{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 9. Tensors on $T_pM$\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": [ "**Warning**. In this notebook there are many repetitions with respect to the notebook [9a. Tensors on modules](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09aManifold_Tensors_onModules.ipynb).\n", "Although mathematically tensors on modules are generalizations of tensors on the tangent spaces, the SageMath Manifolds code in the present notebook differs significantly from that in [notebook 9a](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09aManifold_Tensors_onModules.ipynb).\n", "\n", "
\n", "\n", "### Differentials of functions, linear forms, covectors\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If $M$ is a smooth manifold, $p ∈ M$, $T_pM$ is the tangent space to $M$ at $p$, and $f ∈ C^∞(M)$, then the **differential of $f$ at $p$**, denoted by $d f_p$, is\n", "defined by\n", "\n", "\\begin{equation}\n", "df_p(X_p)=X_p(f),\\quad\\mbox{for } X_p\\in T_pM.\n", "\\label{} \\tag{9.1}\n", "\\end{equation}\n", "\n", "Let us show some examples of scalar functions and differentials." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.1**\n", "\n", "\n", "Define a 3-dimensional manifold $M,$ a point $p\\in M$, the tangent space at $T_pM$, a tangent vector $v\\in T_pM$ and some scalar function $f$ on $M$. " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the 3-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\left({x^{0}}, {x^{1}}, {x^{2}}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left({x^{0}}, {x^{1}}, {x^{2}}\\right)$" ], "text/plain": [ "(x0, x1, x2)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%display latex\n", "N = 3\n", "M = Manifold(N, 'M') # manifold M of dimension N\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "p = M.point(X[:], name='p') # generic point p of M\n", "Tp = M.tangent_space(p) # tangent space at p\n", " # components of tangent vector v:\n", "vn = [var('v'+str(i), latex_name='v'+'^'+str(i)) for i in range(N)]\n", "v = Tp(vn, name='v') # tangent vector v\n", "f = M.scalar_field(function('f')(*X), name='f') # scalar funct.\n", "print(p) # information on p\n", "p.coord() # coordinates of p" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector v at Point p on the 3-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\left[{v^0}, {v^1}, {v^2}\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[{v^0}, {v^1}, {v^2}\\right]$" ], "text/plain": [ "[v0, v1, v2]" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(v) # information on v\n", "v.comp()[:] # components of v" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} f:& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left({x^{0}}, {x^{1}}, {x^{2}}\\right) & \\longmapsto & f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} f:& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left({x^{0}}, {x^{1}}, {x^{2}}\\right) & \\longmapsto & f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\end{array}$" ], "text/plain": [ "f: M → ℝ\n", " (x0, x1, x2) ↦ f(x0, x1, x2)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.disp() # show scalar function f" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let us define the differential of $f$ at $p$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "df_p is a\n", "Linear form df on the Tangent space at Point p on the 3-dimensional differentiable manifold M\n", "Components of df_p:\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\left[\\frac{\\partial}{\\partial {x^{0}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right), \\frac{\\partial}{\\partial {x^{1}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right), \\frac{\\partial}{\\partial {x^{2}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\frac{\\partial}{\\partial {x^{0}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right), \\frac{\\partial}{\\partial {x^{1}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right), \\frac{\\partial}{\\partial {x^{2}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right)\\right]$" ], "text/plain": [ "[diff(f(x0, x1, x2), x0), diff(f(x0, x1, x2), x1), diff(f(x0, x1, x2), x2)]" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# continuation\n", "df = f.differential() # differential of f\n", "dfp = df.at(p) # differential of f at p\n", "print(\"df_p is a\") \n", "print(dfp) # information on dfp\n", "print(\"Components of df_p:\")\n", "dfp.components()[:] # components of differential at p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and check that the value of $df$ at $p$ on the tangent vector $v$\n", "is equal to $v(f)$:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bool(dfp(v)==v(f)) # check if df_p(v)=v(f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Linearity of $df_p$\n", "\n", "
\n", "\n", "Recall that for a vector space (or a module) $V$ the map $t:V\\to R$ is linear iff\n", "\n", "$$ t (av + bw ) = at(v) + bt(w ),\\quad\\text{for}\\quad v,w\\in V,\\quad a,b\\in R.$$\n", "\n", "\n", "To check that $df_p: T_pM\\to R$ is linear let us note that \n", "\n", "$$d f_p (av_p + bw_p ) = (av_p + bw_p )( f )\n", "= av_p ( f ) + bw_p ( f )\n", "= a d f_p (v_p ) + b d f_p (w_p ).$$
\n", "\n", "
\n", "\n", "### Cotangent space $T^*_pM$\n", "\n", "
\n", "\n", "The space of **linear forms** on $T_pM$ is by definition the **dual space** of $T_pM$ and is denoted by $T_p^*M$. Elements of $T_p^*M$ are called **covectors** or **covariant vectors** and $T_p^*M$ is called the **cotangent space** to $M$ at $p$.
