{ "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 10.8, Release Date: 2025-12-18'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sage.version.banner" ] }, { "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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", "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 10.8", "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": "3.12.3" } }, "nbformat": 4, "nbformat_minor": 4 }