUqVnYhfOFaLMqJ5ibcNcV+bsWZiuiWBmXLLEPkF4/z4usPnzJZ+iLCxahO8stS/3zh1ofK1aWdZ0795FJLhYnEGuRtA6HaIJTQl1C9gtRMW8a7l8kVFRKLeYMSP8snv3Wt5nxAhcp1KnJ5w5A023bVvXqL8t1lC2p3PV+/dYZIudYz7/XJqGA2pHbFQud8N2mXC+EB05EheJuzbqPnZMv3oXi8I72sy6d288rJ0VuOMow4dDgG3fLs14ERGoRlSypPVF4hMSsHDJlg0P4YkT7auJa44bN/A7Hzxo1+52C9HQUBx3/Xq7jmsSnQ5jFiiA4KHBg62vRxsfjwCzbNmkM0nev490nlq1XMMSIwhwKRQv7liaRlQUAtUKFMDv3LatOnPFpeKzz5irVlV6FnbjfCF64oRy0YVykbQofIUKzJs2SbdyDgmBEHWVlnI6HW6MtGlt63RhDEFAwYmMGe2rePXuHQoBpEmDIK1Vq6T7XVauhMZgp3C2W4gy6yOTpeLYMTzIxIe2scpMlggNhd+0ShXHhV5oKIRR8eLqqEhkDWKnlv37pRkvNhZR2cWKYdwmTdRTtUwqYmNxb0+dqvRM7Mb5QjQhAatVW/tEqpGkRairV4e/Vw6/zcqVOIap3o9qIzoaATBZsyLQwl6mT8f33rHDsfk8eoTeoUSoQ/y//zley7RHD3RWsROHhGjPnoiKdQSxVF2zZjgvVatCmDrCxYvwSX/5pf33QVgYCrrnyIGoZlfgzRtozZ07Sz92fDzqJpcrp6+ffeCAa/iHLbF/P77TlStKz8RunC9EmVFjM08el6tM8R9Ji8LXry//Ra3Twc9XqpTrFLJ++xYLjEyZ7OsesmcPzMJSVkQ6fRqF1MUqMrNm2d9Cq3hxRFTaiUNCdNUqaMH27BsXh4dy5co4D+XK4XqWysWyfj3GXbzY9n1DQzGvzJldy4TZpw989jaWfrQJnQ6LSWcs2p3FF19A03bh76CMEBWd7zt3KnJ4u4mNTVwUvlkzx82VtnDlCoI3bOmXqTTv3kH4Z8xoW53Te/cgfJs3l8d/fvMmfM2pUyN9Y/Bg27SekBBcA7/9ZvcUHBKiN2/i+LbkTL57xzxvHroEEaGQw/798jzAhgyBX9WW3zwoCAvT7NldSzMJDLR/0WAPYucYsdlAhQoo8uEKgVeGvH4NN4uL11RXRogyo4h3s2aKHd4moqIQbZo/PzSj9u2ZL1xQZi4jRsDX6CpmLmb4DBs2hJnPmof++/fQjooXl7/RclAQ8/jxSC8ggsBfssRyuzGxUfXjx3Yf2iEhKkYGT55sfrvoaNRwbdcOD6xUqeBbd8TEbg1xcbDQ5MhhXaefp0/xe+fJ41rdnuLjIcSqVlVGiB07hsWQ2Dnmf/9zPJDRWcyfj+vR0r2mcpQToqtWQSA58BCSHcOi8F5eMD3cuKHsnN6/hzBv1sy1TCBRUZhzmjRI0TCFIKDaS4YMzj3XkZEwaTZrht/aywv/Xr/eeODQyJFI/3DgN3BIiDJDSzeWo5qQgMVKz576Iv1VqkALldPcmJTgYFyrNWro++Ma4/59BCQVKuRai0NmnFNPT+VNz2fOIP2LCOdx2TL5q4c5giBA6MvhQ3YyygnR9+9h4hs/XrEpmOTNG/SOzJwZK6VvvlHXzb1jh139KxUnNhYakbe36RqZs2Yp/91CQqCN+vtjLqlTI5hj6lSY7+PiUMKyY0eHDuOwEBWrJcXHQ3tbvBjnN3NmzLtYMaT2iOURleDsWSycvv7a+Oc3byJqukQJaKOuxNOnKNs3aJDSM9Fz5Qpzp05QUPLkgbbnrOpdtnD0qGsFSppBOSHKjC7muXOrx/xgqSi8WhAE9CvMm9f1ak3GxyOJ3NMTieWGmtz+/Xh/3Djl5peUhw9R17NVK71WlyED5tm6Ncrg2Zm/a7cQff8emseIEZhP9uz6tmh16mABeOaMeiwVa9Zgfj//nPj9w4cRvV2unPPauElJ27Z4fqmxV+jt24iQ9vJCNsT06dbnWDuDLl2giarlGnUAZYWoWGNSqqR8e0laFH7sWPUXNnj0CIJeTatga0lIQIoTEUzkUVHQ9DNnZm7aVL0BEvHxiO7t2xdzT5NG33UlRw74fQcNQueOv//G9f3ypclFokkhqtMhSvXmTazY161Dqb8WLZgLF9Yf08NDHyC0d686NQ6RAQMg5AMD8eBcuBAP+EaNYPlxNXbtco2uIw8f4npNnZrZzw+WP6XzbkNCcC3Mm6fsPCTCef1ETVGzJlGmTOgN6Gzu3SOaNYto7Vr09hsyBH0pM2d2/lzsYcEC9Cb880+i5s2Vno3tbN5M1LMnUcmSRDExRPHxROfOqf/8z5uHHqqvXxM9fYp+n4avu3fR59SQzJnRSzRLFiJPTyIiikhIIL8zZyi8Rg3y9fYmCg9HL9HXr/W9SUXy50cPzbJl8bdcOaLSpYnq1SMqU4bo11+d9OXtJC6OqFEjovv3ierWJdqyBT1RZ80i8vZWena28fIlel/WrEn0xx/O7WNrLy9e4Lr9+WfMt18/nP9cuZw/lzlziCZMwJyyZnX+8SVGeSEqNut+8ICocGHnHPP6dTRS3rwZD7YRI4j69CHKkME5x5cKQSBq2ZLozBmiK1eI8uZVeka2c+kSmmtHReFa6NFD6RlZpl07orAwoiNHjH8eE4Pm84YNtsVXWBj0SCKKiIsjv99+o/DPPyffVKmwkMuRgyh79sR/c+Uy3cB58GCiv/6CcFI7Fy9C8Oh0RKtXu8ZvnRSdjqhxYyyUrlwhypZN6RnZRmgo0Y8/Ei1eTBQbS9S7N9GoUUQFCjjn+IKARXONGkTr1zvnmHKjrCLM6EPo5+ccP5ixovBqjmCzhtBQBBDUq6deM6g55szRVxHy9kYnBzX7SQSBE7Jl490VKnChQoU4bdq0XKRIEZ44cSLH2lgEw+HAImbkBxKp36f4zz+omZ0jB37nfv2UnpF9TJ4Mf/jRo0rPxDHCwtBWMEsW/B5ffWVfqUdbOXgQ16st+cMqR3khygx/ZK5c8gUYGeZSlSiBQAe1BDNJwbFjuLEnTVJ6JrZx4ADmPWYMfo9Bg/Abde+uriAIQ+7fZybi+Y0b8/79+/nBgwe8a9cuzpEjBw+3sQWcJEL02TNpyiLKhU6nzwesUwfCfsUKzHn1aqVnZxtHjuB6tbNrjyoRO8fkyuWczjEdO6KZhJoXyjaiDiF69Spuqq1bpRvTWFUPKYvCq40pUxBoInUrKrl4+BCr4E8/TfybrF2L4K58+aQr5C0lYpHxJA3Vf/jhBy5cuLBNQ0kiRJmRizlypGNjyMH9+/qmDEOGJC5X+c03CHaxpxykEoSEIBK3QQP3fIZER8MyJ2fnmFevoPUuXCjtuAqjDiHKjHZHH3/s+Dg6HXIM3am+pDUkJOAGz51b/ZHFHz6geHqRIsYjM588YW7cGL9fnz7qSuPp0wcr6SR89913XKVKFbO7xsTEcHh4+H+vZ8+eSSNEO3VCTqta0OmQs5o+PaKJjZk+Y2KYa9ZEmpbaTdE6HTqoZM/u3GIVShAbi/aNhp1jpGpuP3MmMiCSLEBdHfUI0f/9Dz/a/fv27R8fj+oyZco4ryi82njxAjd6kybq7dcqCMgRS5/efH1UQYB/1McHFVjUkpRdrlyywgH3799nX19fXrlypdldJ02axESU7OWwEF20CFqduapAzuLxY6T6ECG1wlybuBcvYEasW1fd7pXZs/F99u1TeibOIyFB2s4xOh0WVF9+KeUsVYF6hGhUFAqO29oiLSYGPhalisKrjb17cR5++EHpmRhn3jzMb/Nm67Y3NAkOGgQtViZMCTnx5UfEgocHfOr/8uLFCy5WrBj36tXL4viyaaLnz+P8BAY6No4jCAKaM2TMCPPy339bt9+JEzDxqTXfOTAQ83OH1o32oNOhTrTYa9Zey96+fcpfozKhHiHKjBspRw7rWn19+ADber58yheFVxujRuHGV9sFe+gQghdGjbJtPzE4JW1aPKDXrZNF0w4NDeVbt26ZfD0VA2L+jWJ88eIFlyhRgrt168Y6O+YjmU80Lg6a/Zw5jo1jL6dPowwiEaI8bQ0KW7IE+/76qzzzs5c3b1DTt1YtdWvKzkAQIAjFBa2tMSZt2zKXL++WlkF1CdFbtyAQk5YHMyQ8HPVVs2dHxZNu3VDZRUOPWNs1e3b7zeNS8/gxSrw1bmx/H9l791AbVqoG0rYyYQLOqSDw8+fPuXjx4ty5c2dOsDPQRDIhygz3RZs2jo9jC4aNzitUsK0tmyGCgGL5adMqX8hdJCYGZuYsWdTdJEMJjh1j/uQTfbaDpc4xd+/iWb10qfPm6ETUJUSZcVMWKJBcG339GsW0M2WC/0dtReHVRmgoWksVL658ma8PH5g/+gi+zdevHR/v2DG9ealNG+fktzHD19e69X8m3IYNG/Lz5885KCjov5ctSCpEx41DHqYzVvrv3sGakCYNAtlWr3Y8YjU6Gr9pgQLKt8bS6RCslSYN8ls1jGOYd2+uc0y3bgggc/WcfBOoT4jevAltdNky/D8oCOH7Pj6oFTtkCPPz58rO0VW4fx+aU61a8DkrgSCgPm66dMyXLkk3rk6HQLL8+fU+NSkEtCni43EN/vADr1mzxqTf1BYkFaJ//ulYYJ41xMUh6jZbNpiPJ00yHzhkK0+f4npt0MB+a4UUjBiBZ5CUKXfujGHnmNy5E3eOuX0bLhxnNSxXAPUJUWYk/ObOjei+tGnRPcMVisKrkTNnIMDatlUmv23hQjzcf/tNnvGjophnzEBAS8aMzMOGIUVGai5cwPeQUDORVIi+eYP5rV3r+FhJef8eEcCFC+NB+dVXiKyVg6NHYfobNkye8S2xaBHO448/KnN8V+b2beYePbCoFTvHdOyIuBU1RI7LhDqF6M6duJB9fFCaKixM6Rm5Nn/8gdXgoEHOdewfOYIHoo2VfOwiOBgLrUyZcMwuXaQNNBPTSCQ0SUkqRJmZS5dGHqtUvHyZ+Jx+/rn5tCSpEBdeGzbIfyxDtm/HIkEpAe4uPHqEso6pUuF3HDBA6RnJijqFqCAgETtPHre1ozudpUtxQTur/dCTJzDNNWzoXNPc+/d4CBcqhO/boAHzX385Hs3buTPM4hIiuRDt1QsBPo5y9So0ilSpoN0PHy6Pdm8KQYAfTWoXgDlOnoTV67PP1Jtj7Wq0aoXrx81jV9QpRJmZ79yB9vTTT0rPxH0YM8Y5PRCjolAxqmBB5YKa4uPxPatVw3cuXRrC1d7qOPnzw1cmIZIL0dWroUnZU3c4Kor5999RhpEI33fuXOVqGEdFIRitcGF5fd3MeNZkzYr0DW3RLg3Xr1vOtHAT1CtEmVGIXNNGpUOng0kuTRrpSnklRRBQlSRtWuaLF+U5hi0IAjpGtGkDX42XFwTFr79aX07w6VNZmsdLLkRv38Y8ra05HB+Pogg9eiDugAiLjg0b1JEX+egRhNvHH8vnz3/1CoK6VCnXbA6uVjp2hDXIxs5Groi6hei9e3jouVnBYkWJiUFOYebMzOfOST/+Tz/hYbx+vfRjO8rr14j6FhPG06VDVOGuXeZv9k2bZGk3JrkQFQQInYkTzW9z+jT84zlz4nsVL44WX3fuSDMPKTl4EBYpOSoGvXnDXLkySg8+eiT9+CmVK1dwXa1apfRMnIK6hSgzVsm5cimXouGOhIXB5+zrK20e3LFj0PaGDpVuTLl4/BhFOypUwA2fJQv8nitXosOMIYMGoVi+xEguRJmZW7ZkbtQo8Xtv3qApw4AB+B5EiH4fOhQLKbVXkZk7V3o3RHAwfvusWZ3nd00ptGuH60wN1gwnoH4hev8+tNEFC5SeiXsREYGi0unToxyfozx9ipKNSuf42cO1a8zffQdTpqcnHtiFCzP37o3UnAoVEOgiMbII0ZkzEdX+55/w4VauDN8UEepLf/MNtDtXauclCFjg+Pjgt3KU58+ZS5bE4lzO3pkpkUuXcK0Z1Jd2dzyYmUntfPUV0Z49RA8fEqVPr/Rs3IeoKKK2bYmOHSPavp2oWTP7xomJIapblyg4mOj8eaLs2aWdpzN59w7n49AhvG7exPt58hA1bUpUrpz+lTMnkYeH3YeKiIggPz8/Cg8PJ19fX3sHIbpxg+j6dfw9fJjo2jV8ljs3UaNGRA0b4lWwoN1zVZwPH4j8/fH33DmizJntG+fRI5wTnQ6/b7Fi0s4zpdO2La7FW7eIvL2Vno1TcA0h+vAhUcmSRLNnEw0bpvRs3IvYWKJOnbBI2biRqH172/ZnJurVC/uePElUubI881SKrVuJOnbEw+HJEwjVmBh8ljVrYqFarBhRjhx4Zctm8SFilRAVBAj2kBCi0FAIAVFoXr9O9PQptvP0xPHLliXatYto/HiiyZMdEvKq4+FDoqpViWrWJNq9m8jLy7b979whatyYKE0aCFBXXlSokYsXiapUIfr1V6Lu3ZWejdNwDSFKRNS7N26chw+JfHyUno17ER+Pi/7333EDdO1q/b5LlxINGEC0di1Rt27yzVEppk0jmjuX6O1bCCqdDtegKMTE1507+MyQrFkhULNnx98sWRI9+CPi4shv9WoK79WLfFOnxoIkPFwvMMW/ScctVAjC0lCAlypFlDYtPq9Rg6h4caL16+U9N0qwfz8sAt99R/T999bvd+0aBGi2bEQHD0JL15CWVq1wH9y4kWK0UCJXEqKPH+PBMHMm0YgRSs/G/dDpiL7+muh//yP6+Wf82xL//EPUoAFR//5ECxfKPkVFaNYM2uC+fea3i40levkSgs/U6+1bCMp/idDpyO/qVQqvUIF8vbzwWaZMeqFrKIDFf+fNS5Qxo/m5DB1KtHMntFZ3ZNYsorFj4YJo29by9ufPE336KVGBAkR//+3a7ga1cv48UbVqWLjZsgh3B5Rzx9rB11+jJuPbt0rPxD3R6RDBaU3t0OfPkSJRt677RuHpdCh5N3WqLMPLEljEjChWIpTtc0cEgblDB+YMGSy3QfznH0Sh16yplQ+VC0FA7nWpUq4VsCYRnkoLcZuYPBkr/u++U3om7omnJ9FPPxGNHEk0ZAjRqFHJTYlE+A3atydKlYpoyxb8dUdu3YI/MiBA6ZnYhr8//p46pew85MLDg2jNGvg027aFCdwY27cTffIJ/PR//w0tX0N6du6EmX3mTNv91G6AawnRPHngo1q+nOjsWaVn4554eCCAa948vJo3JwoL03/OTDRwINHly3hI5cyp2FRl5+RJPBSqV1d6JraRNy8EzMmTSs9EPjJkINqxg+jVK/jzBUH/mSAQTZiAhV6LFkR//WXZBK5hH+/fEw0aRNSyJVHr1krPRhFcS4gSwf/20UdEffsSJSQoPRv3xMMDUdD79mGxUr26PtVjxQqiVauwkKlWTdl5yk1gIFHFinhgS8iSJUuoTJkyVE3O8+fvj/m7M8WLE23YgIDDadPwXng4HubTp8N3ummTlhYnJ5MnE715Q7RokXtFgtuC0vZkuzh7FgnkWs8/+bl/n7lcOfifZs1CZw83b230H8WLM3/7rWzDy+YTZUYT5FSpUkbd6e+/hw942TIUUfDzY96zR+lZuT+XLqEQzqxZSs9EUVwnOjcpYlrF7dswX2nIR2SkPpe0QAGEsYvpFO5KSAhM1Rs3EnXuLMshJCm2YIpLl+ALPHGCqHZtacdWG4IAv/Xp00RFixLt3QstVUM+BAHWjshIXGvuGhdhBa5nzhWZPh35okOHKj0T9yd1aqRnZMyI5P4uXeALcWdEU6irBRWJlC+P+8PdTbrMCGg5cwZmdy8voly5lJ6V+7NyJc75smUpWoASubIQzZSJaP58FAiwlMOn4RiDB6MayYEDqIZz6BBRrVpE9+8rPTP5CAwkypePKH9+pWdiH97eKLrgzkI0MhLVpMaPJ5o0Cf77oCCiL79MHGikIS3BwURjxqAca506Ss9GcVxXiBJBI2rUCKbd6GilZ+OeiEFES5bgodyqFVagcXEILNq8WekZysPJk66rhYoEBECIuqjHxixXrmAht38/onQnTSIqXRrJ/jt2IKhIQx5GjIDGP3u20jNRBa4tRD08UHbu+XOiGTOUno37cfo0Fih9+6Lsokjp0lj1f/wx/IWdOhG9fq3cPKUmJgYVWMR8S1fF3x9lA93JYpCQgEjcatVw/585Q9Smjf7zVq2IJk6Edrp3r2LTdFuOHMFCZc4clFDUcNHo3KRMnIhIxFu3lJ6J+xAUxJwnD7O/v+WG1VmyoA3azp3Om5+cnDyJaM/z52U9jKzRucyo0EPE/L//yTO+s7lxg7lqVbSrGzcODeaNodMxt2iBalP37zt3ju5MTAyin+vUwTnWYGZXq1hkirFjETXav797mq6cTVwcUYcOOJdbtyKwyBSdOqHgdI0a0Ai6d09cnMEVOXkSuYUVKig9E8fIlAmF6l296IJOB82ncmX4QU+dQmBhmjTGt/f0JFq3DjVy27TBPhqOM2cO0YMHCCbydA/RIQXucSbSpoXP7sgRJF9rOMbQoTDXbttmXbeLXLkQcPS//xH98Qe6irhysFdgIBYF7hB16OpFF+7eRfDK6NGolHXxonUVpDJlQjm6x4/Rqk9bXDvG/fswo48YgYWZxn+4hxAlQpeGTp2Ihg93fU1ISX75BX7mJUsQuGEtHh6Iirx+HUK0aVN0gomIkG+ucsDsHkFFIgEBsBS8e6f0TGxDENAZqFIl5OweP46WdOnSWT9GmTJo7bdlC/bVsA9mxEbkyoVyihqJUdqeLCkvXjBnzMjct6/SM3FNzpxhTp2a+ZtvHBtHEJh//hlVjgoWZN63T5LpOYV79+BHdELFG9l9oszMd+/i++zdK98xpObuXeZ69TDvgQOZIyMdG2/cOPhR//5bkumlODZvxm+xe7fSM1El7qOJEqFA/fTp6Id5+LDSs3EtgoOJ2rVDZ/pFixwby8OD6JtviK5eJSpShKhJE/TlvHFDmrnKieg/tEULVzPFiiGK0hVMum/fontQ2bJET57gHv7pJxSNcISpU/WR5O7aY1UuQkORJ962LYr5ayTDvYQoEYKLGjZEY9iQEKVn4xrExyNpXadDIJGpgA1bKVwYhRm2boVvq0IFpMsEB0szvhwEBuIh7i5tszw84BdVc3BRbCwKpxQtCnfClCloeNCggTTje3mhfGOmTFgoRkVJM667Iwhw0SQkEC1erPRsVIv7CVEvL+QxCQJRt25a5RJrGDYMOaFbt0KblxIPD7SkunkTfqktW1DXdMYMdRbIcCd/qEhAAPIp1db1iBkVx8qUQQ/bzp2J7t1DtL0tvk9ryJwZRRju3oWVRAs0ssy8eci1XbdO+ueCO6G0PVk29u+HHT+FdxiwyJo1+g4YzuDNG+YhQ5DXmz///9s787goy/WNX4MriZAp2rFwxRQlJbd0tDS1PB6OdbLcS80ls9LS3PpZWWZ76lGj5aSZR8X0qGXY5r7NuKciooZ6QlJB3BhUQITn98d1JlABYXjnfeedub+fz3www5l7tvd6nue+7+tWasECz+k5u3CB04F06qvUJSeqlFJbtvA93rPHvY9TEux2pdq2ZVyRkewB1YNvvuFjzpihz+OZlW3blCpbVqkJE4yOxOPxXhFVSqmJEzmqx2YzOhLPZOdOpSpUUGroUBYD6UlCglI9evCC1rKlUps26fv4BfHjj4wnIUGXh9NNRK9c4aJl9mz3Pk5xOH5cqV69+Do3a6bU2rX6xzBuHK8L69fr/9hm4Px5FgS2bavU1atGR+PxeN9xbn6mTGG/X9++LFoQ8jhzhvmhiAjmO/QeqBsayj7UzZv52B06sPhowwbjjtrsdqB6debmvAl/fxaMGVlclJDAeoVGjYCtW4F584A9e+h9rTfvvst8a69enEok5KEU+2rT0phH9oZeaTfj3SJarhw/COnp0nCdn+xsXkCysylkWhUSucIDDzAfGx1ND+ROnYCWLfm+ZWfrG4vNxiIcvRcUemCE6YKz5/bxx4GGDZlzf+MN5iUHDWL9ghGULQt88w1Hp/Xo4Zm5eaP49FPmjufNA2rXNjoaU+DdIgrQDvCrr+heEhVldDSewdixvLj95z+eMdDcz4+nBfv3cypH1apAv37crU6fro9hw7VrLL4xu+l8YVitbBs5edL9j+Ws8rZaORD88GHgX//irm/SpNK3rGhB1aoUi/h4VozLApvDtceMAUaNut7UXygao8+TdWPkSBoJ/Pqr0ZEYy/z5zEdFRRkdSdHs26fUgAEsbggMZB4rKcl9j7d7N18XHfLnn3zyiQoLC1P33HOPPjlRpZQ6dYrPb+lS9z1GerpSs2YpVa8eH6tjRzboe0rhWEEsWsRYPSFfbCQOh1KhoUo1b164sb9QIL4jopmZ/ICEhvID44vs3q1UxYpKDRqkfyGRqyQlKTV+vFJBQRTUp59Wau9e7R9n5kwusjIytL/vQtCtsMhJ3bqsjNaaU6foClSlCgt2+vZ1+wQcTXn5ZX62Nm82OhJjyM1Vql8/ur3pVFTnTViU8qFzjIQEToJ47DH2Pnlj7qswUlOZa6xeHdiyhab9ZiI9HZg7F5gxg8eCTZvSUKNPHx7Zl5Y+fYCkJF1NCRwOB4KCgpCWlobAwED3P+BTTzEfuXNn6e8rPZ3HodHRwNq1nHozbBiPAs2WS7t2jY5G8fEsdrr7bqMj0pevvmLNyOLF/B4IJcL7c6L5adCAloCLFnHiiK9w7RrN+TMzgRUrzCegAFC5Mi3hjh3jxJhGjYDJk3nB7tCB72tpKrCdRUXejNXKvJerjj1ZWXzte/fmYmzgQBblREVxATJtmvkEFGCh0ZIlLLB74gk+T1/h4EFOxxk6VATUVYzeChvC4MFK+fvr1+BtNKNH87jKE3oxtcThYI63a1cajJcrp1T37myov3y5+Pdz4gTzYt9+67ZQC0L349x9+/g8S/I5yMlRasMGpYYN43Gts7/zww/5unkTu3axb3rIEPOkO0rD5ctKNWnCW0m+L8J1+KaIXrqkVOPGSoWHsxHdm1m4kBe+WbOMjsS9JCfzOd5/P59vQADzpzExt86BL17Mf5OcrE+s/0N3Eb12ja/Le+8V/XvZ2XQUeuUVpe66i69N3bpKTZqkVFycPrEahdPB6/PPjY7E/Qwd6lubCTfhWznR/MTFAa1asYdt4ULvnNS+dy99U3v25PG1r+SAjx1jrm7RIuDIEfYjtm7Nxv7OnTmhJX9v7KhR9AhNSNA1TN1zogBzf/7+HJ7uJDeX34d16zg5ZdMm5jyDg3l0268f0KaN73x+XnyRLTkbN3rvEb8zDzp3LjB4sNHRmBrfFVGAfZK9ewPjxwPvv290NNpy9iwLiapWpUOM1obeZkAp4OjRPHFYvx44d4454fbt80R1+HAWKumcJzdERCdPpkPV9u15r8mGDSw8q1CBr0unTnxdWrRgvtDXuHqVz//oURYaeZv5+i+/AJGRzIN+9pnvLI7chG+LKMBm/ldeoVPHiBFGR6MN165xhuf+/cDu3eYs9nAHubmccZp/x3X5Mv9fs2bAgAHAvffyVqOG2y8uuoloejorTw8c4BSdNWv492XK8DSmc2cKp9VqzqIzd5CczEVE7drckZYvb3RE2rB3L/Dgg0DHjqyu9sVFksaIiCrFqs9PPuGH6tFHjY6o9Iwbx1aQNWu0m8nojWRncyX+0kvctcfFsYIZ4A4+PPz6W5MmHKmlEZqLaEYG3YHi4lh1GRfHW2Ii/7/FwrFjBw/yCPvttwG9dsBmZPt2Vn4PHszPidlJTOSxfEgITx88wTnKC5BliMXC3WhSEku8N25k/sysfPMN53bOmCECeivKlaOl4O230/JPKeD48TzxiYvjxebzz2llB9AmMTycYvSXv7DVI/8tONg9u7lr13hEn5rK4QFnzvDPycl5wnnsWN783Fq1GGfv3nkLgEaN2M/ZtCnbXERAi6ZNG7bvDBvGRdaQIUZH5DrnzwPduvH9j4kRAdUQ2Yk6ycjIy4Ns22bOSR6xsfzi9+jhe2YSrtKtG1+nH38s/HeysmhSkF9cDx8GUlI47eJGAgMppvnFtYCctCMrC0FffIG04cMRmL/Q6epVCmR+wSyoB7ZiRd73PffkCaVT4IsSyOee4/Sc+PgiXhjhT4YPZ75882ZOhTIbmZnAI4/w/bbb+XkRNENEND9nz+ZV49ntQLVqxsZTEs6f52o5KIjGAbfdZnREnk9uLnDHHTz+njTJtfvIyrp5d+j8c/5bAQ38jmvXEHToENLCwhCYf6JJ2bJ5IpxfjG/8c0CAawulBQuY/z13js9fKJqsLOYQk5JYaFSjhtERFZ/cXA53+P571gJ4a7WxgYiI3sixY2yBCA3lh84MVa05OZzFuXs3v+R16hgdkTmIi2MR0fr1hhx9G1KdC/AzHhoK/PADPzfCrTl1ioVGDRrwumCWOZtjxzJdtXw52/kEzfHC5shSUr8+sGoVsG8fvVmduTBPZtIk+pcuWSICWhLs9rweUl+iXj3uZo0c0m02atbkeLdt21jNbwZmzaIV48yZIqBuRES0IFq3piCtXMn5ep68WV+6FPjgA+DDD4EuXYyOxlzYbEBEhO8VWVgsNOHQ0WzfK2jXjsI0ezbw738bHU3RrFjBroOxY4GRI42OxqsRES2M7t3Z9jJrFitdPZEDB4BnnmFV8ZgxRkdjPmw2Xhh9EauV01yys42OxFw89xy/c8OHM3XiidhsPEXr1YsLbMGtiIgWxYgRwIQJPL5ZutToaK7n/HlOnw8NBebMkUrckpKSwtygrxZaWK1sc4mNNToSc2Gx0JglPJxV8KmpRkd0PUeOsNe9dWtWFHujnamHIa/wrXj3XXqHPv0053B6Ajk5XGlevAh8953vHUdqgTMf6Ks70RYt6MIjedGSU7Eij0szMtiHe+2a0RGRlBS2bNWoweuCuE/pgojorfDzo1mz1coVnhYDjUvL668Dq1fTWKFuXaOjMSd2O51bfG0As5MKFdgSJSLqGiEh9N7evJmnVUZz7hwFNCODwxQ0dNYSikZEtDhUqMCVXePGLN4xcke6bBnw3ns0zH/4YePiMDu+nA91YrVKcVFp6NCB7SPTp3NqkFEkJ+f1sf78s3hl64yIaHEJCuL0g5Ytga5d80y89eTgQWDQIBYMjB2r/+N7C5mZLAoxKB8aFRWFxo0bo1WrVoY8/p+0a8cLb1KSsXGYmZEjmeoZOpQDH/Tmjz8o5ufOcVfcrJn+Mfg4YrZQUjIygCeeYMP1smWs4tWDCxdYLODvz141yYO6js3GkV979gDNmxsWhmFmC05SUoA772Q7V69e+j++t5CRwQXJxYvArl0cXqAHx4/TqlQpXo/MaFXqBchOtKT4+3PaS2Qkq/P0qNrNyQGeeoqrzW+/FQEtLTYbX8OmTY2OxFhq1OCFV450S4e/PwuNHA5a7Olh0HL4MPDAA3RO2rxZBNRARERdoUIFimevXvzSuLvx+s03WSyweLF8WbTAbqeRuMxS5JG2FBeVnjp1uKNft851H+biEhvLmaBVqlBAa9Vy7+MJRSIi6iply1I8Bw8GBg4EvvjCPY/z7bfA1Klstena1T2P4UsoRdHw9aIiJ+3acVCzczi54DqdO9M57IMPWLnrDnbuZBFRSAjHNt55p3seRyg2shQvDWXKUDz9/elkcuUKMHq0dvcfH89pG08+6Rll9N7A0aNskPdVk4UbsVp5/LhrFy/OQukYM4aDIJ55BggLoymDVmzZwjRSeDhH991+u3b3LbiM7ERLi58fDZ4nTuQX6J13tLnftDQ6EtWuDcybJ45EWmGz8bVs08boSDwD5+xROdLVBouFDmL16/P7e+GCNve7Zg1Polq2ZI+4CKjHIDtRLbBYeNxaqRLw2ms8GnvnHdeFLzeXhUSpqdwhBARoG68vY7NxeLVchEiZMlxQiIhqR6VKTMO0bElnsZgYvs6uEhPD06guXdgRYIbxjD6E7ES1wmKhgE6bRjOEl192ffrLlCmc9RgdTW9cQTskH3ozVivbpnJzjY7Ee6hXj4WAP//MwkBXWbKEXQDdu1OYRUA9DhFRrRkzhgbVs2Zx0kNJy91XrgTeeovFRN26uSdGX+X8eeaZJR96Pe3a8bU5csToSLyLrl15QjV1KgWwpHz9NX27+/ShxWf58pqHKJQeEVF3MGIEvwBz57IN5tKl4v27w4fpftKjB/Dqq24N0SfZvp0/ZSd6Pa1bM7cvR7raM2ECj2IHDAAOHSrev1EK+OgjFicNHQrMny/tWB6MiKi7GDiQDdirV3Pnc/x40b/vcLAQISSEAiyFRNpjswHVq/OoTcgjMBC4914xXXAHFgsLA2vXBh5/nAWDRXHlCushxo/nQvrzz2WcmYcj7447eewx7n6uXAFatQLWri3493JzuVJNTqbRfeXKuobpMzjzobJAuRkxXXAfAQE8zk1O5ve8sNxzYiLtKL/7jrnQd9+Vz6oJEBF1N02asMLWaVw/Y8bNBUdTpwLffw8sWgQ0aGBMnN5OdjawY4fkQwujXTvmRM+eNToS76RBA36/Y2L4fb+RTZt4jbhwgYsZ8TI2DSKielClCpujX3mFhUcDB9K0GgBWrQImT2YxUWSksXF6M/v35xmFCzfjXFw488aC9kRGsvJ+8mR+7wEuqD/5hO0rTZtywS2TWEyFiKhelClDS7BFi2gJ9uCDXH32789cqLv9Nn0dm42exxpPbcnKykJERAQsFgv27dun6X3rSp06tJCTI1338n//xzRP//5AXBwLh0aOBF58kaMWq1UzOkKhhIiI6k2/frygp6TQa/P221l9J8UD7sVu53FZhQqa3u348eNRs2ZNTe/TECwW7tKluMi9+PnRczs4mFXRixbx+z9jhlTgmhS5chtB8+bAggXs+zp9mk3ZgvtQiuKg8VHuTz/9hNWrV+Pjjz/W9H4Nw2qlwXl2ttGReDcHD7JKNyuLZvUDBhgdkVAKRESNokMHNrg/+yzN64cPB65eNToq7yQpCTh5UtOiopSUFAwbNgwLFizAbbfdptn9GorVCmRmAmY+lvZ05szhd79hQxZyvfSS0REJpURE1EgqVmRRwZw57CXr1Ill8IK2OI8oNRJRpRQGDRqE5557Di1btiz2v8vKyoLD4bju5lE0b87jbsmLas/Vq8ALLwDDhgFDhgDr14ulp5cgIuoJDBnCIqNjx5i327XL6Ii8C7udLQbBwUX+2ptvvgmLxVLkbffu3Zg9ezYcDgdeLaGr1HvvvYegoKA/byEhIaV5VtpTvjz7mSUvqi0pKay+/fJLjk787DOx8PMiLEq56pIuaM7Jk7T827+fU2Befrl00x8E0rw52wbmzSvy186ePYuzt+iTrFOnDvr06YOYmBhY8jXC5+TkoEyZMujfvz/mz59f4L/NyspCVlbWn//tcDgQEhKCtLQ0BAYGluAJuZEJE4CFC4E//pBGfy1Yu5b2fdnZwPLl0mLlhYiIehqZmSyD/+c/efz49ddy7FMaLl0CgoK4Axg6VJO7PHHixHVHsadOnULXrl2xbNky3H///bj77ruLdT8OhwNBQUGeJaLff88WjMREoFYto6MxL5cuAePG0bavUydW4BbzcyGYC6mp9jQqVgSmT6fP5qBB3EF98AHw/PPSBuMKO3bQZk3DoqJaN4hLwP/mvdavX7/YAuqxtG3LnzabiKirbNrE3WdKCmseRoyQ764XI++sp/LAAzzWHTSIzdgPP8zdgVAy7HY6RjVqZHQk5iA4GLjnHikucoUrV5iC6diRu87YWBYTiYB6NfLuejIBAUBUFLBmDZCQwEkbc+a4PuzbF7HZuLty44WsTp06UEohIiLCbY+hK2JGX3LsdiAigmmD6dOBjRuB+vWNjkrQARFRM9ClC3DgANCzJ0vkIyNZhCQUTU4OsG2bFHOUFKuVpyDFnYPry2RmcmzZAw8AVauyx3b0aNl9+hDyTpuFoCAO+V61il/U8HBWUcqutHDi4zmnVSa3lAyrlQuQnTuNjsSz2b0baNECmDmTY8u2bqWJguBTiIiajchIGldHRgJPP82WmJQUo6PyTGw2tgi1bm10JOYiLIyeznKkWzBXrwKvvw60acNCwD172Bok7Wg+iYioGbnjDu5Cly+nUISHczKMcD12O3DffYC32PLphZ8f88giojezbx8NKd5/nyPNtm/n90/wWUREzUyPHjSz7tCBQ3z79JGhyvlxg+m8z2C1Mp+cm2t0JJ5Bdjbw9tsUUICuYq+/DpQrZ2xcguGIiJqd4GDuQqOjgdWraW83bRonRPgyycnA8eOSD3UVqxW4eBE4dMjoSIxFKdYhNG0KvPUWMHEiBdRbKrGFUiMi6g1YLEDfvpwK0a8f8zONG1NcfbXwyHkUKSLqGq1b81jXl4909+5lZXz37kDNmiwkevtt8b0VrkNE1JsIDmZf6YEDLA7p1Qto3555G1/DbqfjjtkdhIwiIIBuWb4ooidP0uSkRQvO+121ih64svsUCkBE1BsJC8v74l++zCKRPn2A3383OjL9kHxo6WnXzrcmuly6BLzxBlMiP/7IBWlsLCvhxYxfKAQRUW+mc2eW33/1FbB5M63vJkwA0tKMjsy9ZGTwectRbumwWumUlZpqdCTuJSeHY8pCQ4EPP6R139Gj9LwtK/biQtGIiHo7ZcrQDDshAXj1VRpih4ZylZ2dbXR07mHPHj432YmWDuciZNs2Y+NwJ7/8wmPaZ59l/vO332ic4ClTdQSPR0TUV6hUiX1tCQkslBg5kl68MTHeV3xks/H53nuv0ZGYm1q1WFDjjXnRAweAv/6VtypV6M60cKFMrhFKjIior1GzJo93f/2VRTePPspj3717jY5MO+x2usnIUVzpsFi8Ly+anEz/6YgI4NgxYMUKji5z9n8KQgkREfVVIiI4HWbVKlYgtmgB9O9P43EzoxRFVPKh2mC1si/y6lWjIykdKSk0RwgNpXDOmEGjkscfl6IhoVSIiPoyFgsrD2NjmSPdupXi+sgjzBWZ8Zg3IYGuTR6cD42KikLjxo3Rygy7H6uVxh1mPak4dIg7z9q1KZwjRrBoaNQo6fcUNEFEVKB12YgRPN5avBg4d465ombNgPnzzbULsdm4OGjTxuhICuWFF15AfHw8du3aZXQotyYigibrZjrSVYrzPP/+d5qO/PAD8OabQFIS8NFHzIEKgkaIiAp5lC3LftLdu4ENG1hkMWgQULcuDbcvXDA6wlvjNOQPCjI6Eu+gfHm6F5mhuCg7m4vAli2Bhx4CEhOBr79mf/TEiSKeglsQERVuxmIBOnZkvjQ+Hvjb31jZGxLCHjpPNm2QfKj2WK1cnHjq8b7DAUyfDtSvT9vLqlWZjoiNBQYOlGNbwa2IiApFExbGRvTERGD0aGDBAl6snDtWT+L8eebAPDgfakqsVla1JiYaHcn1JCUB48ZxcTdxInef+/ZxEMMjj0jBkKALIqJC8bjzTppvnzgBzJ5NAW3VijvWmBjPGJnlNAUQEdWWtm3501Pyonv3Ak89BdSrxwXeiBHAf//L/H2zZkZHJ/gYIqJCyahUCXj+eU6MWb6cRUePPsoCjtmzgTNnjIvNZgNq1GAOV9COatWAhg2NzYtmZrI1pXNnoHlzVpJ//DF3o++/D9x1l3GxCT6NiKjgGmXKcCi43U7xatIEGDOGZg7duvHYNz1d35jsdu5C5RhPe6xW/UU0JwdYtw4YMoQnIU88QZP4JUvYpvLSS0DlyvrGJAg3ICIqlB6rlbvS06fpzXvpEjBgAHeFffvyuNfdbTLZ2bRuk6Ii92C1slDH3QsjpZgqGDOGuc4uXegoNGoU8907dnDEn7hRCR6CRSlPLbkTTE1iItsNoqPpU3rHHUDPnnRFateOA5+1ZOdO4P77mRf14B7R/DgcDgQFBSEtLQ2Bnm54fugQj+zXrKGwac1vv/GzEh1Nw4waNYDevfl5adVKThcEj0V2ooJ7qF2bFZOxsbw9+yzw00/Agw8yZzlxIsVVK+x2oEIF4L77tLtPIY+GDdlnqeWR7unTdBFq1Yr3P306F1irVwN//AHMnMkeVRFQwYORnaigH7m5vAgvWgQsXcqWlPBw7jb69qXwukrPnmzD2LJFu3jdjKl2ogAtInNygJ9/dv0+Ll5kgVB0NLB+Pd2yIiPZ3xkZCfj7axauIOiB7EQF/fDzA9q3Bz77jLuQmBiK6JQpQJ063IVMmcJCpZLMOnWazktri3uxWnlcnpNT/H+jFCu5P/0U+Mc/WCA0dCj//ssvaQy/YgXw5JMioIIpkZ2oYDyXLgErVwLLltHz9OJFICCAR7+dOrGtoWnTwvOov//OI+KVK9luYxJMtxPduJGGBrGxRc9qPXGCu8z161lde+oUC4HatGFFd+/erOIWBC9ARFTwLHJyOOvUeQHeuhXIyGCv4kMPUVA7deJIK2euLDqaR8Kpqfw9k2A6Eb18mZ7EUVHA8OF5f5+aSq/ldev4vh09yvfmvvvyFkHt23NhJAhehoio4NlkZfEI0SmqO3ZQaENCeHHu3BlYu5a/c+SI0dGWCNOJKEBz99BQOgY535PYWP6/Ro3yRLNDB3rYCoKXIyIqmIv0dGDz5rxdj3OIeFAQL+wdOvDot359j+8lNIWIKsWCrdhYFm3Nncv/Bq5fyDz0kLgGCT6JiKhgblJTgfHj+fPwYc5EBdjuEhbGwqX8t1q1PKZlwuNE9Nw5IC4OOHiQP5035wi8atWYC61UCZg2DWjQwGNeS0EwChFRwbs4c+ZmEYiL47gsgDZxTZrcLK7Vq+suCIaJqMPBEXfO18b5ejl3mOXKsW/T+do0acJb/fram2QIgskRERW8H6XYvH/gwPXCGh/PnCvAXZZTNMLCWD1avToQHMyfgYGai6zbRDQzkzvzM2f48/Rp7tKdz/vECf6enx/zm06hdD7/Bg0opIIg3BIRUcF3ycnh8e+Nu9bffru5F7J8eYppYTen2DpvFSve8uGLLaLZ2cDZsxRFpzDm/3njnwvyt61d+/qdd5MmLASS3kxBKBUiooJwIzk5dFO6UZwKu6Wl3XwfAQGFCtSVzExcuXIF6UqhXm4u0qpVQ6CfX8E73aws9s3eiL8//WXzi7fzzzf+XXBwsURdEISSIyIqCKUlK6vg3aHzqLgQHBkZCJo6FWmvvYbAwkSubNmChbFSJTc8EUEQSoqIqCAYhMdV5wqCUGKk1E4QBEEQXEREVBAEQRBcRERUEARBEFxERFQQBEEQXEREVBAEQRBcRERUEARBEFxERFQQBEEQXET6RAXBIJRSSE9PR+XKlWGRaSiCYEpERAVBEATBReQ4VxAEQRBcRERUEARBEFxERFQQBEEQXEREVBAEQRBcRERUEARBEFxERFQQBEEQXEREVBAEQRBc5P8BhyK4RKlclTsAAAAASUVORK5CYII=", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.plot(stereoN, number_values=15, ranges={th: (pi/8,pi)})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Points on $\\mathbb{S}^2$\n", "\n", "To create a point on $\\mathbb{S}^2$, we use SageMath's ***parent / element*** syntax, i.e. the call operator `S2(...)` acting on the parent `S2`, with the point's coordinates in some chart as argument. \n", "\n", "For instance, we declare the **North pole** (resp. the **South pole**) as the point of coordinates $(0,0)$ in the chart $(V,(x',y'))$ (resp. in the chart $(U,(x,y))$):" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point N on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "N = S2((0,0), chart=stereoS, name='N')\n", "print(N)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point S on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "S = S2((0,0), chart=stereoN, name='S')\n", "print(S)" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^2$" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N.parent()" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^2$" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We have of course

" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in S2" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{False}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in U" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in V" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{False}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce some point $p$ of stereographic coordinates $(x,y) = (1,2)$:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ "p = S2((1,2), chart=stereoN, name='p')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$p$ lies in the open subset $A$:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p in A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Charts acting on points" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By definition, a chart maps points to pairs of real numbers (the point's coordinates): " ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(1, 2\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(1, 2\\right)$" ], "text/plain": [ "(1, 2)" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN(p) # by definition of p" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{1}{5}, \\frac{2}{5}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{1}{5}, \\frac{2}{5}\\right)$" ], "text/plain": [ "(1/5, 2/5)" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS(p)" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(2 \\, \\arctan\\left(\\frac{1}{5} \\, \\sqrt{5}\\right), \\arctan\\left(2\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(2 \\, \\arctan\\left(\\frac{1}{5} \\, \\sqrt{5}\\right), \\arctan\\left(2\\right)\\right)$" ], "text/plain": [ "(2*arctan(1/5*sqrt(5)), arctan(2))" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher(p)" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0\\right)$" ], "text/plain": [ "(0, 0)" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS(N)" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [], "source": [ "#stereoN(N) ## returns an error" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Maps between manifolds: the embedding of $\\mathbb{S}^2$ into $\\mathbb{R}^3$\n", "\n", "Let us first declare $\\mathbb{R}^3$ as the 3-dimensional Euclidean space, denoting the Cartesian coordinates by\n", "$(X,Y,Z)$:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathbb{R}^3,(X, Y, Z)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathbb{R}^3,(X, Y, Z)\\right)$" ], "text/plain": [ "Chart (R^3, (X, Y, Z))" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3. = EuclideanSpace(name='R^3', latex_name=r'\\mathbb{R}^3', metric_name='h')\n", "cartesian = R3.cartesian_coordinates()\n", "cartesian" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As an Euclidean space, `R3` is considered by Sage as a smooth manifold:" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Join of Category of smooth manifolds over Real Field with 53 bits of precision and Category of connected manifolds over Real Field with 53 bits of precision and Category of complete metric spaces\n" ] } ], "source": [ "print(R3.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding $\\Phi: \\mathbb{S}^2 \\longmapsto \\mathbb{R}^3$ is then defined via the method `diff_map` by providing the standard formulas relating the stereographic coordinates to the ambient Cartesian ones when considering the **stereographic projection** from the point $(0,0,1)$ (North pole) or $(0, 0, -1)$ (South pole) to the equatorial plane $Z=0$:" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [], "source": [ "Phi = S2.diff_map(R3, {(stereoN, cartesian): \n", " [2*x/(1+x^2+y^2), 2*y/(1+x^2+y^2),\n", " (x^2+y^2-1)/(1+x^2+y^2)],\n", " (stereoS, cartesian): \n", " [2*xp/(1+xp^2+yp^2), 2*yp/(1+xp^2+yp^2),\n", " (1-xp^2-yp^2)/(1+xp^2+yp^2)]},\n", " name='Phi', latex_name=r'\\Phi')" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \\frac{2 \\, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \\frac{2 \\, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\\right) \\end{array}$" ], "text/plain": [ "Phi: S^2 → R^3\n", "on U: (x, y) ↦ (X, Y, Z) = (2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1), (x^2 + y^2 - 1)/(x^2 + y^2 + 1))\n", "on V: (xp, yp) ↦ (X, Y, Z) = (2*xp/(xp^2 + yp^2 + 1), 2*yp/(xp^2 + yp^2 + 1), -(xp^2 + yp^2 - 1)/(xp^2 + yp^2 + 1))" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display()" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Hom}\\left(\\mathbb{S}^2,\\mathbb{R}^3\\right)\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Hom}\\left(\\mathbb{S}^2,\\mathbb{R}^3\\right)$" ], "text/plain": [ "Set of Morphisms from 2-dimensional differentiable manifold S^2 to Euclidean space R^3 in Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.parent()" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Set of Morphisms from 2-dimensional differentiable manifold S^2 to Euclidean space R^3 in Category of smooth manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(Phi.parent())" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.parent() is Hom(S2, R3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$\\Phi$ maps points of $\\mathbb{S}^2$ to points of $\\mathbb{R}^3$:

" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(N) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\Phi\\left(N\\right)\$" ], "text/latex": [ "$\\displaystyle \\Phi\\left(N\\right)$" ], "text/plain": [ "Point Phi(N) on the Euclidean space R^3" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N1 = Phi(N)\n", "print(N1)\n", "N1" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0, 1\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0, 1\\right)$" ], "text/plain": [ "(0, 0, 1)" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cartesian(N1)" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(S) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\Phi\\left(S\\right)\$" ], "text/latex": [ "$\\displaystyle \\Phi\\left(S\\right)$" ], "text/plain": [ "Point Phi(S) on the Euclidean space R^3" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S1 = Phi(S)\n", "print(S1)\n", "S1" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0, -1\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0, -1\\right)$" ], "text/plain": [ "(0, 0, -1)" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cartesian(S1)" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(p) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\Phi\\left(p\\right)\$" ], "text/latex": [ "$\\displaystyle \\Phi\\left(p\\right)$" ], "text/plain": [ "Point Phi(p) on the Euclidean space R^3" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p1 = Phi(p)\n", "print(p1)\n", "p1" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3}\\right)$" ], "text/plain": [ "(1/3, 2/3, 2/3)" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cartesian(p1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\Phi$ has been defined in terms of the stereographic charts $(U,(x,y))$ and $(V,(x',y'))$, but we may ask its expression in terms of spherical coordinates. This triggers a computation involving the transition map $(A,(x,y))\\rightarrow (A,(\\theta,\\phi))$:" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\end{array}$" ], "text/plain": [ "Phi: S^2 → R^3\n", "on A: (x, y) ↦ (X, Y, Z) = (2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1), (x^2 + y^2 - 1)/(x^2 + y^2 + 1))" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(stereoN_A, cartesian)" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}$" ], "text/plain": [ "Phi: S^2 → R^3\n", "on A: (th, ph) ↦ (X, Y, Z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(spher, cartesian)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use $\\Phi$ to draw the grid of spherical coordinates $(\\theta,\\phi)$ in terms of the Cartesian coordinates $(X,Y,Z)$ of $\\mathbb{R}^3$:" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_spher = spher.plot(chart=cartesian, mapping=Phi, number_values=11, \n", " color='blue', label_axes=False)\n", "graph_spher" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We may also use the embedding $\\Phi$ to display the stereographic coordinate grid in terms of the Cartesian coordinates in $\\mathbb{R}^3$. First for the stereographic coordinates from the North pole:

" ] }, { "cell_type": "code", "execution_count": 85, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = stereoN.plot(chart=cartesian, mapping=Phi, number_values=25, \n", " label_axes=False)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

and then have a view with the stereographic coordinates from the South pole superposed (in green):

" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph += stereoS.plot(chart=cartesian, mapping=Phi, number_values=25, \n", " color='green', label_axes=False)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may add the points $N$, $S$ and $p$ to the graphic, thanks to the method `plot` of points:" ] }, { "cell_type": "code", "execution_count": 87, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph += N.plot(chart=cartesian, mapping=Phi, color='red', \n", " label_offset=0.05)\n", "graph += S.plot(chart=cartesian, mapping=Phi, color='green', \n", " label_offset=0.05)\n", "graph += p.plot(chart=cartesian, mapping=Phi, color='blue', \n", " label_offset=0.05)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Tangent spaces\n", "\n", "The **tangent space** to the manifold $\\mathbb{S}^2$ at the point $p$ is" ] }, { "cell_type": "code", "execution_count": 88, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent space at Point p on the 2-dimensional differentiable manifold S^2\n" ] }, { "data": { "text/html": [ "\$\\displaystyle T_{p}\\,\\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle T_{p}\\,\\mathbb{S}^2$" ], "text/plain": [ "Tangent space at Point p on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp = S2.tangent_space(p)\n", "print(Tp)\n", "Tp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$T_p \\mathbb{S}^2$ is a vector space over $\\mathbb{R}$ (represented here by Sage's symbolic ring SR):" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Category of finite dimensional vector spaces over Symbolic Ring\n" ] } ], "source": [ "print(Tp.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Its dimension equals the manifold's dimension:

" ] }, { "cell_type": "code", "execution_count": 90, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 2\$" ], "text/latex": [ "$\\displaystyle 2$" ], "text/plain": [ "2" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dim(Tp)" ] }, { "cell_type": "code", "execution_count": 91, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dim(Tp) == dim(S2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The tangent space at $p$ is the **fiber over** $p$ of the **tangent bundle** $T\\mathbb{S}^2$:" ] }, { "cell_type": "code", "execution_count": 92, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp is S2.tangent_bundle().fiber(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The vector space $T_p \\mathbb{S}^2$ is endowed with bases inherited from the coordinate frames defined around $p$:" ] }, { "cell_type": "code", "execution_count": 93, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right), \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right), \\left(\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right), \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right), \\left(\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right]$" ], "text/plain": [ "[Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional differentiable manifold S^2,\n", " Basis (∂/∂xp,∂/∂yp) on the Tangent space at Point p on the 2-dimensional differentiable manifold S^2,\n", " Basis (∂/∂th,∂/∂ph) on the Tangent space at Point p on the 2-dimensional differentiable manifold S^2]" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp.bases()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On the contrary, since $(V,(x',y'))$ is the only chart defined so far around the point $N$, \n", "we have a unique predefined basis in $T_N \\mathbb{S}^2$:" ] }, { "cell_type": "code", "execution_count": 94, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right]$" ], "text/plain": [ "[Basis (∂/∂xp,∂/∂yp) on the Tangent space at Point N on the 2-dimensional differentiable manifold S^2]" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T_N = S2.tangent_space(N)\n", "T_N.bases()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To shorten some writings, there is the concept of default basis:" ] }, { "cell_type": "code", "execution_count": 95, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)$" ], "text/plain": [ "Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 95, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp.default_basis()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An element of $T_p\\mathbb{S}^2$ is constructed via SageMath's *parent/element* syntax, i.e. via the call method of the parent:" ] }, { "cell_type": "code", "execution_count": 96, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector v at Point p on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "v = Tp((-2, 3), name='v')\n", "print(v)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Equivalently, one can use the method `tangent_vector` of manifolds:" ] }, { "cell_type": "code", "execution_count": 97, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v == S2.tangent_vector(p, -2, 3, name='v')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One has of course:" ] }, { "cell_type": "code", "execution_count": 98, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v in Tp" ] }, { "cell_type": "code", "execution_count": 99, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{p}\\,\\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle T_{p}\\,\\mathbb{S}^2$" ], "text/plain": [ "Tangent space at Point p on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The vector $v$ expanded in the default basis of $T_p \\mathbb{S}^2$:" ] }, { "cell_type": "code", "execution_count": 100, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = -2 \\frac{\\partial}{\\partial x } + 3 \\frac{\\partial}{\\partial y }\$" ], "text/latex": [ "$\\displaystyle v = -2 \\frac{\\partial}{\\partial x } + 3 \\frac{\\partial}{\\partial y }$" ], "text/plain": [ "v = -2 ∂/∂x + 3 ∂/∂y" ] }, "execution_count": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Expansion in other bases:" ] }, { "cell_type": "code", "execution_count": 101, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = -\\frac{18}{25} \\frac{\\partial}{\\partial {x'} } -\\frac{1}{25} \\frac{\\partial}{\\partial {y'} }\$" ], "text/latex": [ "$\\displaystyle v = -\\frac{18}{25} \\frac{\\partial}{\\partial {x'} } -\\frac{1}{25} \\frac{\\partial}{\\partial {y'} }$" ], "text/plain": [ "v = -18/25 ∂/∂xp - 1/25 ∂/∂yp" ] }, "execution_count": 101, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display(Tp.bases()[1])" ] }, { "cell_type": "code", "execution_count": 102, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = -\\frac{4}{15} \\, \\sqrt{5} \\frac{\\partial}{\\partial {\\theta} } + \\frac{7}{5} \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle v = -\\frac{4}{15} \\, \\sqrt{5} \\frac{\\partial}{\\partial {\\theta} } + \\frac{7}{5} \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "v = -4/15*sqrt(5) ∂/∂th + 7/5 ∂/∂ph" ] }, "execution_count": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display(Tp.bases()[2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Tangent vectors are endowed with a method `plot`:" ] }, { "cell_type": "code", "execution_count": 103, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 103, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph += v.plot(chart=cartesian, mapping=Phi, scale=0.2, width=0.5)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Differential of a smooth map\n", "\n", "The differential of the map $\\Phi$ at the point $p$ is" ] }, { "cell_type": "code", "execution_count": 104, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Generic morphism:\n", " From: Tangent space at Point p on the 2-dimensional differentiable manifold S^2\n", " To: Tangent space at Point Phi(p) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle {\\mathrm{d}\\Phi}_{p}\$" ], "text/latex": [ "$\\displaystyle {\\mathrm{d}\\Phi}_{p}$" ], "text/plain": [ "Generic morphism:\n", " From: Tangent space at Point p on the 2-dimensional differentiable manifold S^2\n", " To: Tangent space at Point Phi(p) on the Euclidean space R^3" ] }, "execution_count": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p = Phi.differential(p)\n", "print(dPhi_p)\n", "dPhi_p" ] }, { "cell_type": "code", "execution_count": 105, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{p}\\,\\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle T_{p}\\,\\mathbb{S}^2$" ], "text/plain": [ "Tangent space at Point p on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p.domain()" ] }, { "cell_type": "code", "execution_count": 106, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{\\Phi\\left(p\\right)}\\,\\mathbb{R}^3\$" ], "text/latex": [ "$\\displaystyle T_{\\Phi\\left(p\\right)}\\,\\mathbb{R}^3$" ], "text/plain": [ "Tangent space at Point Phi(p) on the Euclidean space R^3" ] }, "execution_count": 106, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p.codomain()" ] }, { "cell_type": "code", "execution_count": 107, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Hom}\\left(T_{p}\\,\\mathbb{S}^2,T_{\\Phi\\left(p\\right)}\\,\\mathbb{R}^3\\right)\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Hom}\\left(T_{p}\\,\\mathbb{S}^2,T_{\\Phi\\left(p\\right)}\\,\\mathbb{R}^3\\right)$" ], "text/plain": [ "Set of Morphisms from Tangent space at Point p on the 2-dimensional differentiable manifold S^2 to Tangent space at Point Phi(p) on the Euclidean space R^3 in Category of finite dimensional vector spaces over Symbolic Ring" ] }, "execution_count": 107, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The image by $\\mathrm{d}\\Phi_p$ of the vector $v\\in T_p\\mathbb{S}^2$ introduced above is" ] }, { "cell_type": "code", "execution_count": 108, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\mathrm{d}\\Phi}_{p}\\left(v\\right)\$" ], "text/latex": [ "$\\displaystyle {\\mathrm{d}\\Phi}_{p}\\left(v\\right)$" ], "text/plain": [ "Vector dPhi_p(v) at Point Phi(p) on the Euclidean space R^3" ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p(v)" ] }, { "cell_type": "code", "execution_count": 109, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector dPhi_p(v) at Point Phi(p) on the Euclidean space R^3\n" ] } ], "source": [ "print(dPhi_p(v))" ] }, { "cell_type": "code", "execution_count": 110, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 110, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p(v) in R3.tangent_space(Phi(p))" ] }, { "cell_type": "code", "execution_count": 111, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\mathrm{d}\\Phi}_{p}\\left(v\\right) = -\\frac{10}{9} e_{ X } + \\frac{1}{9} e_{ Y } + \\frac{4}{9} e_{ Z }\$" ], "text/latex": [ "$\\displaystyle {\\mathrm{d}\\Phi}_{p}\\left(v\\right) = -\\frac{10}{9} e_{ X } + \\frac{1}{9} e_{ Y } + \\frac{4}{9} e_{ Z }$" ], "text/plain": [ "dPhi_p(v) = -10/9 e_X + 1/9 e_Y + 4/9 e_Z" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPhi_p(v).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Algebra of scalar fields\n", "\n", "The set $C^\\infty(\\mathbb{S}^2)$ of all smooth functions $\\mathbb{S}^2\\rightarrow \\mathbb{R}$ has naturally the structure of a commutative algebra over $\\mathbb{R}$. $C^\\infty(\\mathbb{S}^2)$ is therefore returned by the method `scalar_field_algebra()` of manifolds:" ] }, { "cell_type": "code", "execution_count": 112, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 112, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CS = S2.scalar_field_algebra()\n", "CS" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Since the algebra internal product is the pointwise multiplication, it is clearly commutative, so that $C^\\infty(\\mathbb{S}^2)$ belongs to Sage's category of commutative algebras:

" ] }, { "cell_type": "code", "execution_count": 113, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces\n" ] } ], "source": [ "print(CS.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The base ring of the algebra $C^\\infty(\\mathbb{S}^2)$ is the field $\\mathbb{R}$, which is represented here by Sage's Symbolic Ring (SR):

" ] }, { "cell_type": "code", "execution_count": 114, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\text{SR}\$" ], "text/latex": [ "$\\displaystyle \\text{SR}$" ], "text/plain": [ "Symbolic Ring" ] }, "execution_count": 114, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CS.base_ring()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Elements of $C^\\infty(\\mathbb{S}^2)$ are of course (smooth) scalar fields:

" ] }, { "cell_type": "code", "execution_count": 115, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(CS.an_element())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

This example element is the constant scalar field that takes the value 2:

" ] }, { "cell_type": "code", "execution_count": 116, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} & \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} & \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}$" ], "text/plain": [ "S^2 → ℝ\n", "on U: (x, y) ↦ 2\n", "on V: (xp, yp) ↦ 2\n", "on A: (th, ph) ↦ 2" ] }, "execution_count": 116, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CS.an_element().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

A specific element is the zero one:

" ] }, { "cell_type": "code", "execution_count": 117, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field zero on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "f = CS.zero()\n", "print(f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Scalar fields map points of $\\mathbb{S}^2$ to real numbers:

" ] }, { "cell_type": "code", "execution_count": 118, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0, 0\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0, 0\\right)$" ], "text/plain": [ "(0, 0, 0)" ] }, "execution_count": 118, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(N), f(S), f(p)" ] }, { "cell_type": "code", "execution_count": 119, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} 0:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & 0 \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & 0 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} 0:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & 0 \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & 0 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$" ], "text/plain": [ "zero: S^2 → ℝ\n", "on U: (x, y) ↦ 0\n", "on V: (xp, yp) ↦ 0\n", "on A: (th, ph) ↦ 0" ] }, "execution_count": 119, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Another specific element is the algebra unit element, i.e. the constant scalar field 1:

" ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field 1 on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "f = CS.one()\n", "print(f)" ] }, { "cell_type": "code", "execution_count": 121, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(1, 1, 1\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(1, 1, 1\\right)$" ], "text/plain": [ "(1, 1, 1)" ] }, "execution_count": 121, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(N), f(S), f(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A generic scalar field is defined by its coordinate expression in some chart(s); for instance:" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} f:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ W : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} f:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ W : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}$" ], "text/plain": [ "f: S^2 → ℝ\n", "on U: (x, y) ↦ 1/(x^2 + y^2 + 1)\n", "on W: (xp, yp) ↦ (xp^2 + yp^2)/(xp^2 + yp^2 + 1)\n", "on A: (th, ph) ↦ -1/2*cos(th) + 1/2" ] }, "execution_count": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = S2.scalar_field({stereoN: 1/(1+x^2+y^2)}, name='f')\n", "f.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that Sage has used the transition map between the two stereographic charts on $W$ to express $f$ in terms of the coordinates $(x',y')$ on $W$. Let us this expression to extend $f$ to the whole of $V$: " ] }, { "cell_type": "code", "execution_count": 123, "metadata": {}, "outputs": [], "source": [ "f.add_expr_by_continuation(stereoS, W)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then $f$ is well defined in all $\\mathbb{S}^2 = U \\cup V$:" ] }, { "cell_type": "code", "execution_count": 124, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} f:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} f:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}$" ], "text/plain": [ "f: S^2 → ℝ\n", "on U: (x, y) ↦ 1/(x^2 + y^2 + 1)\n", "on V: (xp, yp) ↦ (xp^2 + yp^2)/(xp^2 + yp^2 + 1)\n", "on A: (th, ph) ↦ -1/2*cos(th) + 1/2" ] }, "execution_count": 124, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.display()" ] }, { "cell_type": "code", "execution_count": 125, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(N)" ] }, { "cell_type": "code", "execution_count": 126, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Scalar fields map the manifold's points to real numbers:

" ] }, { "cell_type": "code", "execution_count": 127, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 127, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(N)" ] }, { "cell_type": "code", "execution_count": 128, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 1\$" ], "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(S)" ] }, { "cell_type": "code", "execution_count": 129, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{1}{6}\$" ], "text/latex": [ "$\\displaystyle \\frac{1}{6}$" ], "text/plain": [ "1/6" ] }, "execution_count": 129, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We may define the restrictions of $f$ to the open subsets $U$ and $V$:

" ] }, { "cell_type": "code", "execution_count": 130, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ W : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ W : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}$" ], "text/plain": [ "f: U → ℝ\n", " (x, y) ↦ 1/(x^2 + y^2 + 1)\n", "on W: (xp, yp) ↦ (xp^2 + yp^2)/(xp^2 + yp^2 + 1)\n", "on A: (th, ph) ↦ -1/2*cos(th) + 1/2" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fU = f.restrict(U)\n", "fU.display()" ] }, { "cell_type": "code", "execution_count": 131, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} f:& V & \\longrightarrow & \\mathbb{R} \\\\ & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ W : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} f:& V & \\longrightarrow & \\mathbb{R} \\\\ & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ W : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}$" ], "text/plain": [ "f: V → ℝ\n", " (xp, yp) ↦ (xp^2 + yp^2)/(xp^2 + yp^2 + 1)\n", "on W: (x, y) ↦ 1/(x^2 + y^2 + 1)\n", "on A: (th, ph) ↦ -1/2*cos(th) + 1/2" ] }, "execution_count": 131, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fV = f.restrict(V)\n", "fV.display()" ] }, { "cell_type": "code", "execution_count": 132, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{1}{6}, 1\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{1}{6}, 1\\right)$" ], "text/plain": [ "(1/6, 1)" ] }, "execution_count": 132, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fU(p), fU(S)" ] }, { "cell_type": "code", "execution_count": 133, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(U\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(U\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the Open subset U of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 133, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fU.parent()" ] }, { "cell_type": "code", "execution_count": 134, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(V\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(V\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the Open subset V of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 134, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fV.parent()" ] }, { "cell_type": "code", "execution_count": 135, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 135, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CU = U.scalar_field_algebra()\n", "fU.parent() is CU" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

A scalar field on $\\mathbb{S}^2$ can be coerced to a scalar field on $U$, the coercion being simply the restriction:

" ] }, { "cell_type": "code", "execution_count": 136, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 136, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CU.has_coerce_map_from(CS)" ] }, { "cell_type": "code", "execution_count": 137, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 137, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fU == CU(f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The arithmetic of scalar fields (operations in the algebra $C^\\infty(\\mathbb{S}^2)$):" ] }, { "cell_type": "code", "execution_count": 138, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} g:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & -\\frac{2 \\, x^{2} + 2 \\, y^{2} + 1}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & -\\frac{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{1}{4} \\, \\cos\\left({\\theta}\\right)^{2} + \\frac{1}{2} \\, \\cos\\left({\\theta}\\right) - \\frac{3}{4} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} g:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & -\\frac{2 \\, x^{2} + 2 \\, y^{2} + 1}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & -\\frac{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{1}{4} \\, \\cos\\left({\\theta}\\right)^{2} + \\frac{1}{2} \\, \\cos\\left({\\theta}\\right) - \\frac{3}{4} \\end{array}$" ], "text/plain": [ "g: S^2 → ℝ\n", "on U: (x, y) ↦ -(2*x^2 + 2*y^2 + 1)/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)\n", "on V: (xp, yp) ↦ -(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2)/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1)\n", "on A: (th, ph) ↦ 1/4*cos(th)^2 + 1/2*cos(th) - 3/4" ] }, "execution_count": 138, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = f*f - 2*f\n", "g.set_name('g')\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vector fields\n", "\n", "The set $\\mathfrak{X}(\\mathbb{S}^2)$ of all smooth vector fields on $\\mathbb{S}^2$ is a module over the algebra $C^\\infty(\\mathbb{S}^2)$. It is obtained by the method `vector_field_module()`:" ] }, { "cell_type": "code", "execution_count": 139, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathfrak{X}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle \\mathfrak{X}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Module X(S^2) of vector fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 139, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XS = S2.vector_field_module()\n", "XS" ] }, { "cell_type": "code", "execution_count": 140, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Module X(S^2) of vector fields on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(XS)" ] }, { "cell_type": "code", "execution_count": 141, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 141, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XS.base_ring()" ] }, { "cell_type": "code", "execution_count": 142, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbf{Modules}_{C^{\\infty}\\left(\\mathbb{S}^2\\right)}\$" ], "text/latex": [ "$\\displaystyle \\mathbf{Modules}_{C^{\\infty}\\left(\\mathbb{S}^2\\right)}$" ], "text/plain": [ "Category of modules over Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 142, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XS.category()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$\\mathfrak{X}(\\mathbb{S}^2)$ is not a free module:

" ] }, { "cell_type": "code", "execution_count": 143, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{False}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 143, "metadata": {}, "output_type": "execute_result" } ], "source": [ "isinstance(XS, FiniteRankFreeModule)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

because $\\mathbb{S}^2$ is not a parallelizable manifold:

" ] }, { "cell_type": "code", "execution_count": 144, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{False}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 144, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.is_manifestly_parallelizable()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

On the contrary, the set $\\mathfrak{X}(U)$ of smooth vector fields on $U$ is a free module:

" ] }, { "cell_type": "code", "execution_count": 145, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 145, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XU = U.vector_field_module()\n", "isinstance(XU, FiniteRankFreeModule)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

because $U$ is parallelizable:

" ] }, { "cell_type": "code", "execution_count": 146, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 146, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U.is_manifestly_parallelizable()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Due to the introduction of the stereographic coordinates $(x,y)$ on $U$, a basis has already been defined on the free module $\\mathfrak{X}(U)$, namely the coordinate basis $(\\partial/\\partial x, \\partial/\\partial y)$:

" ] }, { "cell_type": "code", "execution_count": 147, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Bases defined on the Free module X(U) of vector fields on the Open subset U of the 2-dimensional differentiable manifold S^2:\n", " - (U, (∂/∂x,∂/∂y)) (default basis)\n" ] } ], "source": [ "XU.print_bases()" ] }, { "cell_type": "code", "execution_count": 148, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (U, (∂/∂x,∂/∂y))" ] }, "execution_count": 148, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eU = XU.default_basis()\n", "eU" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly

" ] }, { "cell_type": "code", "execution_count": 149, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(V, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(V, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (V, (∂/∂xp,∂/∂yp))" ] }, "execution_count": 149, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XV = V.vector_field_module()\n", "eV = XV.default_basis()\n", "eV" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the point of view of the open set $U$, `eU` is also the default vector frame:" ] }, { "cell_type": "code", "execution_count": 150, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 150, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eU is U.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly:" ] }, { "cell_type": "code", "execution_count": 151, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 151, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eV is V.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`eU` is also the default vector frame on $\\mathbb{S}^2$ (although not defined on the whole $\\mathbb{S}^2$), for it is the first vector frame defined on an open subset of $\\mathbb{S}^2$:" ] }, { "cell_type": "code", "execution_count": 152, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 152, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eU is S2.default_frame()" ] }, { "cell_type": "code", "execution_count": 153, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right), \\left(V, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right), \\left(W, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right), \\left(W, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right), \\left(A, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right), \\left(A, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right), \\left(A, \\left(\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right), \\left(V, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right), \\left(W, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right), \\left(W, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right), \\left(A, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right), \\left(A, \\left(\\frac{\\partial}{\\partial {x'} },\\frac{\\partial}{\\partial {y'} }\\right)\\right), \\left(A, \\left(\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]$" ], "text/plain": [ "[Coordinate frame (U, (∂/∂x,∂/∂y)),\n", " Coordinate frame (V, (∂/∂xp,∂/∂yp)),\n", " Coordinate frame (W, (∂/∂x,∂/∂y)),\n", " Coordinate frame (W, (∂/∂xp,∂/∂yp)),\n", " Coordinate frame (A, (∂/∂x,∂/∂y)),\n", " Coordinate frame (A, (∂/∂xp,∂/∂yp)),\n", " Coordinate frame (A, (∂/∂th,∂/∂ph))]" ] }, "execution_count": 153, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce a vector field on $\\mathbb{S}^2$ by providing its components in the\n", "frame `eU`:" ] }, { "cell_type": "code", "execution_count": 154, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = \\frac{\\partial}{\\partial x } -2 \\frac{\\partial}{\\partial y }\$" ], "text/latex": [ "$\\displaystyle v = \\frac{\\partial}{\\partial x } -2 \\frac{\\partial}{\\partial y }$" ], "text/plain": [ "v = ∂/∂x - 2 ∂/∂y" ] }, "execution_count": 154, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v = S2.vector_field(1, -2, frame=eU, name='v')\n", "v.display(eU)" ] }, { "cell_type": "code", "execution_count": 155, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathfrak{X}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle \\mathfrak{X}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Module X(S^2) of vector fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 155, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On $W$, we can express $v$ in terms of the $(x',y')$ coordinates:" ] }, { "cell_type": "code", "execution_count": 156, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = \\left( -{x'}^{2} + 4 \\, {x'} {y'} + {y'}^{2} \\right) \\frac{\\partial}{\\partial {x'} } + \\left( -2 \\, {x'}^{2} - 2 \\, {x'} {y'} + 2 \\, {y'}^{2} \\right) \\frac{\\partial}{\\partial {y'} }\$" ], "text/latex": [ "$\\displaystyle v = \\left( -{x'}^{2} + 4 \\, {x'} {y'} + {y'}^{2} \\right) \\frac{\\partial}{\\partial {x'} } + \\left( -2 \\, {x'}^{2} - 2 \\, {x'} {y'} + 2 \\, {y'}^{2} \\right) \\frac{\\partial}{\\partial {y'} }$" ], "text/plain": [ "v = (-xp^2 + 4*xp*yp + yp^2) ∂/∂xp + (-2*xp^2 - 2*xp*yp + 2*yp^2) ∂/∂yp" ] }, "execution_count": 156, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.restrict(W).display(stereoS.restrict(W).frame(), stereoS.restrict(W))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We extend the definition of $v$ to $V$ thanks to the above expression:" ] }, { "cell_type": "code", "execution_count": 157, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = \\left( -{x'}^{2} + 4 \\, {x'} {y'} + {y'}^{2} \\right) \\frac{\\partial}{\\partial {x'} } + \\left( -2 \\, {x'}^{2} - 2 \\, {x'} {y'} + 2 \\, {y'}^{2} \\right) \\frac{\\partial}{\\partial {y'} }\$" ], "text/latex": [ "$\\displaystyle v = \\left( -{x'}^{2} + 4 \\, {x'} {y'} + {y'}^{2} \\right) \\frac{\\partial}{\\partial {x'} } + \\left( -2 \\, {x'}^{2} - 2 \\, {x'} {y'} + 2 \\, {y'}^{2} \\right) \\frac{\\partial}{\\partial {y'} }$" ], "text/plain": [ "v = (-xp^2 + 4*xp*yp + yp^2) ∂/∂xp + (-2*xp^2 - 2*xp*yp + 2*yp^2) ∂/∂yp" ] }, "execution_count": 157, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.add_comp_by_continuation(eV, W, chart=stereoS)\n", "v.display(eV)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At this stage, the vector field $v$ is defined on the whole manifold $\\mathbb{S}^2$: it has expressions in each of the two frames `eU` and `eV`, which cover $\\mathbb{S}^2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "According to the hairy ball theorem, $v$ has to vanish somewhere. This occurs at the North pole:" ] }, { "cell_type": "code", "execution_count": 158, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector v at Point N on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "vN = v.at(N)\n", "print(vN)" ] }, { "cell_type": "code", "execution_count": 159, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = 0\$" ], "text/latex": [ "$\\displaystyle v = 0$" ], "text/plain": [ "v = 0" ] }, "execution_count": 159, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vN.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$v|_N$ is the zero vector of the tangent vector space $T_N\\mathbb{S}^2$:

" ] }, { "cell_type": "code", "execution_count": 160, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{N}\\,\\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle T_{N}\\,\\mathbb{S}^2$" ], "text/plain": [ "Tangent space at Point N on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 160, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vN.parent()" ] }, { "cell_type": "code", "execution_count": 161, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 161, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vN.parent() is S2.tangent_space(N)" ] }, { "cell_type": "code", "execution_count": 162, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 162, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vN == S2.tangent_space(N).zero()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

On the contrary, $v$ is non-zero at the South pole:

" ] }, { "cell_type": "code", "execution_count": 163, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field v on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "vS = v.at(S)\n", "print(v)" ] }, { "cell_type": "code", "execution_count": 164, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = \\frac{\\partial}{\\partial x } -2 \\frac{\\partial}{\\partial y }\$" ], "text/latex": [ "$\\displaystyle v = \\frac{\\partial}{\\partial x } -2 \\frac{\\partial}{\\partial y }$" ], "text/plain": [ "v = ∂/∂x - 2 ∂/∂y" ] }, "execution_count": 164, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vS.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us plot the vector field $v$ is terms of the stereographic chart $(U,(x,y))$, with the South pole $S$ superposed:

" ] }, { "cell_type": "code", "execution_count": 165, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 27 graphics primitives" ] }, "execution_count": 165, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.plot(chart=stereoN, chart_domain=stereoN, max_range=4, \n", " number_values=5, scale=0.5, aspect_ratio=1) \\\n", "+ S.plot(stereoN, size=30, label_offset=0.2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The vector field appears homogeneous because its components w.r.t. the frame $\\left(\\frac{\\partial}{\\partial x}, \\frac{\\partial}{\\partial y}\\right)$ are constant:

" ] }, { "cell_type": "code", "execution_count": 166, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v = \\frac{\\partial}{\\partial x } -2 \\frac{\\partial}{\\partial y }\$" ], "text/latex": [ "$\\displaystyle v = \\frac{\\partial}{\\partial x } -2 \\frac{\\partial}{\\partial y }$" ], "text/plain": [ "v = ∂/∂x - 2 ∂/∂y" ] }, "execution_count": 166, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display(stereoN.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

On the contrary, once drawn in terms of the stereographic chart $(V, (x',y'))$, $v$ does no longer appears homogeneous:

" ] }, { "cell_type": "code", "execution_count": 167, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 82 graphics primitives" ] }, "execution_count": 167, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.plot(chart=stereoS, chart_domain=stereoS, max_range=4, scale=0.02, \n", " aspect_ratio=1) \\\n", "+ N.plot(chart=stereoS, size=30, label_offset=0.2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Finally, a 3D view of the vector field $v$ is obtained via the embedding $\\Phi$:

" ] }, { "cell_type": "code", "execution_count": 168, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 168, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_v = v.plot(chart=cartesian, mapping=Phi, chart_domain=spher, \n", " number_values=11, scale=0.2)\n", "graph_spher + graph_v" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly, let us draw the first vector field of the stereographic frame from the North pole, namely $\\frac{\\partial}{\\partial x}$:

" ] }, { "cell_type": "code", "execution_count": 169, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (U, (∂/∂x,∂/∂y))" ] }, "execution_count": 169, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN.frame()" ] }, { "cell_type": "code", "execution_count": 170, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial x }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial x }$" ], "text/plain": [ "Vector field ∂/∂x on the Open subset U of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 170, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ex = stereoN.frame()[1]\n", "ex" ] }, { "cell_type": "code", "execution_count": 171, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 171, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_ex = ex.plot(chart=cartesian, mapping=Phi, chart_domain=spher,\n", " number_values=11, scale=0.4, width=1, \n", " label_axes=False)\n", "graph_spher + graph_ex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