The vector space operations in $T_p^*M$ are defined in a natural way\n", "\n", "$$(\\alpha_p+\\beta_p)(v_p)=\\alpha_p(v_p)+\\beta_p(v_p),\\quad\n", "(aα_p )(v_p ) = a(α_p (v_p )),$$\n", "\n", "for $\\ \\ α_p , β_p ∈ T_p^∗ M,\\ \\ v_p ∈ T_p M\\ $ and $a ∈ R$.\n", "\n", "To check what kind of object is $df_p$ we can use the `parent` method:\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Dual of the Tangent space at Point p on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "# continuation\n", "# mathematical object of which \"dfp\" is an element.\n", "print(dfp.parent())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.2**\n", "\n", "If $x^1,\\ldots,x^n$ are local coordinates on $M$, then differentials\n", "$dx^i_p$ are covectors from $T_p^*M$. From (9.1) it follows that if the tangent vector is expressed as $X_p=X_p(x^i)\\frac{\\partial}{\\partial x^i}\\big|_p$, then\n", "\n", "\\begin{equation}\n", "\\displaystyle dx^i_p(X_p)=X_p(x^i),\n", "\\label{}\\tag{9.1'}\n", "\\end{equation}\n", "\n", "i.e. $\\ dx^i_p$ is the covector that maps the tangent vector at $p$ to its $i$-th component in the basis $\\displaystyle \\big\\{\\frac{\\partial}{\\partial x^j}\\big|_p\\big\\}_{j=1}^n.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.3**\n", "\n", "Let us define the scalar functions representing the i-th coordinate of the point." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left({x^{0}}, {x^{1}}, {x^{2}}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left({x^{0}}, {x^{1}}, {x^{2}}\\right)$" ], "text/plain": [ "(x0, x1, x2)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 3 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M of dimension N\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "x0, x1, x2 = X[:]\n", "p = M.point((x0, x1, x2), name='p') # point of M\n", "fx0 = M.scalar_field(x0, name='fx0') # fx0: x-->x^0\n", "fx1 = M.scalar_field(x1, name='fx1') # fx1: x-->x^1\n", "fx2 = M.scalar_field(x2, name='fx2') # fx2: x-->x^2\n", "fx0.expr(), fx1.expr(), fx2.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### And next their differentials:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\mathrm{d}fx0 = \\mathrm{d} {x^{0}}, \\mathrm{d}fx1 = \\mathrm{d} {x^{1}}, \\mathrm{d}fx2 = \\mathrm{d} {x^{2}}\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\mathrm{d}fx0 = \\mathrm{d} {x^{0}}, \\mathrm{d}fx1 = \\mathrm{d} {x^{1}}, \\mathrm{d}fx2 = \\mathrm{d} {x^{2}}\\right]$" ], "text/plain": [ "[dfx0 = dx0, dfx1 = dx1, dfx2 = dx2]" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dx0 = fx0.differential() # differential of fx0\n", "dx1 = fx1.differential() # differential of fx1\n", "dx2 = fx2.differential() # differential of fx2\n", "\n", "dx0_p = dx0.at(p) # differential of fx0 at p\n", "dx1_p = dx1.at(p) # differential of fx1 at p\n", "dx2_p = dx2.at(p) # differential of fx2 at p\n", "\n", "[dx0_p.disp(), dx1_p.disp(), dx2_p.disp()] # show all three" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can check that the just defined differentials, when applied to the tangent vector $v$ give its i-th component $v^i$." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left({v^0}, {v^1}, {v^1}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left({v^0}, {v^1}, {v^1}\\right)$" ], "text/plain": [ "(v0, v1, v1)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp = M.tangent_space(p) # tangent space at p\n", " # variables with superscripts:\n", "vn = [var('v'+str(i), latex_name='v'+'^'+str(i)) for i in range(N)]\n", "v = Tp(vn, name='v') # tangent vector at p\n", "(dx0_p(v), dx1_p(v), dx1_p(v)) # values of differentials of xi \n", " # at p on a tangent vector v" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Bases in the cotangent space \n", "\n", "
\n", "\n", "Using the definition of the basis tangent vectors\n", "$\\frac{\\partial}{\\partial x^i}\\Big|_p(f)=D_i\\Big|_{\\phi(p)}(f\\circ\\phi^{-1})$ and the fact that\n", "$(x^j\\circ\\phi^{-1}(x^1(p),\\ldots,x^n(p))=(x^j\\circ\\phi^{-1})(\\phi(p))=x^j(p),\\ $ we see that\n", "$\\frac{\\partial}{\\partial x^i}\\Big|_p(x^j)=\\delta_i^j$ and consequently $dx^i_p\\big(\\frac{\\partial}{\\partial x^j}\\big|_p\\big)=\n", "\\frac{\\partial}{\\partial x^j}\\big|_p(x^i)=\\delta_j^i$. \n", "The last relation implies that:\n", "
\n", "\n", "**The family $\\{dx^i_p\\}_{i=1}^n$ is a basis of $T_p^*M$**.
