{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 13. Tensor fields\n", "\n", "This notebook is part of the [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html) by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland)." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.6, Release Date: 2022-05-15'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Linear differential forms (1- forms, covector fields) on a manifold\n", "\n", "
\n", "\n", "Let $M$ be a smooth manifold.\n", "A **linear differential form or 1-form or covector field** $α$ on $M$ is a map that assigns to each $p ∈ M$ an element\n", "$α( p) ∈ T_p^∗M$ (the space $T_p^∗M$ was defined in [notebook 9](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09Manifold_LinearFormsTensors_onTp.ipynb)) . The 1-form $α( p)$ will also be denoted by $α_p$, it is \n", "smooth (of class $C^∞$ ) if for all $X ∈ \\mathfrak{X}(M)$ the function $α(X)$ defined by\\begin{equation}\n", "(α(X))( p) = α_p (X_p ),\\quad\\text{for }\\ p\\in M\n", "\\label{}\\tag{13.1}\n", "\\end{equation}\n", "is smooth (of class $C^∞$ ).\n", "\n", "The set of all smooth 1-forms on $M$ is denoted by $\\Omega^1 (M)$. The\n", "set $\\Omega^1(M)$ is a **module over** $C^∞ (M)$ with the algebraic operations given by\n", "\\begin{equation}\n", "\\begin{matrix}\n", "(α + β)_p ≡ α_p + β_p ,\\\\\n", "( f α)_p ≡ f ( p)α_p ,\n", "\\end{matrix}\n", "\\label{}\\tag{13.2}\n", "\\end{equation}\n", "for $α, β ∈ \\Omega^1 (M)$ and $f ∈ C^∞ (M).$\n", "\n", "
\n", "\n", "**Example 13.1**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us define a 1-form on a 3-dimensional manifold." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mbox{1-form on the 3-dimensional differentiable manifold M}\$" ], "text/latex": [ "$\\displaystyle \\mbox{1-form on the 3-dimensional differentiable manifold M}$" ], "text/plain": [ "1-form on the 3-dimensional differentiable manifold M" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%display latex\n", "N = 3 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "a = M.diff_form(1) # 1-form on M\n", "a # print information on a" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\Omega^{1}\\left(M\\right)\$" ], "text/latex": [ "$\\displaystyle \\Omega^{1}\\left(M\\right)$" ], "text/plain": [ "Module Omega^1(M) of 1-forms on the 3-dimensional differentiable manifold M" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# mathematical object of which \"a\" is an element.\n", "a.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Differential of scalar functions\n", "\n", "
\n", "\n", "If $f ∈ C^∞ (M)$, the **differential**\n", "of $f$ denoted by $d f$ is given by $$d f ( p) = d f_p,\\quad\\text{for}\\ p\\in M,$$ where $df_p$ is the differential at point defined in (9.1).\n", "The differential $df$ belongs to $\\Omega^1(M).$ If $X ∈ \\mathfrak{X}(M),$\n", "then from (13.1),(11.1),(9.1) it follows that\n", "\n", "$$(df(X))(p)=df_p(X_p)=X_p(f)=(Xf)(p),$$\n", "for $p ∈ M$ i.e.,\n", "\\begin{equation}\n", "df(X)=X(f).\n", "\\label{}\\tag{13.3}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.2**\n", "\n", "Define a scalar function $f$ and its differential $df$. The differential is an example of 1-form." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mbox{1-form on the 3-dimensional differentiable manifold M}\$" ], "text/latex": [ "$\\displaystyle \\mbox{1-form on the 3-dimensional differentiable manifold M}$" ], "text/plain": [ "1-form on the 3-dimensional differentiable manifold M" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# continuation\n", "f = M.scalar_field()\n", "f.differential()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### The differential of function is a derivation\n", "\n", "
\n", "\n", "The map $d : C^∞ (M) →\\Omega^1 (M)$, which sends $f$ into $df$, satisfies\n", "\\begin{equation}\n", "\\begin{matrix}\n", "d(a f + bg) = ad f + bdg,\\\\\n", "d( f g) = f dg + gd f,\n", "\\end{matrix}\n", "\\label{}\\tag{13.4}\n", "\\end{equation}\n", "for $f, g ∈ C^∞ (M)$ and $a, b ∈ R.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To check the first equality note, that for $X\\in \\mathfrak{X}(M)$\n", "\n", "$$\n", "d(a f + bg)(X) = X(a f + bg) = aX f + bXg\n", "= a\\,d f (X) + b\\,dg(X) = (a\\,d f + b\\,dg)(X).\n", "$$\n", "\n", "The second one follows from\n", "\n", "$$\n", "d( f g)(X) = X( f g) = f Xg + gX f\n", "= f dg(X) + gd f (X) = ( f dg + gd f )(X).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Differentials of coordinate functions form local bases for $\\Omega^1(M)$\n", "\n", "
\n", "\n", "If $(U, φ)$ is a chart on $M$, then as we have checked in [notebook 9](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09Manifold_LinearFormsTensors_onTp.ipynb),\n", "the differentials of the\n", "coordinate functions $x^1 , x^2 , . . . , x^n$ computed at $p\\in U$\n", "form the basis of $T_p^*M$.
\n", "The formula (9.1') implies that on $U$ we have\n", "$$\n", "dx^i(X)=X(x^i).\n", "$$\n", "\n", "Since from (8.5) we have on $U$: $X=X(x^i)\\frac{\\partial}{\\partial x^i},$ the covector field $dx^i$ assigns to the vector field $X$ its $i$-th component function $X(x^i)$ in the frame $\\{\\frac{\\partial}{\\partial x^j}\\}_{j=1}^n$.
\n", "If $α\\in\\Omega^1(M)$, using (9.2), we obtain for $p\\in M$\n", "and $α(p) ∈ T_p^∗ (M)$ \n", "\n", "$$\\alpha(p)=\\alpha(p)\\big(\\frac{\\partial}{\\partial x^i}\\Big|_p\\big)dx^i_p=\\Big[\\alpha\\big(\\frac{\\partial}{\\partial x^i}\\big) dx^i\\Big](p).$$\n", "\n", "Thus\n", "\n", "\\begin{equation}\n", "\\alpha=\\alpha\\big(\\frac{\\partial}{\\partial x^i}\\big) dx^i.\n", "\\label{}\\tag{13.5}\n", "\\end{equation}\n", "\n", "If we denote $\\alpha\\big(\\frac{\\partial}{\\partial x^i}\\big)$ by $\\alpha_i$ then in local coordinates every element of $\\Omega^1(M)$ is of the form\n", "\n", "$$\\alpha_idx^i.$$\n", "\n", "**Remark**. Let us underscore that the last expressions holds only locally (in the domain of a local chart).\n", "\n", "
\n", "\n", "### Coframe\n", "\n", "
\n", "\n", "Recall that in [notebook 11](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/11Manifold_Vect_Fields.ipynb) we defined frames.\n", "\n", "
\n", "\n", "If $U$ is an open subset of a manifold $M$ (for example a coordinate neighborhood) then the **coframe** on $U$ is the sequence $e$ of covector fields on $U$ such that for each $p∈U, \\ e(p)$ is a basis of the cotangent space $T^*_pU$.