For the second vector field of the stereographic frame from the North pole, namely $\\frac{\\partial}{\\partial y}$, we get

" ] }, { "cell_type": "code", "execution_count": 172, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial y }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial y }$" ], "text/plain": [ "Vector field ∂/∂y on the Open subset U of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 172, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ey = stereoN.frame()[2]\n", "ey" ] }, { "cell_type": "code", "execution_count": 173, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 173, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_ey = ey.plot(chart=cartesian, mapping=Phi, chart_domain=spher,\n", " number_values=11, scale=0.4, width=1, color='red', \n", " label_axes=False)\n", "graph_spher + graph_ey" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may combine the two graphs, to get a 3D view of the vector frame associated with the stereographic coordinates from the North pole:" ] }, { "cell_type": "code", "execution_count": 174, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 174, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_frame = graph_spher + graph_ex + graph_ey \\\n", " + N.plot(cartesian, mapping=Phi, label_offset=0.05, size=5) \\\n", " + S.plot(cartesian, mapping=Phi, label_offset=0.05, size=5)\n", "graph_frame + sphere(color='lightgrey', opacity=0.4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The same scene rendered with Tachyon:" ] }, { "cell_type": "code", "execution_count": 175, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(graph_frame + sphere(opacity=0.5), viewer='tachyon', figsize=10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vector fields acting on scalar fields\n", "\n", "$v$ and $f$ are both fields defined on the whole sphere (respectively a vector field and a scalar field). By the very definition of a vector field, $v$ acts on $f$:" ] }, { "cell_type": "code", "execution_count": 176, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field v(f) on the 2-dimensional differentiable manifold S^2\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} v\\left(f\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & -\\frac{2 \\, {\\left({x'}^{3} - 2 \\, {x'}^{2} {y'} + {x'} {y'}^{2} - 2 \\, {y'}^{3}\\right)}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{1}{2} \\, {\\left({\\left(\\cos\\left({\\phi}\\right) - 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right) + 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} v\\left(f\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & -\\frac{2 \\, {\\left({x'}^{3} - 2 \\, {x'}^{2} {y'} + {x'} {y'}^{2} - 2 \\, {y'}^{3}\\right)}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{1}{2} \\, {\\left({\\left(\\cos\\left({\\phi}\\right) - 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right) + 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\end{array}$" ], "text/plain": [ "v(f): S^2 → ℝ\n", "on U: (x, y) ↦ -2*(x - 2*y)/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)\n", "on V: (xp, yp) ↦ -2*(xp^3 - 2*xp^2*yp + xp*yp^2 - 2*yp^3)/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1)\n", "on A: (th, ph) ↦ 1/2*((cos(ph) - 2*sin(ph))*cos(th) - cos(ph) + 2*sin(ph))*sin(th)" ] }, "execution_count": 176, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vf = v(f)\n", "print(vf)\n", "vf.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Values of $v(f)$ at the North pole, at the South pole and at point $p$:" ] }, { "cell_type": "code", "execution_count": 177, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 177, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vf(N)" ] }, { "cell_type": "code", "execution_count": 178, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 178, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vf(S)" ] }, { "cell_type": "code", "execution_count": 179, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{1}{6}\$" ], "text/latex": [ "$\\displaystyle \\frac{1}{6}$" ], "text/plain": [ "1/6" ] }, "execution_count": 179, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vf(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1-forms\n", "\n", "A 1-form on $\\mathbb{S}^2$ is a field of linear forms on the tangent spaces. For instance it can be the differential of a scalar field:" ] }, { "cell_type": "code", "execution_count": 180, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} f:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} f:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{1}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{{x'}^{2} + {y'}^{2}}{{x'}^{2} + {y'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{1}{2} \\, \\cos\\left({\\theta}\\right) + \\frac{1}{2} \\end{array}$" ], "text/plain": [ "f: S^2 → ℝ\n", "on U: (x, y) ↦ 1/(x^2 + y^2 + 1)\n", "on V: (xp, yp) ↦ (xp^2 + yp^2)/(xp^2 + yp^2 + 1)\n", "on A: (th, ph) ↦ -1/2*cos(th) + 1/2" ] }, "execution_count": 180, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.display()" ] }, { "cell_type": "code", "execution_count": 181, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form df on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "df = diff(f)\n", "print(df)" ] }, { "cell_type": "code", "execution_count": 182, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}f = \\left( -\\frac{2 \\, x}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x + \\left( -\\frac{2 \\, y}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} y\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\left( -\\frac{2 \\, x}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x + \\left( -\\frac{2 \\, y}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} y$" ], "text/plain": [ "df = -2*x/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx - 2*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy" ] }, "execution_count": 182, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.display() # display w.r.t. the default frame" ] }, { "cell_type": "code", "execution_count": 183, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}f = \\left( \\frac{2 \\, {x'}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\right) \\mathrm{d} {x'} + \\left( \\frac{2 \\, {y'}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\right) \\mathrm{d} {y'}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\left( \\frac{2 \\, {x'}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\right) \\mathrm{d} {x'} + \\left( \\frac{2 \\, {y'}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\right) \\mathrm{d} {y'}$" ], "text/plain": [ "df = 2*xp/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1) dxp + 2*yp/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1) dyp" ] }, "execution_count": 183, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.display(eV)" ] }, { "cell_type": "code", "execution_count": 184, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}f = \\left( \\frac{\\sqrt{x^{2} + y^{2}}}{x^{2} + y^{2} + 1} \\right) \\mathrm{d} {\\theta}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\left( \\frac{\\sqrt{x^{2} + y^{2}}}{x^{2} + y^{2} + 1} \\right) \\mathrm{d} {\\theta}$" ], "text/plain": [ "df = sqrt(x^2 + y^2)/(x^2 + y^2 + 1) dth" ] }, "execution_count": 184, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.display(spher.frame())" ] }, { "cell_type": "code", "execution_count": 185, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}f = \\frac{1}{2} \\, \\sin\\left({\\theta}\\right) \\mathrm{d} {\\theta}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\frac{1}{2} \\, \\sin\\left({\\theta}\\right) \\mathrm{d} {\\theta}$" ], "text/plain": [ "df = 1/2*sin(th) dth" ] }, "execution_count": 185, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.display(spher.frame(), spher) # asking for the components to be shown in the spherical chart" ] }, { "cell_type": "code", "execution_count": 186, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Module Omega^1(S^2) of 1-forms on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(df.parent())" ] }, { "cell_type": "code", "execution_count": 187, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\Omega^{1}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle \\Omega^{1}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Module Omega^1(S^2) of 1-forms on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 187, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The 1-form acting on a vector field:

" ] }, { "cell_type": "code", "execution_count": 188, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field df(v) on the 2-dimensional differentiable manifold S^2\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\mathrm{d}f\\left(v\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & -\\frac{2 \\, {\\left({x'}^{3} - 2 \\, {x'}^{2} {y'} + {x'} {y'}^{2} - 2 \\, {y'}^{3}\\right)}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{1}{2} \\, {\\left({\\left(\\cos\\left({\\phi}\\right) - 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right) + 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\mathrm{d}f\\left(v\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & -\\frac{2 \\, {\\left({x'}^{3} - 2 \\, {x'}^{2} {y'} + {x'} {y'}^{2} - 2 \\, {y'}^{3}\\right)}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{1}{2} \\, {\\left({\\left(\\cos\\left({\\phi}\\right) - 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right) + 2 \\, \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\end{array}$" ], "text/plain": [ "df(v): S^2 → ℝ\n", "on U: (x, y) ↦ -2*(x - 2*y)/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)\n", "on V: (xp, yp) ↦ -2*(xp^3 - 2*xp^2*yp + xp*yp^2 - 2*yp^3)/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1)\n", "on A: (th, ph) ↦ 1/2*((cos(ph) - 2*sin(ph))*cos(th) - cos(ph) + 2*sin(ph))*sin(th)" ] }, "execution_count": 188, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(df(v))\n", "df(v).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us check the identity $\\mathrm{d}f(v) = v(f)$:

" ] }, { "cell_type": "code", "execution_count": 189, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 189, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df(v) == v(f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly, we have $\\mathcal{L}_v f = v(f)$:

" ] }, { "cell_type": "code", "execution_count": 190, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 190, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.lie_derivative(v) == v(f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Curves in $\\mathbb{S}^2$\n", "\n", "In order to define curves in $\\mathbb{S}^2$, we first introduce the field of real numbers $\\mathbb{R}$ as a 1-dimensional smooth manifold with a canonical coordinate chart:" ] }, { "cell_type": "code", "execution_count": 191, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Real number line ℝ\n" ] } ], "source": [ "R. = manifolds.RealLine()\n", "print(R)" ] }, { "cell_type": "code", "execution_count": 192, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Category of smooth connected manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(R.category())" ] }, { "cell_type": "code", "execution_count": 193, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 1\$" ], "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 193, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dim(R)" ] }, { "cell_type": "code", "execution_count": 194, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\Bold{R},(t)\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\Bold{R},(t)\\right)\\right]$" ], "text/plain": [ "[Chart (ℝ, (t,))]" ] }, "execution_count": 194, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us define a loxodrome of the sphere in terms of its parametric equation with respect to the chart spher = $(A,(\\theta,\\phi))$

" ] }, { "cell_type": "code", "execution_count": 195, "metadata": {}, "outputs": [], "source": [ "c = S2.curve({spher: [2*atan(exp(-t/10)), t]}, (t, -oo, +oo), name='c')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Curves in $\\mathbb{S}^2$ are considered as morphisms from the manifold $\\mathbb{R}$ to the manifold $\\mathbb{S}^2$:

" ] }, { "cell_type": "code", "execution_count": 196, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Hom}\\left(\\Bold{R},\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Hom}\\left(\\Bold{R},\\mathbb{S}^2\\right)$" ], "text/plain": [ "Set of Morphisms from Real number line ℝ to 2-dimensional differentiable manifold S^2 in Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 196, "metadata": {}, "output_type": "execute_result" } ], "source": [ "c.parent()" ] }, { "cell_type": "code", "execution_count": 197, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} c:& \\Bold{R} & \\longrightarrow & \\mathbb{S}^2 \\\\ & t & \\longmapsto & \\left(x, y\\right) = \\left(\\cos\\left(t\\right) e^{\\left(\\frac{1}{10} \\, t\\right)}, e^{\\left(\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right)\\right) \\\\ & t & \\longmapsto & \\left({x'}, {y'}\\right) = \\left(\\cos\\left(t\\right) e^{\\left(-\\frac{1}{10} \\, t\\right)}, e^{\\left(-\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right)\\right) \\\\ & t & \\longmapsto & \\left({\\theta}, {\\phi}\\right) = \\left(2 \\, \\arctan\\left(e^{\\left(-\\frac{1}{10} \\, t\\right)}\\right), t\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} c:& \\Bold{R} & \\longrightarrow & \\mathbb{S}^2 \\\\ & t & \\longmapsto & \\left(x, y\\right) = \\left(\\cos\\left(t\\right) e^{\\left(\\frac{1}{10} \\, t\\right)}, e^{\\left(\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right)\\right) \\\\ & t & \\longmapsto & \\left({x'}, {y'}\\right) = \\left(\\cos\\left(t\\right) e^{\\left(-\\frac{1}{10} \\, t\\right)}, e^{\\left(-\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right)\\right) \\\\ & t & \\longmapsto & \\left({\\theta}, {\\phi}\\right) = \\left(2 \\, \\arctan\\left(e^{\\left(-\\frac{1}{10} \\, t\\right)}\\right), t\\right) \\end{array}$" ], "text/plain": [ "c: ℝ → S^2\n", " t ↦ (x, y) = (cos(t)*e^(1/10*t), e^(1/10*t)*sin(t))\n", " t ↦ (xp, yp) = (cos(t)*e^(-1/10*t), e^(-1/10*t)*sin(t))\n", " t ↦ (th, ph) = (2*arctan(e^(-1/10*t)), t)" ] }, "execution_count": 197, "metadata": {}, "output_type": "execute_result" } ], "source": [ "c.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The curve $c$ can be plotted in terms of stereographic coordinates $(x,y)$:

" ] }, { "cell_type": "code", "execution_count": 198, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 198, "metadata": {}, "output_type": "execute_result" } ], "source": [ "c.plot(chart=stereoN, aspect_ratio=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We recover the well-known fact that the graph of a loxodrome in terms of stereographic coordinates is a logarithmic spiral.

\n", "

Thanks to the embedding $\\Phi$, we may also plot $c$ in terms of the Cartesian coordinates of $\\mathbb{R}^3$:

" ] }, { "cell_type": "code", "execution_count": 199, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 199, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_c = c.plot(mapping=Phi, max_range=40, plot_points=200, \n", " thickness=2, label_axes=False)\n", "graph_spher + graph_c" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The tangent vector field (or velocity vector) to the curve $c$ is

" ] }, { "cell_type": "code", "execution_count": 200, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {c'}\$" ], "text/latex": [ "$\\displaystyle {c'}$" ], "text/plain": [ "Vector field c' along the Real number line ℝ with values on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 200, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vc = c.tangent_vector_field()\n", "vc" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$c'$ is a vector field along $\\mathbb{R}$ taking its values in tangent spaces to $\\mathbb{S}^2$:

" ] }, { "cell_type": "code", "execution_count": 201, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field c' along the Real number line ℝ with values on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(vc)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The set of vector fields along $\\mathbb{R}$ taking their values on $\\mathbb{S}^2$ via the differential mapping $c: \\mathbb{R} \\rightarrow \\mathbb{S}^2$ is denoted by $\\mathfrak{X}(\\mathbb{R},c)$; it is a module over the algebra $C^\\infty(\\mathbb{R})$:

" ] }, { "cell_type": "code", "execution_count": 202, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathfrak{X}\\left(\\Bold{R},c\\right)\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathfrak{X}\\left(\\Bold{R},c\\right)$" ], "text/plain": [ "Module X(ℝ,c) of vector fields along the Real number line ℝ mapped into the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 202, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vc.parent()" ] }, { "cell_type": "code", "execution_count": 203, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Modules}_{C^{\\infty}\\left(\\Bold{R}\\right)}\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Modules}_{C^{\\infty}\\left(\\Bold{R}\\right)}$" ], "text/plain": [ "Category of modules over Algebra of differentiable scalar fields on the Real number line ℝ" ] }, "execution_count": 203, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vc.parent().category()" ] }, { "cell_type": "code", "execution_count": 204, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}C^{\\infty}\\left(\\Bold{R}\\right)\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}C^{\\infty}\\left(\\Bold{R}\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the Real number line ℝ" ] }, "execution_count": 204, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vc.parent().base_ring()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

A coordinate view of $c'$:

" ] }, { "cell_type": "code", "execution_count": 205, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {c'} = \\left( \\frac{1}{10} \\, \\cos\\left(t\\right) e^{\\left(\\frac{1}{10} \\, t\\right)} - e^{\\left(\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right) \\right) \\frac{\\partial}{\\partial x } + \\left( \\cos\\left(t\\right) e^{\\left(\\frac{1}{10} \\, t\\right)} + \\frac{1}{10} \\, e^{\\left(\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right) \\right) \\frac{\\partial}{\\partial y }\$" ], "text/latex": [ "$\\displaystyle {c'} = \\left( \\frac{1}{10} \\, \\cos\\left(t\\right) e^{\\left(\\frac{1}{10} \\, t\\right)} - e^{\\left(\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right) \\right) \\frac{\\partial}{\\partial x } + \\left( \\cos\\left(t\\right) e^{\\left(\\frac{1}{10} \\, t\\right)} + \\frac{1}{10} \\, e^{\\left(\\frac{1}{10} \\, t\\right)} \\sin\\left(t\\right) \\right) \\frac{\\partial}{\\partial y }$" ], "text/plain": [ "c' = (1/10*cos(t)*e^(1/10*t) - e^(1/10*t)*sin(t)) ∂/∂x + (cos(t)*e^(1/10*t) + 1/10*e^(1/10*t)*sin(t)) ∂/∂y" ] }, "execution_count": 205, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vc.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us plot the vector field $c'$ in terms of the stereographic chart $(U,(x,y))$:

" ] }, { "cell_type": "code", "execution_count": 206, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 31 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(vc.plot(chart=stereoN, number_values=30, scale=0.5, color='red') +\n", " c.plot(chart=stereoN), aspect_ratio=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

A 3D view of $c'$ is obtained via the embedding $\\Phi$:

" ] }, { "cell_type": "code", "execution_count": 207, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 207, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_vc = vc.plot(chart=cartesian, mapping=Phi, ranges={t: (-20, 20)}, \n", " number_values=30, scale=0.5, color='red', \n", " label_axes=False)\n", "graph_spher + graph_c + graph_vc" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Riemannian metric on $\\mathbb{S}^2$\n", "\n", "The standard metric on $\\mathbb{S}^2$ is that induced by the Euclidean metric of $\\mathbb{R}^3$. The latter\n", "is obtained by:" ] }, { "cell_type": "code", "execution_count": 208, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = \\mathrm{d} X\\otimes \\mathrm{d} X+\\mathrm{d} Y\\otimes \\mathrm{d} Y+\\mathrm{d} Z\\otimes \\mathrm{d} Z\$" ], "text/latex": [ "$\\displaystyle h = \\mathrm{d} X\\otimes \\mathrm{d} X+\\mathrm{d} Y\\otimes \\mathrm{d} Y+\\mathrm{d} Z\\otimes \\mathrm{d} Z$" ], "text/plain": [ "h = dX⊗dX + dY⊗dY + dZ⊗dZ" ] }, "execution_count": 208, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h = R3.metric()\n", "h.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The metric $g$ on $\\mathbb{S}^2$ is the pullback of $h$ by the embedding $\\Phi$:" ] }, { "cell_type": "code", "execution_count": 209, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Riemannian metric g on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "g = S2.metric('g')\n", "g.set( Phi.pullback(h) )\n", "print(g)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that we could have defined $g$ intrinsically, i.e. by providing its components in the two frames `stereoN` and `stereoS`. Instead, we have chosen to get it as the pullback of $h$, since the pullback computation is implemented in SageMath.\n", "\n", "The metric is a symmetric tensor field of type (0,2):" ] }, { "cell_type": "code", "execution_count": 210, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Module T^(0,2)(S^2) of type-(0,2) tensors fields on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(g.parent())" ] }, { "cell_type": "code", "execution_count": 211, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 2\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 2\\right)$" ], "text/plain": [ "(0, 2)" ] }, "execution_count": 211, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.tensor_type()" ] }, { "cell_type": "code", "execution_count": 212, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "symmetry: (0, 1); no antisymmetry\n" ] } ], "source": [ "g.symmetries()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric in terms of the default frame on $\\mathbb{S}^2$ (stereoN):" ] }, { "cell_type": "code", "execution_count": 213, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y\$" ], "text/latex": [ "$\\displaystyle g = \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y$" ], "text/plain": [ "g = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx⊗dx + 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy⊗dy" ] }, "execution_count": 213, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may factorize the metric components:" ] }, { "cell_type": "code", "execution_count": 214, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y\$" ], "text/latex": [ "$\\displaystyle g = \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y$" ], "text/plain": [ "g = 4/(x^2 + y^2 + 1)^2 dx⊗dx + 4/(x^2 + y^2 + 1)^2 dy⊗dy" ] }, "execution_count": 214, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.apply_map(factor, frame=eU, keep_other_components=True)\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A matrix view of the components of $g$ in the manifold's default frame:" ] }, { "cell_type": "code", "execution_count": 215, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rr}\n", "\\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}} & 0 \\\\\n", "0 & \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}}\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rr}\n", "\\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}} & 0 \\\\\n", "0 & \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}}\n", "\\end{array}\\right)$" ], "text/plain": [ "[4/(x^2 + y^2 + 1)^2 0]\n", "[ 0 4/(x^2 + y^2 + 1)^2]" ] }, "execution_count": 215, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "code", "execution_count": 216, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}}\$" ], "text/latex": [ "$\\displaystyle \\frac{4}{{\\left(x^{2} + y^{2} + 1\\right)}^{2}}$" ], "text/plain": [ "4/(x^2 + y^2 + 1)^2" ] }, "execution_count": 216, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[1,1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Display in terms of the vector frame `eV` = $(V, (\\partial_{x'}, \\partial_{y'}))$:" ] }, { "cell_type": "code", "execution_count": 217, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\frac{4}{{\\left({x'}^{2} + {y'}^{2} + 1\\right)}^{2}} \\mathrm{d} {x'}\\otimes \\mathrm{d} {x'} + \\frac{4}{{\\left({x'}^{2} + {y'}^{2} + 1\\right)}^{2}} \\mathrm{d} {y'}\\otimes \\mathrm{d} {y'}\$" ], "text/latex": [ "$\\displaystyle g = \\frac{4}{{\\left({x'}^{2} + {y'}^{2} + 1\\right)}^{2}} \\mathrm{d} {x'}\\otimes \\mathrm{d} {x'} + \\frac{4}{{\\left({x'}^{2} + {y'}^{2} + 1\\right)}^{2}} \\mathrm{d} {y'}\\otimes \\mathrm{d} {y'}$" ], "text/plain": [ "g = 4/(xp^2 + yp^2 + 1)^2 dxp⊗dxp + 4/(xp^2 + yp^2 + 1)^2 dyp⊗dyp" ] }, "execution_count": 217, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.apply_map(factor, frame=eV, keep_other_components=True)\n", "g.display(eV)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Expression of the metric tensor in terms of spherical coordinates:" ] }, { "cell_type": "code", "execution_count": 218, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = dth⊗dth + sin(th)^2 dph⊗dph" ] }, "execution_count": 218, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(spher.frame(), chart=spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The metric acts on vector field pairs, resulting in a scalar field:" ] }, { "cell_type": "code", "execution_count": 219, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(v,v) on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(g(v,v))" ] }, { "cell_type": "code", "execution_count": 220, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(\\mathbb{S}^2\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 220, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(v,v).parent()" ] }, { "cell_type": "code", "execution_count": 221, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} g\\left(v,v\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{20}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{20 \\, {\\left({x'}^{4} + 2 \\, {x'}^{2} {y'}^{2} + {y'}^{4}\\right)}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 5 \\, \\cos\\left({\\theta}\\right)^{2} - 10 \\, \\cos\\left({\\theta}\\right) + 5 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} g\\left(v,v\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{20}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{20 \\, {\\left({x'}^{4} + 2 \\, {x'}^{2} {y'}^{2} + {y'}^{4}\\right)}}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 5 \\, \\cos\\left({\\theta}\\right)^{2} - 10 \\, \\cos\\left({\\theta}\\right) + 5 \\end{array}$" ], "text/plain": [ "g(v,v): S^2 → ℝ\n", "on U: (x, y) ↦ 20/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)\n", "on V: (xp, yp) ↦ 20*(xp^4 + 2*xp^2*yp^2 + yp^4)/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1)\n", "on A: (th, ph) ↦ 5*cos(th)^2 - 10*cos(th) + 5" ] }, "execution_count": 221, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(v,v).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Levi-Civita connection** associated with the metric $g$:" ] }, { "cell_type": "code", "execution_count": 222, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Riemannian metric g on the 2-dimensional differentiable manifold S^2\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\nabla_{g}\$" ], "text/latex": [ "$\\displaystyle \\nabla_{g}$" ], "text/plain": [ "Levi-Civita connection nabla_g associated with the Riemannian metric g on the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 222, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla = g.connection()\n", "print(nabla)\n", "nabla" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a test, we verify that $\\nabla_g$ acting on $g$ results in zero:" ] }, { "cell_type": "code", "execution_count": 223, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\nabla_{g} g = 0\$" ], "text/latex": [ "$\\displaystyle \\nabla_{g} g = 0$" ], "text/plain": [ "nabla_g(g) = 0" ] }, "execution_count": 223, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla(g).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The nonzero Christoffel symbols of $g$ (skipping those that can be deduced by symmetry on the last two indices) w.r.t. two charts:" ] }, { "cell_type": "code", "execution_count": 224, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, x} \\, x \\, x }^{ \\, x \\phantom{\\, x} \\phantom{\\, x} } & = & -\\frac{2 \\, x}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, x \\, y }^{ \\, x \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{2 \\, y}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, y \\, y }^{ \\, x \\phantom{\\, y} \\phantom{\\, y} } & = & \\frac{2 \\, x}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} } & = & \\frac{2 \\, y}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, y }^{ \\, y \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{2 \\, x}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & -\\frac{2 \\, y}{x^{2} + y^{2} + 1} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, x} \\, x \\, x }^{ \\, x \\phantom{\\, x} \\phantom{\\, x} } & = & -\\frac{2 \\, x}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, x \\, y }^{ \\, x \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{2 \\, y}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, y \\, y }^{ \\, x \\phantom{\\, y} \\phantom{\\, y} } & = & \\frac{2 \\, x}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} } & = & \\frac{2 \\, y}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, y }^{ \\, y \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{2 \\, x}{x^{2} + y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & -\\frac{2 \\, y}{x^{2} + y^{2} + 1} \\end{array}$" ], "text/plain": [ "Gam^x_xx = -2*x/(x^2 + y^2 + 1) \n", "Gam^x_xy = -2*y/(x^2 + y^2 + 1) \n", "Gam^x_yy = 2*x/(x^2 + y^2 + 1) \n", "Gam^y_xx = 2*y/(x^2 + y^2 + 1) \n", "Gam^y_xy = -2*x/(x^2 + y^2 + 1) \n", "Gam^y_yy = -2*y/(x^2 + y^2 + 1) " ] }, "execution_count": 224, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display(chart=stereoN)" ] }, { "cell_type": "code", "execution_count": 225, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$" ], "text/plain": [ "Gam^th_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_th,ph = cos(th)/sin(th) " ] }, "execution_count": 225, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display(chart=spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\nabla_g$ acting on the vector field $v$:" ] }, { "cell_type": "code", "execution_count": 226, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(v) of type (1,1) on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(nabla(v))" ] }, { "cell_type": "code", "execution_count": 227, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\nabla_{g} v = \\left( -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} x + \\left( -\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y + \\left( \\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x + \\left( -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} y\$" ], "text/latex": [ "$\\displaystyle \\nabla_{g} v = \\left( -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} x + \\left( -\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y + \\left( \\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x + \\left( -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} y$" ], "text/plain": [ "nabla_g(v) = -2*(x - 2*y)/(x^2 + y^2 + 1) ∂/∂x⊗dx - 2*(2*x + y)/(x^2 + y^2 + 1) ∂/∂x⊗dy + 2*(2*x + y)/(x^2 + y^2 + 1) ∂/∂y⊗dx - 2*(x - 2*y)/(x^2 + y^2 + 1) ∂/∂y⊗dy" ] }, "execution_count": 227, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla(v).display(stereoN.frame())" ] }, { "cell_type": "code", "execution_count": 228, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rr}\n", "-\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1} & -\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} \\\\\n", "\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} & -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1}\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rr}\n", "-\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1} & -\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} \\\\\n", "\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1} & -\\frac{2 \\, {\\left(x - 2 \\, y\\right)}}{x^{2} + y^{2} + 1}\n", "\\end{array}\\right)$" ], "text/plain": [ "[-2*(x - 2*y)/(x^2 + y^2 + 1) -2*(2*x + y)/(x^2 + y^2 + 1)]\n", "[ 2*(2*x + y)/(x^2 + y^2 + 1) -2*(x - 2*y)/(x^2 + y^2 + 1)]" ] }, "execution_count": 228, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla(v)[:]" ] }, { "cell_type": "code", "execution_count": 229, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle -\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1}\$" ], "text/latex": [ "$\\displaystyle -\\frac{2 \\, {\\left(2 \\, x + y\\right)}}{x^{2} + y^{2} + 1}$" ], "text/plain": [ "-2*(2*x + y)/(x^2 + y^2 + 1)" ] }, "execution_count": 229, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla(v)[1,2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Curvature\n", "\n", "The Riemann tensor associated with the metric $g$:" ] }, { "cell_type": "code", "execution_count": 230, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold S^2\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Riem}\\left(g\\right) = \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Riem}\\left(g\\right) = \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x$" ], "text/plain": [ "Riem(g) = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) ∂/∂x⊗dy⊗dx⊗dy - 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) ∂/∂x⊗dy⊗dy⊗dx - 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) ∂/∂y⊗dx⊗dx⊗dy + 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) ∂/∂y⊗dx⊗dy⊗dx" ] }, "execution_count": 230, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem = g.riemann()\n", "print(Riem)\n", "Riem.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The components of the Riemann tensor in the default frame on $\\mathbb{S}^2$:" ] }, { "cell_type": "code", "execution_count": 231, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, x} \\, y \\, x \\, y }^{ \\, x \\phantom{\\, y} \\phantom{\\, x} \\phantom{\\, y} } & = & \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, x} \\, y \\, y \\, x }^{ \\, x \\phantom{\\, y} \\phantom{\\, y} \\phantom{\\, x} } & = & -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, y} \\, x \\, x \\, y }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, y} \\, x \\, y \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, y} \\phantom{\\, x} } & = & \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, x} \\, y \\, x \\, y }^{ \\, x \\phantom{\\, y} \\phantom{\\, x} \\phantom{\\, y} } & = & \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, x} \\, y \\, y \\, x }^{ \\, x \\phantom{\\, y} \\phantom{\\, y} \\phantom{\\, x} } & = & -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, y} \\, x \\, x \\, y }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, y} \\, x \\, y \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, y} \\phantom{\\, x} } & = & \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\end{array}$" ], "text/plain": [ "Riem(g)^x_yxy = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) \n", "Riem(g)^x_yyx = -4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) \n", "Riem(g)^y_xxy = -4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) \n", "Riem(g)^y_xyx = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) " ] }, "execution_count": 231, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The components in the frame associated with spherical coordinates:

" ] }, { "cell_type": "code", "execution_count": 232, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\sin\\left({\\theta}\\right)^{2} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -1 \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & 1 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\sin\\left({\\theta}\\right)^{2} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -1 \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & 1 \\end{array}$" ], "text/plain": [ "Riem(g)^th_ph,th,ph = sin(th)^2 \n", "Riem(g)^th_ph,ph,th = -sin(th)^2 \n", "Riem(g)^ph_th,th,ph = -1 \n", "Riem(g)^ph_th,ph,th = 1 " ] }, "execution_count": 232, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(spher.frame(), chart=spher)" ] }, { "cell_type": "code", "execution_count": 233, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Module T^(1,3)(S^2) of type-(1,3) tensors fields on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(Riem.parent())" ] }, { "cell_type": "code", "execution_count": 234, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "no symmetry; antisymmetry: (2, 3)\n" ] } ], "source": [ "Riem.symmetries()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Riemann tensor associated with the Euclidean metric $h$ on $\\mathbb{R}^3$ is identically zero:" ] }, { "cell_type": "code", "execution_count": 235, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Riem}\\left(h\\right) = 0\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Riem}\\left(h\\right) = 0$" ], "text/plain": [ "Riem(h) = 0" ] }, "execution_count": 235, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.riemann().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Ricci tensor and the Ricci scalar:" ] }, { "cell_type": "code", "execution_count": 236, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( \\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y$" ], "text/plain": [ "Ric(g) = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx⊗dx + 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy⊗dy" ] }, "execution_count": 236, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric = g.ricci()\n", "Ric.display()" ] }, { "cell_type": "code", "execution_count": 237, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & 2 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}$" ], "text/plain": [ "r(g): S^2 → ℝ\n", "on U: (x, y) ↦ 2\n", "on V: (xp, yp) ↦ 2\n", "on A: (th, ph) ↦ 2" ] }, "execution_count": 237, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R = g.ricci_scalar()\n", "R.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence we recover the fact that $(\\mathbb{S}^2,g)$ is a Riemannian manifold of constant positive curvature.\n", "\n", "In dimension 2, the Riemann curvature tensor is entirely determined by the Ricci scalar $R$ according to\n", "$$ R^i_{\\ \\, jlk} = \\frac{R}{2} \\left( \\delta^i_{\\ \\, k} g_{jl} - \\delta^i_{\\ \\, l} g_{jk} \\right)$$\n", "Let us check this formula here, under the form $R^i_{\\ \\, jlk} = -R g_{j[k} \\delta^i_{\\ \\, l]}$:" ] }, { "cell_type": "code", "execution_count": 238, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 238, "metadata": {}, "output_type": "execute_result" } ], "source": [ "delta = S2.tangent_identity_field()\n", "Riem == - R*(g*delta).antisymmetrize(2,3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly the relation $\\mathrm{Ric} = (R/2)\\; g$ must hold:" ] }, { "cell_type": "code", "execution_count": 239, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 239, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == (R/2)*g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Manifold orientation and volume 2-form" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In order to introduce the volume 2-form associated with the metric $g$, we need to define an orientation on $\\mathbb{S}^2$ first. We choose the orientation so that the vector frame $(\\partial/\\partial x', \\partial/\\partial y')$ of the stereographic coordinates from the South pole is right-handed. This is somewhat natural, because the triplet $(\\partial/\\partial x', \\partial/\\partial y', n)$, where $n$ is the unit outward normal to $\\mathbb{S}^2$, is right-handed with respect to the standard orientation of $\\mathbb{R}^3$. On the contrary the triplet\n", "$(\\partial/\\partial x, \\partial/\\partial y, n)$ formed from stereographic coordinates from the North pole is left-handed (see the above plot). Actually, we can check that $(\\partial/\\partial x, \\partial/\\partial y)$\n", "and $(\\partial/\\partial x', \\partial/\\partial y')$ lead to two opposite orientations, because the transition map\n", "$(x, y) \\mapsto (x', y')$ has a negative Jacobian determinant:" ] }, { "cell_type": "code", "execution_count": 240, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle -\\frac{1}{x^{4} + 2 \\, x^{2} y^{2} + y^{4}}\$" ], "text/latex": [ "$\\displaystyle -\\frac{1}{x^{4} + 2 \\, x^{2} y^{2} + y^{4}}$" ], "text/plain": [ "-1/(x^4 + 2*x^2*y^2 + y^4)" ] }, "execution_count": 240, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_to_S.jacobian_det()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the orientation via the method `set_orientation()` with a list of right-handed vector frames, whose domains form an open cover of $\\mathbb{S}^2$. We therefore provide `eV` = $(\\partial/\\partial x', \\partial/\\partial y')$ (domain: $V$) and the \"reversed\" frame $(\\partial/\\partial y, \\partial/\\partial x)$ on $U$:" ] }, { "cell_type": "code", "execution_count": 241, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(f_{1} = \\frac{\\partial}{\\partial y }, f_{2} = \\frac{\\partial}{\\partial x }\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(f_{1} = \\frac{\\partial}{\\partial y }, f_{2} = \\frac{\\partial}{\\partial x }\\right)$" ], "text/plain": [ "(f_1 = ∂/∂y, f_2 = ∂/∂x)" ] }, "execution_count": 241, "metadata": {}, "output_type": "execute_result" } ], "source": [ "reU = U.vector_frame('f', (eU[2], eU[1]))\n", "reU[1].display(eU), reU[2].display(eU)" ] }, { "cell_type": "code", "execution_count": 242, "metadata": {}, "outputs": [], "source": [ "S2.set_orientation([eV, reU])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **volume 2-form** or **Levi-Civita tensor** associated with $g$ is then" ] }, { "cell_type": "code", "execution_count": 243, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-form eps_g on the 2-dimensional differentiable manifold S^2\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\epsilon_{g} = \\left( -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\$" ], "text/latex": [ "$\\displaystyle \\epsilon_{g} = \\left( -\\frac{4}{x^{4} + y^{4} + 2 \\, {\\left(x^{2} + 1\\right)} y^{2} + 2 \\, x^{2} + 1} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$" ], "text/plain": [ "eps_g = -4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx∧dy" ] }, "execution_count": 243, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eps = g.volume_form()\n", "print(eps)\n", "eps.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice the minus sign in the the above expression, which reflects the fact that the default frame $(\\partial/\\partial x, \\partial/\\partial y)$ is left-handed. On the contrary, we have" ] }, { "cell_type": "code", "execution_count": 244, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\epsilon_{g} = \\left( \\frac{4}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\right) \\mathrm{d} {x'}\\wedge \\mathrm{d} {y'}\$" ], "text/latex": [ "$\\displaystyle \\epsilon_{g} = \\left( \\frac{4}{{x'}^{4} + {y'}^{4} + 2 \\, {\\left({x'}^{2} + 1\\right)} {y'}^{2} + 2 \\, {x'}^{2} + 1} \\right) \\mathrm{d} {x'}\\wedge \\mathrm{d} {y'}$" ], "text/plain": [ "eps_g = 4/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1) dxp∧dyp" ] }, "execution_count": 244, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eps.display(eV)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A nicer display is obtained by factorizing the components:" ] }, { "cell_type": "code", "execution_count": 245, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\epsilon_{g} = \\frac{4}{{\\left({x'}^{2} + {y'}^{2} + 1\\right)}^{2}} \\mathrm{d} {x'}\\wedge \\mathrm{d} {y'}\$" ], "text/latex": [ "$\\displaystyle \\epsilon_{g} = \\frac{4}{{\\left({x'}^{2} + {y'}^{2} + 1\\right)}^{2}} \\mathrm{d} {x'}\\wedge \\mathrm{d} {y'}$" ], "text/plain": [ "eps_g = 4/(xp^2 + yp^2 + 1)^2 dxp∧dyp" ] }, "execution_count": 245, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eps.apply_map(factor, frame=eV, keep_other_components=True)\n", "eps.display(stereoS.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The frame associated with spherical coordinates is right-handed and we recover the standard expression of the volume 2-form:" ] }, { "cell_type": "code", "execution_count": 246, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\epsilon_{g} = \\sin\\left({\\theta}\\right) \\mathrm{d} {\\theta}\\wedge \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle \\epsilon_{g} = \\sin\\left({\\theta}\\right) \\mathrm{d} {\\theta}\\wedge \\mathrm{d} {\\phi}$" ], "text/plain": [ "eps_g = sin(th) dth∧dph" ] }, "execution_count": 246, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eps.display(spher.frame(), chart=spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The exterior derivative of the 2-form $\\epsilon_g$:" ] }, { "cell_type": "code", "execution_count": 247, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3-form deps_g on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(diff(eps))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course, since $\\mathbb{S}^2$ has dimension 2, all 3-forms vanish identically:" ] }, { "cell_type": "code", "execution_count": 248, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}\\epsilon_{g} = 0\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}\\epsilon_{g} = 0$" ], "text/plain": [ "deps_g = 0" ] }, "execution_count": 248, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(eps).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Non-holonomic frames\n", "Up to know, all the vector frames introduced on $\\mathbb{S}^2$ have been coordinate frames. Let us introduce a non-coordinate frame on the open subset $A$. To ease the manipulations, we change first the default chart and default frame on $A$ to the spherical coordinate ones:" ] }, { "cell_type": "code", "execution_count": 249, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,(x, y)\\right)$" ], "text/plain": [ "Chart (A, (x, y))" ] }, "execution_count": 249, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.default_chart()" ] }, { "cell_type": "code", "execution_count": 250, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (A, (∂/∂x,∂/∂y))" ] }, "execution_count": 250, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.default_frame()" ] }, { "cell_type": "code", "execution_count": 251, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,({\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,({\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (A, (th, ph))" ] }, "execution_count": 251, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.set_default_chart(spher)\n", "A.set_default_frame(spher.frame())\n", "A.default_chart()" ] }, { "cell_type": "code", "execution_count": 252, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A, \\left(\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A, \\left(\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (A, (∂/∂th,∂/∂ph))" ] }, "execution_count": 252, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the new vector frame $e = \\left(\\frac{\\partial}{\\partial\\theta}, \\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial\\phi}\\right)$:" ] }, { "cell_type": "code", "execution_count": 253, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{\\partial}{\\partial {\\theta} }, \\frac{\\partial}{\\partial {\\phi} }\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{\\partial}{\\partial {\\theta} }, \\frac{\\partial}{\\partial {\\phi} }\\right)$" ], "text/plain": [ "(Vector field ∂/∂th on the Open subset A of the 2-dimensional differentiable manifold S^2,\n", " Vector field ∂/∂ph on the Open subset A of the 2-dimensional differentiable manifold S^2)" ] }, "execution_count": 253, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.frame()[:]" ] }, { "cell_type": "code", "execution_count": 254, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector frame (A, (e_1,e_2))\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\left(A, \\left(e_{1},e_{2}\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A, \\left(e_{1},e_{2}\\right)\\right)$" ], "text/plain": [ "Vector frame (A, (e_1,e_2))" ] }, "execution_count": 254, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d_dth, d_dph = spher.frame()[:]\n", "e = A.vector_frame('e', (d_dth, 1/sin(th)*d_dph))\n", "print(e)\n", "e" ] }, { "cell_type": "code", "execution_count": 255, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(e_{1} = \\frac{\\partial}{\\partial {\\theta} }, e_{2} = \\frac{1}{\\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(e_{1} = \\frac{\\partial}{\\partial {\\theta} }, e_{2} = \\frac{1}{\\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }\\right)$" ], "text/plain": [ "(e_1 = ∂/∂th, e_2 = 1/sin(th) ∂/∂ph)" ] }, "execution_count": 255, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(e[1].display(), e[2].display())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The new frame is an orthonormal frame for the metric $g$:

" ] }, { "cell_type": "code", "execution_count": 256, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 1\$" ], "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 256, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(e[1],e[1]).expr()" ] }, { "cell_type": "code", "execution_count": 257, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 257, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(e[1],e[2]).expr()" ] }, { "cell_type": "code", "execution_count": 258, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 1\$" ], "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 258, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(e[2],e[2]).expr()" ] }, { "cell_type": "code", "execution_count": 259, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1\n", "\\end{array}\\right)$" ], "text/plain": [ "[1 0]\n", "[0 1]" ] }, "execution_count": 259, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[e,:]" ] }, { "cell_type": "code", "execution_count": 260, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = e^{1}\\otimes e^{1}+e^{2}\\otimes e^{2}\$" ], "text/latex": [ "$\\displaystyle g = e^{1}\\otimes e^{1}+e^{2}\\otimes e^{2}$" ], "text/plain": [ "g = e^1⊗e^1 + e^2⊗e^2" ] }, "execution_count": 260, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(e)" ] }, { "cell_type": "code", "execution_count": 261, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\epsilon_{g} = e^{1}\\wedge e^{2}\$" ], "text/latex": [ "$\\displaystyle \\epsilon_{g} = e^{1}\\wedge e^{2}$" ], "text/plain": [ "eps_g = e^1∧e^2" ] }, "execution_count": 261, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eps.display(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is non-holonomic, since its structure coefficients are not identically zero:" ] }, { "cell_type": "code", "execution_count": 262, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[\\left[0, 0\\right], \\left[0, 0\\right]\\right], \\left[\\left[0, -\\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)}\\right], \\left[\\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)}, 0\\right]\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[\\left[0, 0\\right], \\left[0, 0\\right]\\right], \\left[\\left[0, -\\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)}\\right], \\left[\\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)}, 0\\right]\\right]\\right]$" ], "text/plain": [ "[[[0, 0], [0, 0]], [[0, -cos(th)/sin(th)], [cos(th)/sin(th), 0]]]" ] }, "execution_count": 262, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e.structure_coeff()[:]" ] }, { "cell_type": "code", "execution_count": 263, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle -\\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} e_{2}\$" ], "text/latex": [ "$\\displaystyle -\\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} e_{2}$" ], "text/plain": [ "-cos(th)/sin(th) e_2" ] }, "execution_count": 263, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[2].lie_derivative(e[1]).display(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

while we have of course

" ] }, { "cell_type": "code", "execution_count": 264, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[\\left[0, 0\\right], \\left[0, 0\\right]\\right], \\left[\\left[0, 0\\right], \\left[0, 0\\right]\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[\\left[0, 0\\right], \\left[0, 0\\right]\\right], \\left[\\left[0, 0\\right], \\left[0, 0\\right]\\right]\\right]$" ], "text/plain": [ "[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]" ] }, "execution_count": 264, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.frame().structure_coeff()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using SymPy as the symbolic backend\n", "\n", "By default, the symbolic backend used in calculus on manifolds is SageMath's one (Pynac + Maxima), implemented via the symbolic ring `SR`. We can choose to use [SymPy](https://www.sympy.org/) instead:" ] }, { "cell_type": "code", "execution_count": 265, "metadata": {}, "outputs": [], "source": [ "S2.set_calculus_method('sympy')" ] }, { "cell_type": "code", "execution_count": 266, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} & \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{2}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{2 \\left(xp^{2} + yp^{2}\\right)}{xp^{2} + yp^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 1 - \\cos{\\left(th \\right)} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} & \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R} \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\frac{2}{x^{2} + y^{2} + 1} \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\frac{2 \\left(xp^{2} + yp^{2}\\right)}{xp^{2} + yp^{2} + 1} \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & 1 - \\cos{\\left(th \\right)} \\end{array}$" ], "text/plain": [ "S^2 → ℝ\n", "on U: (x, y) ↦ 2/(x**2 + y**2 + 1)\n", "on V: (xp, yp) ↦ 2*(xp**2 + yp**2)/(xp**2 + yp**2 + 1)\n", "on A: (th, ph) ↦ 1 - cos(th)" ] }, "execution_count": 266, "metadata": {}, "output_type": "execute_result" } ], "source": [ "F = 2*f\n", "F.display()" ] }, { "cell_type": "code", "execution_count": 267, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|2/(x**2|\\verb| |\\verb|+|\\verb| |\\verb|y**2|\\verb| |\\verb|+|\\verb| |\\verb|1)|\$" ], "text/latex": [ "$\\displaystyle \\verb|2/(x**2|\\verb| |\\verb|+|\\verb| |\\verb|y**2|\\verb| |\\verb|+|\\verb| |\\verb|1)|$" ], "text/plain": [ "2/(x**2 + y**2 + 1)" ] }, "execution_count": 267, "metadata": {}, "output_type": "execute_result" } ], "source": [ "F.expr()" ] }, { "cell_type": "code", "execution_count": 268, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|<class|\\verb| |\\verb|'sympy.core.mul.Mul'>|\$" ], "text/latex": [ "$\\displaystyle \\verb||$" ], "text/plain": [ "" ] }, "execution_count": 268, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(F.expr())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Back to Sage's default:" ] }, { "cell_type": "code", "execution_count": 269, "metadata": {}, "outputs": [], "source": [ "S2.set_calculus_method('SR')" ] }, { "cell_type": "code", "execution_count": 270, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{2}{x^{2} + y^{2} + 1}\$" ], "text/latex": [ "$\\displaystyle \\frac{2}{x^{2} + y^{2} + 1}$" ], "text/plain": [ "2/(x^2 + y^2 + 1)" ] }, "execution_count": 270, "metadata": {}, "output_type": "execute_result" } ], "source": [ "F.expr()" ] }, { "cell_type": "code", "execution_count": 271, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|<class|\\verb| |\\verb|'sage.symbolic.expression.Expression'>|\$" ], "text/latex": [ "$\\displaystyle \\verb||$" ], "text/plain": [ "" ] }, "execution_count": 271, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(F.expr())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Going further\n", "\n", "See the notebooks [Smooth manifolds, charts and scalar fields](https://nbviewer.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/JNCF2018/jncf18_scalar.ipynb) and [Smooth manifolds, vector fields and tensor fields](https://nbviewer.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/JNCF2018/jncf18_vector.ipynb) from the lectures [Symbolic tensor calculus on manifolds](https://sagemanifolds.obspm.fr/jncf2018/). Many example notebooks are \n", "provided at the [SageManifolds page](https://sagemanifolds.obspm.fr/examples.html).\n", "\n", "See also the series of notebooks by Andrzej Chrzeszczyk: [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html), as well as the tutorial videos by Christian Bär: [Manifolds in SageMath](https://www.youtube.com/playlist?list=PLnrOCYZpQUuJlnQbQ48zgGk-Ks1t145Yw)." ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 10.4", "language": "sage", "name": "sagemath" }, "language": "python", "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.12" } }, "nbformat": 4, "nbformat_minor": 4 }