\n", "\n", "In fact if the linear combination $\\ a_idx^i_p\\ $ is equal to zero covector, then $\\ 0 = (a_i dx^i_p ) (∂/∂ x^j|_p)= a_i\\delta_j^i=a_j\\ $ i.e., the coefficients of this combination vanish, so the family is **linearly independent**.
\n", "If $\\ \\alpha_p\\in T^*_pM$, then for $\\ v_p\\in T_pM\\ $ in the form \n", "$v_p= v_p(x^i)\\frac{\\partial}{\\partial x^i}\\big|_p$ we have\n", "\n", "$$\\alpha_p(v_p)=\\alpha_p\\big( v_p(x^i)\\frac{\\partial}{\\partial x^i}\\big|_p\\big)= v_p(x^i)\\alpha_p\\big(\\frac{\\partial}{\\partial x^i}\\big|_p\\big).$$\n", "\n", "From the definition of the differential it follows $v_p(x^i)=dx^i_p(v_p)$, so\n", "\n", "$$\\alpha_p(v_p)= \\alpha_p\\big(\\frac{\\partial}{\\partial x^i}\\big|_p\\big)dx^i_p(v_p)=\\Big[ \\alpha_p\\big(\\frac{\\partial}{\\partial x^i}\\big|_p\\big)dx^i_p\\Big](v_p),$$ \n", "\n", "consequently\n", "\n", "\\begin{equation}\n", "\\alpha_p= \\alpha_p\\Big( \\frac{\\partial}{\\partial x^i}\\big|_p\\Big)dx^i_p.\n", "\\label{} \\tag{9.2}\n", "\\end{equation}\n", "\n", "Thus **every covector from $T_p^*M$ is a linear combination of** $\\{dx^i_p\\}_{i=1}^n.$

\n", "\n", "
\n", "\n", "**Example 9.4**\n", "\n", "For example the differential of $f$ is a linear combination of $dx^0,dx^1,dx^2$ with coefficients \n", "$\\frac{\\partial f}{\\partial x^0},\\frac{\\partial f}{\\partial x^1}, \\frac{\\partial f}{\\partial x^2}.$

\n", "\n", "**In SageMath Manifolds the index $p$ denoting the point is always dropped**. \n", "\n", "\n", "The function `display` shows the differentials of functions and general linear forms as linear combinations of $dx^i.$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{d}f = \\frac{\\partial}{\\partial {x^{0}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{0}} + \\frac{\\partial}{\\partial {x^{1}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{1}} + \\frac{\\partial}{\\partial {x^{2}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{2}}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\frac{\\partial}{\\partial {x^{0}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{0}} + \\frac{\\partial}{\\partial {x^{1}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{1}} + \\frac{\\partial}{\\partial {x^{2}}}f\\left({x^{0}}, {x^{1}}, {x^{2}}\\right) \\mathrm{d} {x^{2}}$" ], "text/plain": [ "df = diff(f(x0, x1, x2), x0) dx0 + diff(f(x0, x1, x2), x1) dx1 + diff(f(x0, x1, x2), x2) dx2" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# continuation\n", "dfp.disp() # show df_p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.5**\n", "\n", "Let us define a general linear form on the tangent space $T_pM$ of a 3-dimensional manifold." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "N = 3 # dimension of manifold M\n", "M = Manifold(N, 'M') # manifold M of dimension N\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "p = M.point(X[:], name='p') # point of M\n", "Tp = M.tangent_space(p) # tangent space at p\n", "a = Tp.linear_form() # linear form on T_pM\n", "a[:] = var('a', n=3) # coefficients in the basis dx^i" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle a_{0} \\mathrm{d} {x^{0}} + a_{1} \\mathrm{d} {x^{1}} + a_{2} \\mathrm{d} {x^{2}}\$" ], "text/latex": [ "$\\displaystyle a_{0} \\mathrm{d} {x^{0}} + a_{1} \\mathrm{d} {x^{1}} + a_{2} \\mathrm{d} {x^{2}}$" ], "text/plain": [ "a0 dx0 + a1 dx1 + a2 dx2" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.disp() # show a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**The linear form applied to a tangent vector gives a scalar**." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle a_{0} {v^0} + a_{1} {v^1} + a_{2} {v^2}\$" ], "text/latex": [ "$\\displaystyle a_{0} {v^0} + a_{1} {v^1} + a_{2} {v^2}$" ], "text/plain": [ "a0*v0 + a1*v1 + a2*v2" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#continuation # variables wit superscripts:\n", "vn = [var('v'+str(i), latex_name='v'+'^'+str(i)) for i in range(N)]\n", "v = Tp(vn, name='v') # tangent vector v at p\n", "a(v) # value of a on v" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**The value of a tangent vector $v\\in T_pM$ on a linear form $a\\in T_p^*M$ is by definition equal to $a(v)$**." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle a_{0} {v^0} + a_{1} {v^1} + a_{2} {v^2}\$" ], "text/latex": [ "$\\displaystyle a_{0} {v^0} + a_{1} {v^1} + a_{2} {v^2}$" ], "text/plain": [ "a0*v0 + a1*v1 + a2*v2" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v(a) # value of v on a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "\n", "## Tensors in tangent spaces\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Recall, that if $V$ is a vector space or a module, by a **multilinear** or more precisely $k$-**linear form** we mean a function $t:\n", "V^k\\to R$ which is linear in each of its arguments i.e., for $i=1,\\ldots,k$\n", "\n", "$$t(v_1\\ldots,\\alpha v_i+\\beta w_i,\\ldots v_k)=\n", "\\alpha t(v_1\\ldots,v_i,\\ldots v_k)+\\beta t(v_1\\ldots,w_i,\\ldots v_k),\\quad \\alpha,\\beta\\in R,\\quad v_i,w_i\\in V.$$\n", "