\n", "From (13.5) it follows that the covectors $\\ dx^1,\\ldots,dx^n\\ $ define a coframe on the coordinate neighborhood $U.\\ $ Let us recall that the linear independence of $\\ dx^1_p,\\ldots,dx^n_p\\ $ for every $p\\in U$ was\n", "checked in [notebook 9](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09Manifold_LinearFormsTensors_onTp.ipynb) (cf. the argument before the formula (9.2)).\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M,({x^{0}}, {x^{1}}, {x^{2}})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M,({x^{0}}, {x^{1}}, {x^{2}})\\right)$" ], "text/plain": [ "Chart (M, (x0, x1, x2))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 3\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "X" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathrm{d} {x^{0}}, \\mathrm{d} {x^{1}}, \\mathrm{d} {x^{2}}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathrm{d} {x^{0}}, \\mathrm{d} {x^{1}}, \\mathrm{d} {x^{2}}\\right)$" ], "text/plain": [ "(1-form dx0 on the 3-dimensional differentiable manifold M,\n", " 1-form dx1 on the 3-dimensional differentiable manifold M,\n", " 1-form dx2 on the 3-dimensional differentiable manifold M)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.coframe()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Using the local representation we can give concrete examples of 1-forms/covector fields.\n", "\n", "
\n", "\n", "**Example 13.3**\n", "\n", "Define 1-form $\\ a=a_0dx^0+a_1dx^1+a_2dx^2\\ $ in SageMath." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle a = a_{0}\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{0}} + a_{1}\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{1}} + a_{2}\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{2}}\$" ], "text/latex": [ "$\\displaystyle a = a_{0}\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{0}} + a_{1}\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{1}} + a_{2}\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{2}}$" ], "text/plain": [ "a = a0(x0, x1, x2) dx0 + a1(x0, x1, x2) dx1 + a2(x0, x1, x2) dx2" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%display latex\n", "N = 3 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "a = M.diff_form(1,name='a') # 1-form a\n", "ast = ['a'+str(j) for j in range(N)] # list of component names \n", "af = [M.scalar_field(function(ast[j])(*X), name=ast[j]) \n", " for j in range(N)] # list of component functions\n", "a[:] = af # define all components of a\n", "a.disp() # show a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Function arguments can be omitted:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle a = a_{0} \\mathrm{d} {x^{0}} + a_{1} \\mathrm{d} {x^{1}} + a_{2} \\mathrm{d} {x^{2}}\$" ], "text/latex": [ "$\\displaystyle a = a_{0} \\mathrm{d} {x^{0}} + a_{1} \\mathrm{d} {x^{1}} + a_{2} \\mathrm{d} {x^{2}}$" ], "text/plain": [ "a = a0 dx0 + a1 dx1 + a2 dx2" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Manifold.options.omit_function_arguments=True\n", "a.disp() # show abbreviated a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.4**\n", "\n", "Now use the local coordinates to show the differential of a scalar function." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form df on the 3-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}f = \\frac{\\partial\\,f}{\\partial {x^{0}}} \\mathrm{d} {x^{0}} + \\frac{\\partial\\,f}{\\partial {x^{1}}} \\mathrm{d} {x^{1}} + \\frac{\\partial\\,f}{\\partial {x^{2}}} \\mathrm{d} {x^{2}}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\frac{\\partial\\,f}{\\partial {x^{0}}} \\mathrm{d} {x^{0}} + \\frac{\\partial\\,f}{\\partial {x^{1}}} \\mathrm{d} {x^{1}} + \\frac{\\partial\\,f}{\\partial {x^{2}}} \\mathrm{d} {x^{2}}$" ], "text/plain": [ "df = d(f)/dx0 dx0 + d(f)/dx1 dx1 + d(f)/dx2 dx2" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# continuation # scalar function f:\n", "f = M.scalar_field(function('f')(*X), name='f')\n", "df = f.differential() # differential of f\n", "print(df) # show information on df\n", "df.disp() # show df" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To explain the last formula let us note that by (13.5)\n", "$\\ d f = d f (\\frac{∂}{∂ x^i} )dx^i,\\ $ but, by virtue of (13.3), \n", "$\\ d f (\\frac{\\partial}{\\partial x^i})= \\frac{∂}{∂ x^i} f,\\ $\n", "so that\n", "\\begin{equation}\n", "d f = \\frac{\\partial f}{\\partial x^i}dx^i.\n", "\\label{}\\tag{13.6}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Tensor fields\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Tensor fields of type $\\mathbf{(0,k)}$ (covariant tensor fields)\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A **tensor field $t$ of type $(0,k)$** or a **covariant tensor field of rank $k$** on a manifold $M$ is a map $t$ that associates with\n", "each point $p∈M$ a tensor $t(p)=t_p\\in T^{(0,k)}_pM\\ \\ \\ $ ($T^{(0,k)}_pM$ was defined in [notebook 9](09Manifold_LinearFormsTensors_onTp.ipynb)). \n", "
\n", "If $t$\n", "is a tensor field of type $(0, k)$ and $X_1 , . . . , X_k$ are vector fields on $M$, then $t (X_1 , . . . , X_k)$ is the real-valued function given by \n", "\n", "$$[t (X_1 , . . . , X _k )]( p) = t_p (X_1 ( p), . . . , X_k ( p)).$$\n", "\n", "We say that $t$ is smooth if $t (X_1 , . . . , X_k )$ is a smooth function for all\n", "$X_1 , . . . , X_k ∈ \\mathfrak{X}(M)$.
\n", "One can prove that if $x^1,\\ldots,x^n$ are local coordinates on $U$ and \n", "$t_{i_1\\ldots i_k}=t(\\frac{\\partial}{\\partial x^{i_1}},\\ldots,\\frac{\\partial}{\\partial x^{i_k}})$, then $t$ is smooth iff for arbitrary $p\\in M$ there is a coordinate map \n", "$(U,(x^1,\\ldots,x^n))$ around $p$ such that the real functions $t_{i_1\\ldots i_k}$ are smooth. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The operations of tensor addition, scalar multiplication and tensor product are defined pointwise:\n", "\n", "$$\n", "\\begin{matrix}\n", "(at + bs)_p = at_p + bs_p,\\\\\n", "( f t)_p = f ( p)t_p,\\\\\n", "(t ⊗ s)_p = t_p ⊗ s_p,\n", "\\end{matrix}\n", "$$\n", "\n", "for $a, b ∈ R\\ $, $s, t$ -covariant tensor fields on $M$ and $f : M → R.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using these formulas and (9.5) we can check that arbitrary tensor field of type $(0,k)$ can be expressed in local coordinates as follows\n", "\n", "\\begin{equation}\n", "\\begin{matrix}\n", "t=t_{i_1\\ldots i_k}dx^{i_1}\\otimes\\dots\\otimes dx^{i_k},\\\\\n", "t_{i_1\\ldots i_k}=t\\big(\\frac{\\partial}{\\partial x^{i_1}},\\ldots,\\frac{\\partial}{\\partial x^{i_k}}\\big).\n", "\\end{matrix}\n", "\\tag{13.7}\n", "\\end{equation}\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If $X_1 , . . . , X_k$ are vector fields on $M$ and $f\\in C^\\infty(M)$, then from the multi-linearity of $t_p$ on $T_pM$ for $p\\in M$ it follows that\n", "\n", "$$[t (X_1 , . . . , f X_i , . . . , X_k )]( p) =\n", "t_p (X_1 ( p), . . . , ( f X_i )( p), . . . , X_k ( p))\\\\\n", "= t_p (X_1 ( p), . . . , f ( p)X_i ( p), . . . , X_k ( p))\\\\\n", "= f ( p)t_p( X_1 ( p), . . . , X_i ( p), . . . , X_k ( p))\\\\\n", "= f ( p)[t (X_1 , . . . , X_i , . . . , X_k )]( p),\n", "$$\n", "\n", "for $p ∈ M$. We have checked that \n", "\n", "$$t (X_1 , . . . , f X_i , . . . , X_k )=ft (X_1 , . . . , X_i , . . . , X_k ),\\quad i=1,\\ldots,k,$$\n", "\n", "for $X_i\\in \\mathfrak{X}(M)$ and $f\\in C^\\infty(M).$
\n", "\n", "Similarly for $X_i,Y_i\\in\\mathfrak{X}(M)$ we can check that for $i=1,\\ldots,k$\n", "\n", "$$t (X_1 , . . . , X_i + Y_i , . . . , X_k ) = t (X_1 , . . . , X_i , . . . , X_k ) + t (X_1 , . . . , Y_i , . . . , X_k ).$$\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Tensorial property\n", "\n", "