\n", "Assume that $M$ is a smooth manifold." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Space of tensors $T^{(k,m)}_pM$ of mixed type\n", "\n", "
\n", "\n", "is the space of multilinear maps:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ T^{(k,m)}_pM = \\{t_p:\\underbrace{T_p^*M\\times\\cdots\\times T_p^*M}_{k\\ \\; \\mbox{times}}\n", " \\times \\underbrace{T_pM\\times\\cdots\\times T_pM}_{m\\ \\; \\mbox{times}}\\to R\\},$$\n", "\n", "where $T_pM$ is the tangent space at $p\\in M$ and\n", "$T_p^*M$ is the corresponding cotangent space." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since for linear forms $\\alpha$ and tangent vectors $v$ the assignment $\\alpha\\to v(\\alpha)$ defines a linear form $T_p^*M\\to R$, the elements $v\\in T_pM$ can be considered as elements of $T_p^{(1,0)}$. On the other hand $T^{(0,1)}_p$ as the space of linear forms on $T_pM$ is equal to $T_p^*M$. Thus, the tensor spaces generalize the tangent and cotangent spaces. \n", "\n", "
\n", "\n", "**Example 9.6**\n", "\n", "Let us show how to define tensor modules of type (1,0) and (0,1) in SageMath Manifolds." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{p}\\,M\$" ], "text/latex": [ "$\\displaystyle T_{p}\\,M$" ], "text/plain": [ "Tangent space at Point p on the 3-dimensional differentiable manifold M" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 3 # manifold dimension\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", "p = M.point(X[:], name='p') # point on M\n", "Tp = M.tangent_space(p) # tangent space at p\n", "Tp.tensor_module(1,0) # T_p^(1,0)M" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent space at Point p on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "print(Tp.tensor_module(1,0)) # tensors (1,0) type are tangent vectors" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{p}\\,M^*\$" ], "text/latex": [ "$\\displaystyle T_{p}\\,M^*$" ], "text/plain": [ "Free module of type-(0,1) tensors on the Tangent space at Point p on the 3-dimensional differentiable manifold M" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp.tensor_module(0,1) # tensors of type (0,1) are linear forms on T_pM\n", "# SageMath output looks a little strange\n", "# but note that the dual space to V is denoted by V*" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Free module of type-(0,1) tensors on the Tangent space at Point p on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "print(Tp.tensor_module(0,1)) # SageMath naming is ready for arbitrary\n", " # tensors, covector is just (0,1) tensor" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Space $T^{(0,k)}_pM$ - of covariant tensors of rank $k$\n", "is the space of multilinear maps\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ T^{(0,k)}_pM = \\{t_p:\\underbrace{T_pM\\times\\cdots\\times T_pM}_{k\\ \\; \\mbox{times}}\\to R\\},$$\n", "\n", "where $T_pM$ denotes the tangent space at $p\\in M$.\n", "\n", "For k=1 we obtain the space of covectors at $p$, i.e., linear forms on $T_pM.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In $T^{(0,k)}_pM$ we introduce the vector space structure by\n", "\n", "$$ (at_p+bs_p)(v_1\\ldots,v_k)=at_p(v_1\\ldots,v_k)+bs_p(v_1\\ldots,v_k),$$\n", "\n", "where $t_p,s_p\\in T^{(0,k)}_pM$, $v_1,\\ldots,v_k\\in T_pM$ and $a,b\\in R$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $t_p\\in T^{(0,k)}_pM,\\, s_p\\in T^{(0,m)}_pM$ we define the **tensor product** $t_p\\otimes s_p\\in T_p^{(0,k+m)}M$ by\n", "\n", "\\begin{equation}\n", "(t_p ⊗ s_p )(v_1 ,\\ldots , v_{k+m} ) ≡ t_p (v_1 ,\\ldots, v_k ) s_p (v_{k+1} , . . . , v_{k+m} ),\n", "\\label{eq:tensor_product}\\tag{9.3}\n", "\\end{equation}\n", "\n", "for $v_1,\\ldots,v_{k+m}\\in T_pM$.

\n", "In SageMath Manifolds the symbol of tensor product is simply $*$.