\n", "\n", "If $t$ is a map that to each set of $k$ vector fields $X_i\\in\\mathfrak{X}(M)\\ $ on $M$ associates a function $t(X_1,\\ldots,X_k):M\\to R$ with the property that for functions $f,g\\in C^\\infty(M),\\ $ $X_i,Y_i\\in\\mathfrak{X}(M)\\ $ and $i=1,\\ldots,k$\n", "\n", "$$t (X_1 , . . . , fX_i + gY_i , . . . , X_k ) = ft (X_1 , . . . , X_i , . . . , X_k ) +gt (X_1 , . . . , Y_i , . . . , X_k ),$$\n", "\n", "then $t$ is a tensor field of type $(0,k).$\n", "\n", "
\n", "\n", "Note that the last property means that covariant tensor fields of rank $k$ are just multilinear functions on the Cartesian product $\\underbrace{\\mathfrak{X}(M)\\times\\cdots\\times \\mathfrak{X}(M)}_{k\\ \\; \\mbox{times}}$ of $k$ copies of the module $\\mathfrak{X}(M)$ over the ring $C^\\infty(M)$.\n", "\n", "The module of covariant tensor fields of type $(0,k)\\ $ on a manifold $M$ will be denoted by $T^{(0,k)}M$. \n", "\n", "**Warning.** If $M$ is a module -not a manifold, in [notebook 9a](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09aManifold_Tensors_onModules.ipynb) the same notations denote the space of covariant tensors of rank $k$ on the module $M$.\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Covariant tensor fields in components\n", "\n", "
\n", "\n", "Using this property we can give another proof of (13.7). If the vector fields are expressed in the form $\\quad X_i = dx^j (X_i ) \\frac{∂}{∂ x^j} ,\\quad i = 1, . . . , k,$ then\n", "$$t (X_1 , . . . , X_k ) = t(dx^i(X_1 )\\frac{∂}{∂ x^i} , . . . , \n", "dx^m (X_k )\\frac{∂}{∂ x^m})\\\\\n", "= dx^i (X_1 ) · · · dx^m (X_k )\\;t( \\frac{∂}{∂ x^i},\\ldots \\frac{∂}{∂ x^m})\\\\\n", "=\\Big[t(\\frac{∂}{∂ x^i},\\ldots \\frac{∂}{∂ x^m})dx^i \\otimes · · · \\otimes dx^m\\Big] (X_1 , . . . , X_k ).$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.5**\n", "\n", "Define a general (0,2)-type tensor field on a 2-dimensional manifold (4 scalar functions as components)." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field t of type (0,2) on the 2-dimensional differentiable manifold M\n" ] } ], "source": [ "N = 2 # dimension of manifold\n", "M = Manifold(N, 'M') # manifold M of dim. N\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "t = M.tensor_field(0, 2, name='t') # tensor field of type (0,2)\n", "print(t) # show information on t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The components of the tensor field $t$ are scalar functions" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = f_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = f_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = f00 dx0⊗dx0 + f01 dx0⊗dx1 + f10 dx1⊗dx0 + f11 dx1⊗dx1" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x0, x1 = X[:] # coordinates x^0 and x^1 of chart X as the Python variables x0 and x1\n", "f00 = M.scalar_field(function('f00')(x0, x1), name='f00') # scalar functions\n", "f01 = M.scalar_field(function('f01')(x0, x1), name='f01') # defining components\n", "f10 = M.scalar_field(function('f10')(x0, x1), name='f10')\n", "f11 = M.scalar_field(function('f11')(x0, x1), name='f11')\n", "t[0,0] = f00; t[0,1] = f01 # components of tensor field t\n", "t[1,0] = f10; t[1,1] = f11 \n", "t.disp() # show t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $t_{ij}=t(\\frac{\\partial}{\\partial x_i},\\frac{\\partial}{\\partial x_j})$" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{\\partial}{\\partial {x^{0}} }, \\frac{\\partial}{\\partial {x^{1}} }\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{\\partial}{\\partial {x^{0}} }, \\frac{\\partial}{\\partial {x^{1}} }\\right)$" ], "text/plain": [ "(Vector field ∂/∂x0 on the 2-dimensional differentiable manifold M,\n", " Vector field ∂/∂x1 on the 2-dimensional differentiable manifold M)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fr = X.frame();fr[:] # local frame on M" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rr}\n", "f_{00}\\left({x^{0}}, {x^{1}}\\right) & f_{01}\\left({x^{0}}, {x^{1}}\\right) \\\\\n", "f_{10}\\left({x^{0}}, {x^{1}}\\right) & f_{11}\\left({x^{0}}, {x^{1}}\\right)\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rr}\n", "f_{00}\\left({x^{0}}, {x^{1}}\\right) & f_{01}\\left({x^{0}}, {x^{1}}\\right) \\\\\n", "f_{10}\\left({x^{0}}, {x^{1}}\\right) & f_{11}\\left({x^{0}}, {x^{1}}\\right)\n", "\\end{array}\\right)$" ], "text/plain": [ "[f00(x0, x1) f01(x0, x1)]\n", "[f10(x0, x1) f11(x0, x1)]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix(2,2,lambda i,j: t(fr[i],fr[j]).expr()) # compute t(d/dxi,d/dxj)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathrm{d} {x^{0}}, \\mathrm{d} {x^{1}}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathrm{d} {x^{0}}, \\mathrm{d} {x^{1}}\\right)$" ], "text/plain": [ "(1-form dx0 on the 2-dimensional differentiable manifold M,\n", " 1-form dx1 on the 2-dimensional differentiable manifold M)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.coframe()[:] # local coframe" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.6**\n", "\n", "Let us define a more concrete example with variable components (upper indices and powers do not mix well)." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = {x^{0}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}} {x^{1}} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = {x^{0}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}} {x^{1}} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = x0^2 dx0⊗dx0 + x0*x1 dx0⊗dx1 + x0*x1 dx1⊗dx0 + x1^2 dx1⊗dx1" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# continuation\n", "t[:] = matrix(2, 2, lambda i,j: X[i]*X[j]) # t_{ij}=x_i*x_j\n", "t.disp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.7**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An example of a (0,4)-type tensor field (`*` denotes the tensor product):" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t\\otimes t = {x^{0}}^{4} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{1}}^{4} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\)" ], "text/latex": [ "$\\displaystyle t\\otimes t = {x^{0}}^{4} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{3} {x^{1}} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}}^{2} {x^{1}}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + {x^{0}} {x^{1}}^{3} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + {x^{1}}^{4} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t⊗t = x0^4 dx0⊗dx0⊗dx0⊗dx0 + x0^3*x1 dx0⊗dx0⊗dx0⊗dx1 + x0^3*x1 dx0⊗dx0⊗dx1⊗dx0 + x0^2*x1^2 dx0⊗dx0⊗dx1⊗dx1 + x0^3*x1 dx0⊗dx1⊗dx0⊗dx0 + x0^2*x1^2 dx0⊗dx1⊗dx0⊗dx1 + x0^2*x1^2 dx0⊗dx1⊗dx1⊗dx0 + x0*x1^3 dx0⊗dx1⊗dx1⊗dx1 + x0^3*x1 dx1⊗dx0⊗dx0⊗dx0 + x0^2*x1^2 dx1⊗dx0⊗dx0⊗dx1 + x0^2*x1^2 dx1⊗dx0⊗dx1⊗dx0 + x0*x1^3 dx1⊗dx0⊗dx1⊗dx1 + x0^2*x1^2 dx1⊗dx1⊗dx0⊗dx0 + x0*x1^3 dx1⊗dx1⊗dx0⊗dx1 + x0*x1^3 dx1⊗dx1⊗dx1⊗dx0 + x1^4 dx1⊗dx1⊗dx1⊗dx1" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(t*t).disp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.8**\n", "\n", "Defining tensor fields with symbolic components we can use shorter and more general notation. \n", "\n", "Let us define a tensor field of type (0,2) in a more compact way." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = f_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = f_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = f00 dx0⊗dx0 + f01 dx0⊗dx1 + f10 dx1⊗dx0 + f11 dx1⊗dx1" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 2 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "t = M.tensor_field(0,2, name='t') # tensor field of type (0,2)\n", "def fn(i,j): return 'f'+str(i)+str(j) # names for components\n", "def fl(i,j): return 'f'+'_'+'{'+str(i)+str(j)+'}' # latex comp.names \n", "ff=[[M.scalar_field(function(fn(i,j),latex_name=fl(i,j))(*X)) \n", " for j in range(N)] for i in range(N)] # nested list of comp.