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Covariant tensors in local components\n", "\n", "
\n", "\n", "If the tangent vectors $v_i$ are expressed as \n", "$v_i=v_i(x^j)\\frac{\\partial}{\\partial x^j}\\big|_p=dx^j_p(v_i)\\frac{\\partial}{\\partial x^j}\\big|_p, \\ \\ i=1,\\ldots k$, then by the multilinearity of $t_p$ and the definition (9.3) of the tensor product we have for $t_p\\in T^{(0,k)}_p$:
\n", "$$t_p(v_1,\\ldots,v_k)=t_p\\big(dx^{i_1}_p(v_1)\\frac{\\partial}{\\partial x^{i_1}}\\big|_p,\\ldots,dx^{i_k}_p(v_k)\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\big)\\\\\n", "=dx^{i_1}_p(v_1)\\dots dx^{i_k}_p(v_k)t_p\\big(\\frac{\\partial}{\\partial x^{i_1}}\\big|_p,\\ldots,\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\big)\\\\\n", "=\\Big[t_p\\big(\\frac{\\partial}{\\partial x^{i_1}}\\big|_p,\\ldots,\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\big)dx^{i_1}_p\\otimes\\dots\\otimes dx^{i_k}_p\\Big](v_1,\\ldots,v_k),\n", "$$\n", "so" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\\begin{equation}\n", "t_p=t_{i_1\\ldots i_k}dx^{i_1}_p\\otimes\\dots\\otimes dx^{i_k}_p,\\\\\n", "\\label{}\\tag{9.4}\n", "\\end{equation}\n", "where\n", "\\begin{equation}\n", "t_{i_1\\ldots i_k}=t_p\\big(\\frac{\\partial}{\\partial x^{i_1}}\\big|_p,\\ldots,\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\big).\n", "\\label{}\\tag{9.5}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.7**\n", "\n", "Let us check the formula (9.3) in the case of two general tensors from $T^{(0,2)}_pM$ for a 2-dimensional manifold $M$." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "N = 2 # dimension of manifold M\n", "M2 = Manifold(N, 'M2') # manifold M2 of dimension 2\n", "X = M2.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "p = M2.point(X[:], name='p') # p -point with coordinates (x^0,x^1)\n", "Tp = M2.tangent_space(p) # Tp -tangent space at p\n", "t = Tp.tensor((0,2)) # t -tensor from T^{(0,2)}_pM\n", "s = Tp.tensor((0,2)) # s -tensor from T^{(0,2)}_pM" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we define a list of components of four tangent vectors:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[{v^{0}_{0}}, {v^{1}_{0}}\\right], \\left[{v^{0}_{1}}, {v^{1}_{1}}\\right], \\left[{v^{0}_{2}}, {v^{1}_{2}}\\right], \\left[{v^{0}_{3}}, {v^{1}_{3}}\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[{v^{0}_{0}}, {v^{1}_{0}}\\right], \\left[{v^{0}_{1}}, {v^{1}_{1}}\\right], \\left[{v^{0}_{2}}, {v^{1}_{2}}\\right], \\left[{v^{0}_{3}}, {v^{1}_{3}}\\right]\\right]$" ], "text/plain": [ "[[v00, v10], [v01, v11], [v02, v12], [v03, v13]]" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = 4 # number of vectors\n", "vv=[[var('v'+str(i)+str(j), # components of four vectors\n", " latex_name='v'+'^{'+str(i)+'}_{'+str(j)+'}')\n", " for i in range(N)] for j in range(M)]\n", "vv # show four vectors components" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and the vectors:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {v^{0}_{0}} \\frac{\\partial}{\\partial {x^{0}} } + {v^{1}_{0}} \\frac{\\partial}{\\partial {x^{1}} }\$" ], "text/latex": [ "$\\displaystyle {v^{0}_{0}} \\frac{\\partial}{\\partial {x^{0}} } + {v^{1}_{0}} \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "v00 ∂/∂x0 + v10 ∂/∂x1" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v=[Tp(vv[k]) for k in range(M)]; # list of 4 tangent vectors\n", "v[0].disp() # show first tangent vector" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we define the components of two (0,2)-type tensors $t\\ $ and $s$." ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [], "source": [ "tt=[[var('t'+str(i)+str(j)) # components of tensor t\n", " for j in range(N)] for i in range(N)]\n", "ss=[[var('s'+str(i)+str(j)) # components of tensor s\n", " for j in range(N)] for i in range(N)]\n", "\n", "\n", "t[:] = tt # define tensor t components t_{ij}\n", "s[:] = ss # define tensor s components s_{ij}" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t00 dx0⊗dx0 + t01 dx0⊗dx1 + t10 dx1⊗dx0 + t11 dx1⊗dx1" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t.disp() # show t" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle s_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + s_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + s_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + s_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle s_{00} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + s_{01} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + s_{10} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + s_{11} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "s00 dx0⊗dx0 + s01 dx0⊗dx1 + s10 dx1⊗dx0 + s11 dx1⊗dx1" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.disp() # show s" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we are ready to check the formula (9.3)." ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ " # check the formula (9.3)\n", "bool((t*s)(v[0],v[1],v[2],v[3]) == t(v[0],v[1])*s(v[2],v[3]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Algebraic properties of tensor product of covariant tensors\n", "\n", "