\n", "t[:] = ff # define all components\n", "t.disp() # show the result" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we can see, the result is the same as previously." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The functions arguments can be omitted." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = f_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = f_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = f00 dx0⊗dx0 + f01 dx0⊗dx1 + f10 dx1⊗dx0 + f11 dx1⊗dx1" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Manifold.options.omit_function_arguments=True\n", "t.disp()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t\\otimes t = f_{00}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{00} f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{10} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{00} f_{11} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{01} f_{10} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{01} f_{11} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{10} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\)" ], "text/latex": [ "$\\displaystyle t\\otimes t = f_{00}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{00} f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{10} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{00} f_{11} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01}^{2} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{01} f_{10} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{01} f_{11} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} f_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{10} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{00} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{01} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{10} f_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{11}^{2} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t⊗t = f00^2 dx0⊗dx0⊗dx0⊗dx0 + f00*f01 dx0⊗dx0⊗dx0⊗dx1 + f00*f10 dx0⊗dx0⊗dx1⊗dx0 + f00*f11 dx0⊗dx0⊗dx1⊗dx1 + f00*f01 dx0⊗dx1⊗dx0⊗dx0 + f01^2 dx0⊗dx1⊗dx0⊗dx1 + f01*f10 dx0⊗dx1⊗dx1⊗dx0 + f01*f11 dx0⊗dx1⊗dx1⊗dx1 + f00*f10 dx1⊗dx0⊗dx0⊗dx0 + f01*f10 dx1⊗dx0⊗dx0⊗dx1 + f10^2 dx1⊗dx0⊗dx1⊗dx0 + f10*f11 dx1⊗dx0⊗dx1⊗dx1 + f00*f11 dx1⊗dx1⊗dx0⊗dx0 + f01*f11 dx1⊗dx1⊗dx0⊗dx1 + f10*f11 dx1⊗dx1⊗dx1⊗dx0 + f11^2 dx1⊗dx1⊗dx1⊗dx1" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(t*t).disp()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "Manifold.options.omit_function_arguments=False" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Remark**.\n", "If we want to make purely algebraic operations on tensor fields, then we can use symbols, not functions as components (this will not work if we want to use derivatives!)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.9**\n", "\n", "Define (0,3) type tensor field on a two dimensional manifold." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field t3 of type (0,3) on the 2-dimensional differentiable manifold M\n" ] } ], "source": [ "N = 2 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "t3 = M.tensor_field(0,3, name='t3')# tensor field of type (0,3)\n", "print(t3) # information on t3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define a three-dimensional table of symbols $a_{ijk}$" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [], "source": [ "a3 =[[[SR('a'+str(i)+str(j)+str(k)) for k in range(N)] \n", " for j in range(N)] for i in range(N)] # nested list of components" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and all components define with one equality." ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[\\left[a_{000}, a_{001}\\right], \\left[a_{010}, a_{011}\\right]\\right], \\left[\\left[a_{100}, a_{101}\\right], \\left[a_{110}, a_{111}\\right]\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[\\left[a_{000}, a_{001}\\right], \\left[a_{010}, a_{011}\\right]\\right], \\left[\\left[a_{100}, a_{101}\\right], \\left[a_{110}, a_{111}\\right]\\right]\\right]$" ], "text/plain": [ "[[[a000, a001], [a010, a011]], [[a100, a101], [a110, a111]]]" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t3[:] = a3 # define all components as symbols\n", "t3[:] # show components of t3 " ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t3 = a_{000} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + a_{001} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + a_{010} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + a_{011} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + a_{100} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + a_{101} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + a_{110} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + a_{111} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t3 = a_{000} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + a_{001} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + a_{010} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + a_{011} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + a_{100} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + a_{101} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + a_{110} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + a_{111} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t3 = a000 dx0⊗dx0⊗dx0 + a001 dx0⊗dx0⊗dx1 + a010 dx0⊗dx1⊗dx0 + a011 dx0⊗dx1⊗dx1 + a100 dx1⊗dx0⊗dx0 + a101 dx1⊗dx0⊗dx1 + a110 dx1⊗dx1⊗dx0 + a111 dx1⊗dx1⊗dx1" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t3.disp() # show the tensor" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Remember that the components in this example do not depend on coordinates. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.10**\n", "\n", "If we need a proper tensor field, with components depending on coordinates, the previous example should be modified.\n", "\n", "The first cell remains the same:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field t3 of type (0,3) on the 2-dimensional differentiable manifold M\n" ] } ], "source": [ "N = 2 # dimension of manifold\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "t3 = M.tensor_field(0,3, name='t3');print(t3) # tensor field" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "but to define components we need scalar fields: " ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t3 = f_{000}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{001}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{010}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{011}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{100}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{101}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{110}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{111}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t3 = f_{000}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{001}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{010}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{011}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{100}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{101}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{110}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{111}\\left({x^{0}}, {x^{1}}\\right) \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t3 = f000(x0, x1) dx0⊗dx0⊗dx0 + f001(x0, x1) dx0⊗dx0⊗dx1 + f010(x0, x1) dx0⊗dx1⊗dx0 + f011(x0, x1) dx0⊗dx1⊗dx1 + f100(x0, x1) dx1⊗dx0⊗dx0 + f101(x0, x1) dx1⊗dx0⊗dx1 + f110(x0, x1) dx1⊗dx1⊗dx0 + f111(x0, x1) dx1⊗dx1⊗dx1" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def fn(i,j,k): return 'f'+str(i)+str(j)+str(k) # names of components\n", "def fl(i,j,k): return 'f'+'_'+'{'+str(i)+str(j)+str(k)+'}' # latex names\n", " # nested list of component functions\n", "ff = [[[M.scalar_field(function(fn(i,j,k),latex_name=fl(i,j,k))(*X)) \n", " for k in range(N)] for j in range(N)] for i in range(N)]\n", "t3[:] = ff # define all components\n", "t3.disp() # show t3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The functions argument can be omitted:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t3 = f_{000} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{001} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{010} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{011} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{100} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{101} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{110} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{111} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t3 = f_{000} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{001} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{010} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{011} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + f_{100} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + f_{101} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + f_{110} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + f_{111} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t3 = f000 dx0⊗dx0⊗dx0 + f001 dx0⊗dx0⊗dx1 + f010 dx0⊗dx1⊗dx0 + f011 dx0⊗dx1⊗dx1 + f100 dx1⊗dx0⊗dx0 + f101 dx1⊗dx0⊗dx1 + f110 dx1⊗dx1⊗dx0 + f111 dx1⊗dx1⊗dx1" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Manifold.options.omit_function_arguments=True\n", "t3.disp() # output without component arguments" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [], "source": [ "Manifold.options.omit_function_arguments=False" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Unfortunately, (in our system) latex encounters problems if the outputs are to long.\n", "\n", "If the latex formatted output is not needed, the `%display plain` method works properly." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "t3⊗t3 = f000^2 dx0⊗dx0⊗dx0⊗dx0⊗dx0⊗dx0 + f000*f001 dx0⊗dx0⊗dx0⊗dx0⊗dx0⊗dx1 + f000*f010 dx0⊗dx0⊗dx0⊗dx0⊗dx1⊗dx0 + f000*f011 dx0⊗dx0⊗dx0⊗dx0⊗dx1⊗dx1 + f000*f100 dx0⊗dx0⊗dx0⊗dx1⊗dx0⊗dx0 + f000*f101 dx0⊗dx0⊗dx0⊗dx1⊗dx0⊗dx1 + f000*f110 dx0⊗dx0⊗dx0⊗dx1⊗dx1⊗dx0 + f000*f111 dx0⊗dx0⊗dx0⊗dx1⊗dx1⊗dx1 + f000*f001 dx0⊗dx0⊗dx1⊗dx0⊗dx0⊗dx0 + f001^2 dx0⊗dx0⊗dx1⊗dx0⊗dx0⊗dx1 + f001*f010 dx0⊗dx0⊗dx1⊗dx0⊗dx1⊗dx0 + f001*f011 dx0⊗dx0⊗dx1⊗dx0⊗dx1⊗dx1 + f001*f100 dx0⊗dx0⊗dx1⊗dx1⊗dx0⊗dx0 + f001*f101 dx0⊗dx0⊗dx1⊗dx1⊗dx0⊗dx1 + f001*f110 dx0⊗dx0⊗dx1⊗dx1⊗dx1⊗dx0 + f001*f111 dx0⊗dx0⊗dx1⊗dx1⊗dx1⊗dx1 + f000*f010 dx0⊗dx1⊗dx0⊗dx0⊗dx0⊗dx0 + f001*f010 dx0⊗dx1⊗dx0⊗dx0⊗dx0⊗dx1 + f010^2 dx0⊗dx1⊗dx0⊗dx0⊗dx1⊗dx0 + f010*f011 dx0⊗dx1⊗dx0⊗dx0⊗dx1⊗dx1 + f010*f100 dx0⊗dx1⊗dx0⊗dx1⊗dx0⊗dx0 + f010*f101 dx0⊗dx1⊗dx0⊗dx1⊗dx0⊗dx1 + f010*f110 dx0⊗dx1⊗dx0⊗dx1⊗dx1⊗dx0 + f010*f111 dx0⊗dx1⊗dx0⊗dx1⊗dx1⊗dx1 + f000*f011 dx0⊗dx1⊗dx1⊗dx0⊗dx0⊗dx0 + f001*f011 dx0⊗dx1⊗dx1⊗dx0⊗dx0⊗dx1 + f010*f011 dx0⊗dx1⊗dx1⊗dx0⊗dx1⊗dx0 + f011^2 dx0⊗dx1⊗dx1⊗dx0⊗dx1⊗dx1 + f011*f100 dx0⊗dx1⊗dx1⊗dx1⊗dx0⊗dx0 + f011*f101 dx0⊗dx1⊗dx1⊗dx1⊗dx0⊗dx1 + f011*f110 dx0⊗dx1⊗dx1⊗dx1⊗dx1⊗dx0 + f011*f111 dx0⊗dx1⊗dx1⊗dx1⊗dx1⊗dx1 + f000*f100 dx1⊗dx0⊗dx0⊗dx0⊗dx0⊗dx0 + f001*f100 dx1⊗dx0⊗dx0⊗dx0⊗dx0⊗dx1 + f010*f100 dx1⊗dx0⊗dx0⊗dx0⊗dx1⊗dx0 + f011*f100 dx1⊗dx0⊗dx0⊗dx0⊗dx1⊗dx1 + f100^2 dx1⊗dx0⊗dx0⊗dx1⊗dx0⊗dx0 + f100*f101 dx1⊗dx0⊗dx0⊗dx1⊗dx0⊗dx1 + f100*f110 dx1⊗dx0⊗dx0⊗dx1⊗dx1⊗dx0 + f100*f111 dx1⊗dx0⊗dx0⊗dx1⊗dx1⊗dx1 + f000*f101 dx1⊗dx0⊗dx1⊗dx0⊗dx0⊗dx0 + f001*f101 dx1⊗dx0⊗dx1⊗dx0⊗dx0⊗dx1 + f010*f101 dx1⊗dx0⊗dx1⊗dx0⊗dx1⊗dx0 + f011*f101 dx1⊗dx0⊗dx1⊗dx0⊗dx1⊗dx1 + f100*f101 dx1⊗dx0⊗dx1⊗dx1⊗dx0⊗dx0 + f101^2 dx1⊗dx0⊗dx1⊗dx1⊗dx0⊗dx1 + f101*f110 dx1⊗dx0⊗dx1⊗dx1⊗dx1⊗dx0 + f101*f111 dx1⊗dx0⊗dx1⊗dx1⊗dx1⊗dx1 + f000*f110 dx1⊗dx1⊗dx0⊗dx0⊗dx0⊗dx0 + f001*f110 dx1⊗dx1⊗dx0⊗dx0⊗dx0⊗dx1 + f010*f110 dx1⊗dx1⊗dx0⊗dx0⊗dx1⊗dx0 + f011*f110 dx1⊗dx1⊗dx0⊗dx0⊗dx1⊗dx1 + f100*f110 dx1⊗dx1⊗dx0⊗dx1⊗dx0⊗dx0 + f101*f110 dx1⊗dx1⊗dx0⊗dx1⊗dx0⊗dx1 + f110^2 dx1⊗dx1⊗dx0⊗dx1⊗dx1⊗dx0 + f110*f111 dx1⊗dx1⊗dx0⊗dx1⊗dx1⊗dx1 + f000*f111 dx1⊗dx1⊗dx1⊗dx0⊗dx0⊗dx0 + f001*f111 dx1⊗dx1⊗dx1⊗dx0⊗dx0⊗dx1 + f010*f111 dx1⊗dx1⊗dx1⊗dx0⊗dx1⊗dx0 + f011*f111 dx1⊗dx1⊗dx1⊗dx0⊗dx1⊗dx1 + f100*f111 dx1⊗dx1⊗dx1⊗dx1⊗dx0⊗dx0 + f101*f111 dx1⊗dx1⊗dx1⊗dx1⊗dx0⊗dx1 + f110*f111 dx1⊗dx1⊗dx1⊗dx1⊗dx1⊗dx0 + f111^2 dx1⊗dx1⊗dx1⊗dx1⊗dx1⊗dx1" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#(t3*t3).disp() returns latex code string\n", "#(t3*t3)[:] works OK\n", "Manifold.options.omit_function_arguments=True\n", "tt = t3*t3\n", "%display plain\n", "tt.disp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Tensor fields of type $(k,0)$ (contravariant tensor fields of rank $k$)\n", "\n", "
\n", "\n", "A **tensor field of type $(k, 0)$** (or a **contravariant tensor field\n", " of rank $k$**) on a manifold $M$ is a map $t$ that\n", " associates to each point $p ∈ M$ a tensor $t(p)=t_p\\in T^{(k,0)}_pM\\ \\ \\ $ ($T^{(k,0)}_pM$ was defined in [notebook 9](09Manifold_LinearFormsTensors_onTp.ipynb)). \n", "
\n", "The tensor\n", "field $t$ is smooth if for arbitrary 1-forms $α_1 , . . . , α_k$ , the function $t(α_1 , . . . , α_k )$,\n", "defined by $[t (α_1, . . . , α_k )]( p) = t_p( α_1(p), . . . , α_k(p))$ , is smooth. \n", "\n", "The operations of tensor addition, scalar multiplication and tensor product are defined pointwise as in the case of covariant tensor fields.\n", "\n", "
\n", "\n", "### Tensorial property\n", "\n", "
\n", "\n", "A map $t$ is a tensor field of type $(k,0)$ iff it associates to each set of $k$ covector fields $α_1 , . . . , α_k$ on $M$ a function $t(α_1 , . . . , α_k):M\\to R$ with \n", "the property that for functions $f,g:M\\to R$, $\\alpha_1,\\ldots,\\alpha_k,\\beta_1,\\ldots,\\beta_k\\in T^*(M)$, and $i=1,\\ldots,k$\n", "\n", "$$t (\\alpha_1 , . . . , f\\alpha_i + g\\beta_i , . . . , \\alpha_k ) = ft (\\alpha_1 , . . . , \\alpha_i , . . . , \\alpha_k ) +gt (\\alpha_1 , . . . , \\beta_i , . . . , \\alpha_k ).$$\n", "\n", "
\n", "\n", "Note that the last property means that contravariant tensor fields of rank $k$ are just multilinear functions on the Cartesian product $\\underbrace{\\Omega^1(M)\\times\\cdots\\times \\Omega^1(M)}_{k\\ \\; \\mbox{times}}$ of $k$ copies of the module $\\Omega^1(M)$ over the ring $C^\\infty(M)$.\n", "\n", "The module of contravariant tensor fields of type $(k,0)$\n", "on a manifold $M$ will be denoted by $T^{(𝑘,0)}M.$\n", "\n", "\n", "**Warning**. If $M$\n", "is a module -not a manifold, in [notebook 9a](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09aManifold_Tensors_onModules.ipynb) the same notations denote the space of contravariant tensors of rank 𝑘 on the module $M$.\n", "\n", "
\n", "\n", "### Contravariant tensor fields in components\n", "\n", "
\n", "\n", "Using (9.7) we can check that any tensor\n", "field of type $(k,0)$ on $M$ is expressed locally as\n", "\n", "\\begin{equation}\n", "t = t (dx^{i_1} , . . . , dx^{i_k} )\\frac{\\partial}{\\partial x^{i_1}}\\otimes\\dots\\otimes\n", "\\frac{\\partial}{\\partial x^{i_k}}.\n", "\\label{}\\tag{13.8}\n", "\\end{equation}\n", "\n", "One can prove that if $x^1,\\ldots,x^n$ are local coordinates on $U$ and \n", "$\\ t^{i_1\\ldots i_k}=t(dx^{i_1} , . . . , dx^{i_k} )\\ $, then $t$ is smooth iff for arbitrary $p\\in M$ there is a coordinate map \n", "$(U,(x^1,\\ldots,x^n))$ around $p$ such that the real functions $\\ t^{i_1\\ldots i_k}\\ $ are smooth. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.11**\n", "\n", "Let us show an example of tensor field $t\\in T^{(2,0)}M$.
" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field t of type (2,0) on the 2-dimensional differentiable manifold M\n" ] } ], "source": [ "%display latex\n", "N = 2 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "t = M.tensor_field(2,0, name='t'); print(t) # tensor field (2,0) type" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Below, we use superscripts to show that the contravariant tensor fields may look in SageMath Manifolds as in textbooks." ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = f^{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\$" ], "text/latex": [ "$\\displaystyle t = f^{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "t = f00 ∂/∂x0⊗∂/∂x0 + f01 ∂/∂x0⊗∂/∂x1 + f10 ∂/∂x1⊗∂/∂x0 + f11 ∂/∂x1⊗∂/∂x1" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def fn(i,j): return 'f'+str(i)+str(j) # component names\n", "def fl(i,j): return 'f'+'^'+'{'+str(i)+str(j)+'}' # latex names\n", "ff = [[M.scalar_field(function(fn(i,j),latex_name=fl(i,j))(*X)) \n", " for j in range(N)] for i in range(N)] # nested list of comp.\n", "t[:] = ff # define all components\n", "Manifold.options.omit_function_arguments=True # output without comp. arguments\n", "t.disp() # show t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using tensor products it is easy to obtain large outputs." ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t\\otimes t = f^{00}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{01}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{01}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{10}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{10}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{10}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{11}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\)" ], "text/latex": [ "$\\displaystyle t\\otimes t = f^{00}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{01}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{01}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{10}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{10}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{10}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{00}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{01}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{10}\\left({x^{0}}, {x^{1}}\\right) f^{11}\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{11}\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "t⊗t = f00(x0, x1)^2 ∂/∂x0⊗∂/∂x0⊗∂/∂x0⊗∂/∂x0 + f00(x0, x1)*f01(x0, x1) ∂/∂x0⊗∂/∂x0⊗∂/∂x0⊗∂/∂x1 + f00(x0, x1)*f10(x0, x1) ∂/∂x0⊗∂/∂x0⊗∂/∂x1⊗∂/∂x0 + f00(x0, x1)*f11(x0, x1) ∂/∂x0⊗∂/∂x0⊗∂/∂x1⊗∂/∂x1 + f00(x0, x1)*f01(x0, x1) ∂/∂x0⊗∂/∂x1⊗∂/∂x0⊗∂/∂x0 + f01(x0, x1)^2 ∂/∂x0⊗∂/∂x1⊗∂/∂x0⊗∂/∂x1 + f01(x0, x1)*f10(x0, x1) ∂/∂x0⊗∂/∂x1⊗∂/∂x1⊗∂/∂x0 + f01(x0, x1)*f11(x0, x1) ∂/∂x0⊗∂/∂x1⊗∂/∂x1⊗∂/∂x1 + f00(x0, x1)*f10(x0, x1) ∂/∂x1⊗∂/∂x0⊗∂/∂x0⊗∂/∂x0 + f01(x0, x1)*f10(x0, x1) ∂/∂x1⊗∂/∂x0⊗∂/∂x0⊗∂/∂x1 + f10(x0, x1)^2 ∂/∂x1⊗∂/∂x0⊗∂/∂x1⊗∂/∂x0 + f10(x0, x1)*f11(x0, x1) ∂/∂x1⊗∂/∂x0⊗∂/∂x1⊗∂/∂x1 + f00(x0, x1)*f11(x0, x1) ∂/∂x1⊗∂/∂x1⊗∂/∂x0⊗∂/∂x0 + f01(x0, x1)*f11(x0, x1) ∂/∂x1⊗∂/∂x1⊗∂/∂x0⊗∂/∂x1 + f10(x0, x1)*f11(x0, x1) ∂/∂x1⊗∂/∂x1⊗∂/∂x1⊗∂/∂x0 + f11(x0, x1)^2 ∂/∂x1⊗∂/∂x1⊗∂/∂x1⊗∂/∂x1" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Manifold.options.omit_function_arguments=False # output with comp. arguments\n", "(t*t).disp() # show tensor product" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "t⊗t = f00^2 ∂/∂x0⊗∂/∂x0⊗∂/∂x0⊗∂/∂x0 + f00*f01 ∂/∂x0⊗∂/∂x0⊗∂/∂x0⊗∂/∂x1 + f00*f10 ∂/∂x0⊗∂/∂x0⊗∂/∂x1⊗∂/∂x0 + f00*f11 ∂/∂x0⊗∂/∂x0⊗∂/∂x1⊗∂/∂x1 + f00*f01 ∂/∂x0⊗∂/∂x1⊗∂/∂x0⊗∂/∂x0 + f01^2 ∂/∂x0⊗∂/∂x1⊗∂/∂x0⊗∂/∂x1 + f01*f10 ∂/∂x0⊗∂/∂x1⊗∂/∂x1⊗∂/∂x0 + f01*f11 ∂/∂x0⊗∂/∂x1⊗∂/∂x1⊗∂/∂x1 + f00*f10 ∂/∂x1⊗∂/∂x0⊗∂/∂x0⊗∂/∂x0 + f01*f10 ∂/∂x1⊗∂/∂x0⊗∂/∂x0⊗∂/∂x1 + f10^2 ∂/∂x1⊗∂/∂x0⊗∂/∂x1⊗∂/∂x0 + f10*f11 ∂/∂x1⊗∂/∂x0⊗∂/∂x1⊗∂/∂x1 + f00*f11 ∂/∂x1⊗∂/∂x1⊗∂/∂x0⊗∂/∂x0 + f01*f11 ∂/∂x1⊗∂/∂x1⊗∂/∂x0⊗∂/∂x1 + f10*f11 ∂/∂x1⊗∂/∂x1⊗∂/∂x1⊗∂/∂x0 + f11^2 ∂/∂x1⊗∂/∂x1⊗∂/∂x1⊗∂/∂x1" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Manifold.options.omit_function_arguments=True # output without comp. arguments\n", "%display plain\n", "(t*t).disp() # show tensor product" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.12**\n", "\n", "The superscripts in contravariant tensor components are not obligatory. \n", "Let us modify the previous example using subscripts." ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t2 = h_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\$" ], "text/latex": [ "$\\displaystyle t2 = h_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "t2 = h00 ∂/∂x0⊗∂/∂x0 + h01 ∂/∂x0⊗∂/∂x1 + h10 ∂/∂x1⊗∂/∂x0 + h11 ∂/∂x1⊗∂/∂x1" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "reset()\n", "N = 2 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "t2 = M.tensor_field(2,0, name='t2') # tensor field (2,0) type\n", "def hn(i,j): return 'h'+str(i)+str(j) # names for components\n", "def hl(i,j): return 'h'+'_'+'{'+str(i)+str(j)+'}' # latex names\n", "ff = [[M.scalar_field(function(hn(i,j), latex_name=hl(i,j))(*X)) \n", " for j in range(N)] for i in range(N)] # component functions\n", "t2[:] = ff # define all components\n", "Manifold.options.omit_function_arguments=True # output without comp.arg\n", "%display latex\n", "t2.disp() # show the tensor field" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t2\\otimes t2 = h_{00}^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{00} h_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{00} h_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01}^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{01} h_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} h_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} h_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{10}^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{10} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{10} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{11}^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\)" ], "text/latex": [ "$\\displaystyle t2\\otimes t2 = h_{00}^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{00} h_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{00} h_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01}^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{01} h_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} h_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} h_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{10}^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{10} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{00} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{01} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + h_{10} h_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + h_{11}^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "t2⊗t2 = h00^2 ∂/∂x0⊗∂/∂x0⊗∂/∂x0⊗∂/∂x0 + h00*h01 ∂/∂x0⊗∂/∂x0⊗∂/∂x0⊗∂/∂x1 + h00*h10 ∂/∂x0⊗∂/∂x0⊗∂/∂x1⊗∂/∂x0 + h00*h11 ∂/∂x0⊗∂/∂x0⊗∂/∂x1⊗∂/∂x1 + h00*h01 ∂/∂x0⊗∂/∂x1⊗∂/∂x0⊗∂/∂x0 + h01^2 ∂/∂x0⊗∂/∂x1⊗∂/∂x0⊗∂/∂x1 + h01*h10 ∂/∂x0⊗∂/∂x1⊗∂/∂x1⊗∂/∂x0 + h01*h11 ∂/∂x0⊗∂/∂x1⊗∂/∂x1⊗∂/∂x1 + h00*h10 ∂/∂x1⊗∂/∂x0⊗∂/∂x0⊗∂/∂x0 + h01*h10 ∂/∂x1⊗∂/∂x0⊗∂/∂x0⊗∂/∂x1 + h10^2 ∂/∂x1⊗∂/∂x0⊗∂/∂x1⊗∂/∂x0 + h10*h11 ∂/∂x1⊗∂/∂x0⊗∂/∂x1⊗∂/∂x1 + h00*h11 ∂/∂x1⊗∂/∂x1⊗∂/∂x0⊗∂/∂x0 + h01*h11 ∂/∂x1⊗∂/∂x1⊗∂/∂x0⊗∂/∂x1 + h10*h11 ∂/∂x1⊗∂/∂x1⊗∂/∂x1⊗∂/∂x0 + h11^2 ∂/∂x1⊗∂/∂x1⊗∂/∂x1⊗∂/∂x1" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tt2 = t2*t2\n", "tt2.disp() # show tensor product" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.13**\n", "\n", "Let us show an example of (3,0) type tensor field, using superscripts." ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "%display latex \n", "N = 2 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "s3 = M.tensor_field(3,0, name='s3'); # tensor field of type (3,0)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle s3 = f^{000} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{001} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{010} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{011} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{100} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{101} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{110} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{111} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\)" ], "text/latex": [ "$\\displaystyle s3 = f^{000} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{001} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{010} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{011} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{100} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{101} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + f^{110} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + f^{111} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "s3 = f000 ∂/∂x0⊗∂/∂x0⊗∂/∂x0 + f001 ∂/∂x0⊗∂/∂x0⊗∂/∂x1 + f010 ∂/∂x0⊗∂/∂x1⊗∂/∂x0 + f011 ∂/∂x0⊗∂/∂x1⊗∂/∂x1 + f100 ∂/∂x1⊗∂/∂x0⊗∂/∂x0 + f101 ∂/∂x1⊗∂/∂x0⊗∂/∂x1 + f110 ∂/∂x1⊗∂/∂x1⊗∂/∂x0 + f111 ∂/∂x1⊗∂/∂x1⊗∂/∂x1" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def fn(i,j,k): return 'f'+str(i)+str(j)+str(k) # components names\n", "def fl(i,j,k): return 'f'+'^'+'{'+str(i)+str(j)+str(k)+'}' # latex names\n", "ff = [[[M.scalar_field(function(fn(i,j,k),latex_name=fl(i,j,k))(*X)) \n", " for k in range(N)] for j in range(N)] for i in range(N)] # components\n", "s3[:] = ff # define all components\n", "Manifold.options.omit_function_arguments=True # output without comp. args\n", "s3.disp() # show the tensor" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field s3⊗s3 of type (6,0) on the 2-dimensional differentiable manifold M\n" ] } ], "source": [ "# check tensor product \n", "ss3 = s3*s3\n", "print(ss3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Tensor fields of type $(𝑘,m)$ (mixed tensor fields) \n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A tensor field of type $(k,m)$ (a mixed tensor field) on $M$ is a map $t$ \n", "that to each point $p ∈ M$ associates a tensor $t(p)=t_p\\in T^{(k,m)}_pM\\ \\ \\ $ ($T^{(k,m)}_pM$ was defined in [notebook 9](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09Manifold_LinearFormsTensors_onTp.ipynb).\n", "\n", "
\n", "A tensor field of type $(0,0)$ on $M$ is a scalar function $M\\to R$. \n", "\n", "A tensor field\n", "$t$ of type $(k,m)$ is smooth if for $X_1 , . . . , X_m ∈ \\mathfrak{X}(M)$ and $α_1 , . . . , α_k ∈ \\Omega^1 (M)$,\n", "the function $M \\to R$ that to each point $p ∈ M$ associates the value $t_p(α_1 ( p), . . . , α_k ( p),X_1 ( p), . . . , X_m ( p) )$ is smooth.\n", "
\n", "The sum, the product by scalars, the product by real-valued functions, and the\n", "tensor product of mixed tensor fields are defined pointwise as in the case of covariant and contravariant tensors." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{matrix}\n", "(at + bs)_p = at_p + bs_p,\\\\\n", "( f t)_p = f ( p)t_p,\\\\\n", "(t ⊗ s)_p = t_p ⊗ s_p,\n", "\\end{matrix}\n", "$$\n", "\n", "for $a, b ∈ R$, $s, t$ mixed tensor fields on $M$ and $f : M → R$ (addition is defined only for tensors of the same type).\n", "\n", "The set of smooth\n", "tensor fields of type $(k,m)$ on a manifold $M$, will be denoted by $T^{(k,m)}M$.
\n", "$T^{(k,m)}M$ **is a module over the ring $C^∞(M)$**.\n", "\n", "**Warning.** Recall, that in [notebook 9a](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09aManifold_Tensors_onModules.ipynb), the same symbol $\\ T^{(k,m)}M\\ $ denotes the space of mixed tensors on a module $M$.\n", "\n", "
\n", "\n", "### Tensorial property\n", "\n", "
\n", "\n", "As previously, we can check, that tensor fields of type $(k,m)$ are just multilinear functions on the Cartesian product \n", "$\\underbrace{\\Omega^1(M)\\times\\cdots\\times \\Omega^1(M)}_{k\\ \\; \\mbox{times}}\n", "\\times \\underbrace{\\mathfrak{X}(M)\\times\\cdots\\times \\mathfrak{X}(M)}_{m\\ \\; \\mbox{times}}\n", "$ of $k$ copies of the module $\\Omega^1(M)$ and $m$ copies of the module $\\mathfrak{X}(M)$ over the ring $C^\\infty(M)$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### General tensor fields in components\n", "\n", "
\n", "\n", "Generalizing the formulas (13.7) and (13.8) we obtain the following expression for the general tensor field $t\\in T^{(k,m)}M$ in local components\n", "\n", "\\begin{equation}\n", "t=\n", "t\\big(dx^{i_1},..,dx^{i_k},\\frac{\\partial}{\\partial x^{j_1}},..,\\frac{\\partial}{\\partial x^{j_m}}\\big)\n", "\\frac{\\partial}{\\partial x^{i_1}}\\otimes\\dots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\otimes dx^{j_1}\\otimes\\ldots\\otimes dx^{j_m}.\n", "\\label{}\\tag{13.9}\n", "\\end{equation}\n", "\n", "Very often the notation\n", "$$t^{i_1\\ldots i_k}_{j_1\\ldots j_m}=t\\big(dx^{i_1},..,dx^{i_k},\\frac{\\partial}{\\partial x^{j_1}},..,\\frac{\\partial}{\\partial x^{j_m}}\\big)\n", "$$\n", "is used and then \n", "$$t=t^{i_1\\ldots i_k}_{j_1\\ldots j_m}\\frac{\\partial}{\\partial x^{i_1}}\\otimes\\dots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\otimes dx^{j_1}\\otimes\\ldots\\otimes dx^{j_m}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One can prove that if $x^1,\\ldots,x^n$ are local coordinates on $U$ and $t^{i_1\\ldots i_k}_{j_1\\ldots j_m}$ are defined as above,\n", " then $t$ is smooth iff for arbitrary $p\\in M$ there is a coordinate map \n", "$(U,(x^1,\\ldots,x^n))$ around $p$ such that the real functions $t^{i_1\\ldots i_k}_{j_1\\ldots j_m}$ are smooth." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.14**\n", "\n", "Consider a 2-dimensional manifold with global coordinates $x^0,x^1$:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [], "source": [ "%display latex\n", "N = 2 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define a general tensor field of type (1,1).\n", "\n", "In the case of mixed type tensor fields it is important to distinguish the lower and upper indices!" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = t^0_0 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}} + t^0_1 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}} + t^1_0 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}} + t^1_1 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = t^0_0 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}} + t^0_1 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}} + t^1_0 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}} + t^1_1 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = t00 ∂/∂x0⊗dx0 + t01 ∂/∂x0⊗dx1 + t10 ∂/∂x1⊗dx0 + t11 ∂/∂x1⊗dx1" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def fn(i,j): return 't'+str(i)+str(j) # component names\n", "def fl(i,j): return 't'+'^'+str(i)+'_'+str(j) # latex names\n", "ff = [[M.scalar_field(function(fn(i,j),latex_name=fl(i,j))(*X)) \n", " for j in range(N)] for i in range(N)] # list of components\n", "t = M.tensor_field(1,1, name='t') # tensor of (1,1) type\n", "t[:] = ff # define all components\n", "t.disp() # show the tensor" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With this notations, the tensor product can be computed without problems:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t\\otimes t = t^0_0\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^0_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^0_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^1_0\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^1_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^1_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^1_1\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\)" ], "text/latex": [ "$\\displaystyle t\\otimes t = t^0_0\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^0_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^0_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_0\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^0_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^0_1\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^1_0\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^1_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^1_0\\left({x^{0}}, {x^{1}}\\right) t^1_1\\left({x^{0}}, {x^{1}}\\right) \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^1_1\\left({x^{0}}, {x^{1}}\\right)^{2} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t⊗t = t00(x0, x1)^2 ∂/∂x0⊗∂/∂x0⊗dx0⊗dx0 + t00(x0, x1)*t01(x0, x1) ∂/∂x0⊗∂/∂x0⊗dx0⊗dx1 + t00(x0, x1)*t01(x0, x1) ∂/∂x0⊗∂/∂x0⊗dx1⊗dx0 + t01(x0, x1)^2 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx1 + t00(x0, x1)*t10(x0, x1) ∂/∂x0⊗∂/∂x1⊗dx0⊗dx0 + t00(x0, x1)*t11(x0, x1) ∂/∂x0⊗∂/∂x1⊗dx0⊗dx1 + t01(x0, x1)*t10(x0, x1) ∂/∂x0⊗∂/∂x1⊗dx1⊗dx0 + t01(x0, x1)*t11(x0, x1) ∂/∂x0⊗∂/∂x1⊗dx1⊗dx1 + t00(x0, x1)*t10(x0, x1) ∂/∂x1⊗∂/∂x0⊗dx0⊗dx0 + t01(x0, x1)*t10(x0, x1) ∂/∂x1⊗∂/∂x0⊗dx0⊗dx1 + t00(x0, x1)*t11(x0, x1) ∂/∂x1⊗∂/∂x0⊗dx1⊗dx0 + t01(x0, x1)*t11(x0, x1) ∂/∂x1⊗∂/∂x0⊗dx1⊗dx1 + t10(x0, x1)^2 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx0 + t10(x0, x1)*t11(x0, x1) ∂/∂x1⊗∂/∂x1⊗dx0⊗dx1 + t10(x0, x1)*t11(x0, x1) ∂/∂x1⊗∂/∂x1⊗dx1⊗dx0 + t11(x0, x1)^2 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx1" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#t*t works\n", "Manifold.options.omit_function_arguments=False # output with comp.args\n", "tt = t*t # tensor product\n", "tt.disp() # show tensor product" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 13.15**\n", "\n", "Now let us try to define (2,2)-type tensor field on the same two-dimensional manifold" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t = t^{00}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{00}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{00}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{00}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^{01}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{01}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{01}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{01}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^{10}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{10}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{10}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{10}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^{11}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{11}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{11}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{11}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\)" ], "text/latex": [ "$\\displaystyle t = t^{00}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{00}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{00}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{00}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^{01}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{01}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{01}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{01}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^{10}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{10}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{10}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{10}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t^{11}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t^{11}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t^{11}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t^{11}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = t0000 ∂/∂x0⊗∂/∂x0⊗dx0⊗dx0 + t0001 ∂/∂x0⊗∂/∂x0⊗dx0⊗dx1 + t0010 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx0 + t0011 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx1 + t0100 ∂/∂x0⊗∂/∂x1⊗dx0⊗dx0 + t0101 ∂/∂x0⊗∂/∂x1⊗dx0⊗dx1 + t0110 ∂/∂x0⊗∂/∂x1⊗dx1⊗dx0 + t0111 ∂/∂x0⊗∂/∂x1⊗dx1⊗dx1 + t1000 ∂/∂x1⊗∂/∂x0⊗dx0⊗dx0 + t1001 ∂/∂x1⊗∂/∂x0⊗dx0⊗dx1 + t1010 ∂/∂x1⊗∂/∂x0⊗dx1⊗dx0 + t1011 ∂/∂x1⊗∂/∂x0⊗dx1⊗dx1 + t1100 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx0 + t1101 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx1 + t1110 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx0 + t1111 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx1" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# continuation; first define names and latex names of components\n", "def fl(i0,i1,i2,i3): return 't'+'^'+'{'+str(i0)+str(i1)+'}'+'_'+'{'+str(i2)+str(i3)+'}'\n", "def fn(i0,i1,i2,i3): return 't'+str(i0)+str(i1)+str(i2)+str(i3)\n", "# define nested list of components of the tensor\n", "ff = [[[[M.scalar_field(function(fn(i0,i1,i2,i3),latex_name=fl(i0,i1,i2,i3))(*X)) \n", " for i3 in range(N)] for i2 in range(N)] for i1 in range(N)] for i0 in range(N)]\n", "t = M.tensor_field(2,2, name='t') # tensor field of type (2,2)\n", "t[:] = ff # define all components#\n", "Manifold.options.omit_function_arguments=True # output without comp. args\n", "t.disp() # show the tensor" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Tensor product works with this notations (but the output is very long)." ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [], "source": [ "#t*t works\n", "tt = t*t" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "# tt.disp() -latex code\n", "# tt[:] very long output after very long time" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [], "source": [ "# %display plain\n", "# tt.disp() # very-very long output" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is easier to check that the linear operations work:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 3 \\, t^{00}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{00}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{00}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{00}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{01}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{01}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{01}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{01}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{10}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{10}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{10}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{10}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{11}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{11}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{11}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{11}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\)" ], "text/latex": [ "$\\displaystyle 3 \\, t^{00}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{00}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{00}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{00}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{01}_{00} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{01}_{01} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{01}_{10} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{01}_{11} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{10}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{10}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{10}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{10}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{11}_{00} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{11}_{01} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\, t^{11}_{10} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 3 \\, t^{11}_{11} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "3*t0000 ∂/∂x0⊗∂/∂x0⊗dx0⊗dx0 + 3*t0001 ∂/∂x0⊗∂/∂x0⊗dx0⊗dx1 + 3*t0010 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx0 + 3*t0011 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx1 + 3*t0100 ∂/∂x0⊗∂/∂x1⊗dx0⊗dx0 + 3*t0101 ∂/∂x0⊗∂/∂x1⊗dx0⊗dx1 + 3*t0110 ∂/∂x0⊗∂/∂x1⊗dx1⊗dx0 + 3*t0111 ∂/∂x0⊗∂/∂x1⊗dx1⊗dx1 + 3*t1000 ∂/∂x1⊗∂/∂x0⊗dx0⊗dx0 + 3*t1001 ∂/∂x1⊗∂/∂x0⊗dx0⊗dx1 + 3*t1010 ∂/∂x1⊗∂/∂x0⊗dx1⊗dx0 + 3*t1011 ∂/∂x1⊗∂/∂x0⊗dx1⊗dx1 + 3*t1100 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx0 + 3*t1101 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx1 + 3*t1110 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx0 + 3*t1111 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx1" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Manifold.options.omit_function_arguments=True # output without comp.args\n", "(t + 2*t).disp() # show t+2*t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## What's next?\n", "\n", "Take a look at the notebook [Differential k-forms](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/14Manifold_Differential_Forms.ipynb)." ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.6", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 4 }