\n", "\n", "Tensor product has the following properties\n", "\n", "$$\n", "\\begin{matrix}\n", "(a\\,r_p + b\\,s_p ) ⊗ t_p = a\\,r_p ⊗ t_p + b\\,s_p ⊗ t_p ,\\\\\n", "r_p ⊗ (a\\,s_p + b\\,t_p ) = a\\,r_p ⊗ s_p + b\\,r_p ⊗ t_p ,\\\\\n", "(r_p ⊗ s_p ) ⊗ t_p = r_p ⊗ (s_p ⊗ t_p ),\n", "\\end{matrix}\n", "$$\n", "\n", "for $a,b\\in R\\ $ and for arbitrary covariant tensors $r_p,s_p,t_p$ (the addition is defined only for tensors of the same rank $(0,k)$).\n", "\n", "To check the first formula, let us note that for $X_1,\\ldots X_{k+l}\\in T_pM$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "(ar_p+bs_{ p} ) ⊗ t_p (X_1 , . . . , X_{k+l} ) \n", "= (ar_{ p} + bs_{ p} )(X_1 , . . . , X_k ) t_p (X_{k+1} , . . . , X_{k+l} )\\\\\n", "= (ar_{ p} (X_1 , . . . , X_k ) + b s_{ p} (X_1 , . . . , X_k ))\n", "t_p (X_{k+1} , . . . , X_{k+l} ) \\\\\n", "=ar_{ p} (X_1 , . . . , X_k )t_p (X_{k+1}, . . . , X_{k+l} )\n", "+b s_{ p} (X_1 , . . . , X_k )t_p (X_{k+1} , . . . , X_{k+l} )\\\\\n", "= (a r_{ p} ⊗ t_p + b s_{ p} ⊗ t_p) (X_1 , . . . , X_{k+l} ),\n", "$$\n", "\n", "and similarly for the second formula.\n", "\n", "The associativity follows from" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "((r_p ⊗ s_p ) ⊗ t_p)(X_1 , . . . , X_{k+l+m} )\\\\\n", "= (r_p ⊗ s_p )(X_1 , . . . , X_{k+l} )\\; t_p (X_{k+l+1} , . . . , \n", "X_{k+l+m} )\\\\\n", "= (r_p (X_1 , . . . , X_k ) s_p (X_{k+1} , . . . , X_{k+l} )) \n", "t_p (X_{k+l+1} , . . . , X_{k+l+m})\\\\\n", "= r_p (X_1 , . . . , X_k )( s_p (X_{k+1} , . . . , X_{k+l} ) \n", "t_p (X_{k+l+1} , . . . , X_{k+l+m} ))\\\\\n", "= r_p (X_1 , . . . , X_k ) (s_p ⊗ t_p )(X_{k+1} , . . . , X_{k+l+m} )\\\\\n", "= (r_p ⊗ (s_p ⊗ t_p )) (X_1 , . . . , X_{k+l+m} ),\n", "$$\n", "\n", "for $X_1 , . . . , X_{k+l+m} \\in T_pM.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.8**\n", "\n", "Let us show an example of a tensor $t\\in T^{(0,3)}_pM$. First let us define the corresponding 3-dimensional table of components. " ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[\\left[t_{000}, t_{001}\\right], \\left[t_{010}, t_{011}\\right]\\right], \\left[\\left[t_{100}, t_{101}\\right], \\left[t_{110}, t_{111}\\right]\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[\\left[t_{000}, t_{001}\\right], \\left[t_{010}, t_{011}\\right]\\right], \\left[\\left[t_{100}, t_{101}\\right], \\left[t_{110}, t_{111}\\right]\\right]\\right]$" ], "text/plain": [ "[[[t000, t001], [t010, t011]], [[t100, t101], [t110, t111]]]" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 2 # dimension of manifold\n", "st=[[[var('t'+str(i0)+str(i1)+str(i2)) # components of t\n", " for i2 in range(N)] \n", " for i1 in range(N)] \n", " for i0 in range(N)]\n", "st" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we define a manifold $M$, a point $p\\in M$, tangent space $T_pM$\n", "and the tensor $t\\in T^{(0,3)}_pM$." ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = t_{000} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t_{001} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t_{010} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t_{011} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t_{100} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t_{101} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t_{110} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t_{111} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = t_{000} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t_{001} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t_{010} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t_{011} \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + t_{100} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + t_{101} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + t_{110} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + t_{111} \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = t000 dx0⊗dx0⊗dx0 + t001 dx0⊗dx0⊗dx1 + t010 dx0⊗dx1⊗dx0 + t011 dx0⊗dx1⊗dx1 + t100 dx1⊗dx0⊗dx0 + t101 dx1⊗dx0⊗dx1 + t110 dx1⊗dx1⊗dx0 + t111 dx1⊗dx1⊗dx1" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(N, 'M') # 2-dimensional manifold\n", "X = M.chart(' '.join(['x'+str(i)+':x^{'+str(i)+'}' for i in range(N)])) # chart on M\n", "p = M.point(X[:], name='p') # point in M\n", "Tp = M.tangent_space(p) # tangent space T_p\n", "t = Tp.tensor((0,3), name='t') # tensor of type (0,3)\n", "t[:] = st # define components of t\n", "t.disp() # show t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we show how to define a tensor of type (0,3) on a 2-dimensional manifold with concrete components." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t = \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 2 \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 4 \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 5 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 6 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 7 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 8 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\$" ], "text/latex": [ "$\\displaystyle t = \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 2 \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 4 \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 5 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 6 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 7 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 8 \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = dx0⊗dx0⊗dx0 + 2 dx0⊗dx0⊗dx1 + 3 dx0⊗dx1⊗dx0 + 4 dx0⊗dx1⊗dx1 + 5 dx1⊗dx0⊗dx0 + 6 dx1⊗dx0⊗dx1 + 7 dx1⊗dx1⊗dx0 + 8 dx1⊗dx1⊗dx1" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ct=range(1,9) # 1,...,8 -components of t\n", "for i0 in range(N):\n", " for i1 in range(N):\n", " for i2 in range(N):\n", " t[i0,i1,i2]=ct[i0*N*N+i1*N+i2]\n", "t.disp() # show t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define the local basis of $T_pM$." ] }, { "cell_type": "code", "execution_count": 30, "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": [ "Basis (∂/∂x0,∂/∂x1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Xf = X.frame() # default frame\n", "Xfp = Xf.at(p); Xfp # default frame at p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Check that the component $t_{011}$ of the tensor from the previous example is equal to $t(\\frac{\\partial}{\\partial x_0},\\frac{\\partial}{\\partial x_1},\\frac{\\partial}{\\partial x_1})$." ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 4\$" ], "text/latex": [ "$\\displaystyle 4$" ], "text/plain": [ "4" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t(Xfp[0], Xfp[1], Xfp[1]) # value of t(d/dx0,d/dx1,d/dx1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Space $T^{(k,0)}_pM $ of contravariant tensors of rank $k$\n", "is the space of multilinear forms:\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ T^{(k,0)}_pM = \\{t_p:\\underbrace{T_p^*M\\times\\cdots\\times T_p^*M}_{k\\ \\;\\mbox{times}}\n", " \\to R\\},$$\n", "where $T_p^*M$ is the cotangent space at $p\\in M$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In $T^{(k,0)}_pM$ we introduce the vector space structure by\n", "\n", "$$ (at_p+bs_p)(\\alpha_1\\ldots,\\alpha_k)=at_p(\\alpha_1\\ldots,\\alpha_k)+bs_p(\\alpha_1\\ldots,\\alpha_k),$$\n", "\n", "where $t_p,s_p\\in T^{(k,0)}_pM$, $\\alpha_1,\\ldots,\\alpha_k\\in T_p^*M$ and $a,b\\in R$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $t_p\\in T^{(k,0)}_pM,s_p\\in T^{(m,0)}_pM$ we define the **tensor product** $t_p\\otimes s_p$ by\n", "\n", "\\begin{equation}\n", "(t_p ⊗ s_p )(\\alpha_1 ,\\ldots , \\alpha_{k+m} ) = t_p (\\alpha_1 ,\\ldots, \\alpha_k ) s_p (\\alpha_{k+1} , \\ldots , \\alpha_{k+m} ),\n", "\\tag{9.6}\n", "\\end{equation}\n", "\n", "for $\\alpha_1,\\ldots,\\alpha_{k+m}\\in T^*_pM$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### Contravariant tensors in local components\n", "\n", "
\n", "\n", "Let $t_p\\in T^{(k,0)}_pM,\\ \\ α_1 , . . . , α_k ∈ T_p^∗M $ and let all covectors $\\alpha_i$ be expressed as \n", "$α_i = α_i \\big(\\frac{∂}{∂ x^j}\\big|_p\\big) dx^j_ p\\ \\ $ (cf. (9.2)). We have\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$t_p(\\alpha_1,\\ldots,\\alpha_k)=\n", "t_p\\big(\\alpha_1\\big(\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\big)dx^{i_1}_p,\\ldots,\n", "\\alpha_k\\big(\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\big)dx^{i_k}_p \\big)\\\\\n", "=\\alpha_1\\big(\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\big)\\ldots\\alpha_k\\big(\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\big)t_p(dx^{i_1}_p\\dots dx^{i_k}_p).$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $v_p(\\alpha_p)=\\alpha_p(v_p)$ we obtain" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$t_p(\\alpha_1,\\ldots,\\alpha_k)\n", "=\\Big[t_p(dx^{i_1}_p,\\dots, dx^{i_k}_p)\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\otimes\\ldots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\Big](\\alpha_1,\\ldots,\\alpha_k),\n", "$$ so\n", "\\begin{equation}\n", "t_p=t^{i_1\\ldots i_k}_p\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\otimes\\dots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\big|_p,\n", "\\label{}\\tag{9.7}\n", "\\end{equation}\n", "\n", "\\begin{equation}\n", "t_p^{i_1\\ldots i_k}=t_p(dx^{i_1}_p,\\ldots,dx^{i_k}_p).\n", "\\label{}\\tag{9.7'}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example 9.9**\n", "\n", "Let us give an example of a tensor $t_p\\in T^{(3,0)}_pM$.\n", "\n", "First we define the corresponding 3-dimensional table of components with upper indices." ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[\\left[{t^{000}}, {t^{001}}\\right], \\left[{t^{010}}, {t^{011}}\\right]\\right], \\left[\\left[{t^{100}}, {t^{101}}\\right], \\left[{t^{110}}, {t^{111}}\\right]\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[\\left[{t^{000}}, {t^{001}}\\right], \\left[{t^{010}}, {t^{011}}\\right]\\right], \\left[\\left[{t^{100}}, {t^{101}}\\right], \\left[{t^{110}}, {t^{111}}\\right]\\right]\\right]$" ], "text/plain": [ "[[[t000, t001], [t010, t011]], [[t100, t101], [t110, t111]]]" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 2 # dimension of manifold\n", "st=[[[var('t'+str(i0)+str(i1)+str(i2),\n", " latex_name='t'+'^'+'{'+str(i0)+str(i1)+str(i2)+'}') \n", " for i2 in range(N)] \n", " for i1 in range(N)] # components of t\n", " for i0 in range(N)] # with superscripts\n", "st" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we define a two-dimensional manifold, the tangent space and the tensor $t_p\\in T^{(3,0)}_pM$ t with component from the table st." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [], "source": [ "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", "p = M.point(X[:], name='p') # point p in M\n", "Tp = M.tangent_space(p) # tangent space at p\n", "t = Tp.tensor((3,0), name='t') # tensor of type (3,0)\n", "t[:] = st # tensor components" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We are ready to display the general tensor of type (3,0) on a 2-dimensional manifold in components." ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t = {t^{000}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{001}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + {t^{010}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{011}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + {t^{100}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{101}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + {t^{110}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{111}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\)" ], "text/latex": [ "$\\displaystyle t = {t^{000}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{001}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + {t^{010}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{011}} \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + {t^{100}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{101}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + {t^{110}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + {t^{111}} \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "t = t000 ∂/∂x0⊗∂/∂x0⊗∂/∂x0 + t001 ∂/∂x0⊗∂/∂x0⊗∂/∂x1 + t010 ∂/∂x0⊗∂/∂x1⊗∂/∂x0 + t011 ∂/∂x0⊗∂/∂x1⊗∂/∂x1 + t100 ∂/∂x1⊗∂/∂x0⊗∂/∂x0 + t101 ∂/∂x1⊗∂/∂x0⊗∂/∂x1 + t110 ∂/∂x1⊗∂/∂x1⊗∂/∂x0 + t111 ∂/∂x1⊗∂/∂x1⊗∂/∂x1" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t.disp() # show t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here is a tensor of type (3,0) on a 2-dimensional manifold with concrete components:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t = \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 2 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + 3 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 4 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + 5 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 6 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + 7 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 8 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\)" ], "text/latex": [ "$\\displaystyle t = \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 2 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + 3 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 4 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + 5 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 6 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} } + 7 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} } + 8 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }$" ], "text/plain": [ "t = ∂/∂x0⊗∂/∂x0⊗∂/∂x0 + 2 ∂/∂x0⊗∂/∂x0⊗∂/∂x1 + 3 ∂/∂x0⊗∂/∂x1⊗∂/∂x0 + 4 ∂/∂x0⊗∂/∂x1⊗∂/∂x1 + 5 ∂/∂x1⊗∂/∂x0⊗∂/∂x0 + 6 ∂/∂x1⊗∂/∂x0⊗∂/∂x1 + 7 ∂/∂x1⊗∂/∂x1⊗∂/∂x0 + 8 ∂/∂x1⊗∂/∂x1⊗∂/∂x1" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ct=range(1,9) # 1,...,8 - components of t\n", "for i0 in range(N):\n", " for i1 in range(N):\n", " for i2 in range(N):\n", " t[i0,i1,i2]=ct[i0*N*N+i1*N+i2]\n", "t.disp() # show t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us define the basis of $T^*_pM$." ] }, { "cell_type": "code", "execution_count": 36, "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": [ "Dual basis (dx0,dx1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Xcf = X.coframe() # basis of cotangent space\n", "Xcfp = Xcf.at(p); Xcfp # basis of cotangent space at p " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Check that the component $\\ t_{011}\\ $ is equal to $\\ t( dx^0,dx^1, dx^1)$." ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 4\$" ], "text/latex": [ "$\\displaystyle 4$" ], "text/plain": [ "4" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t(Xcfp[0], Xcfp[1], Xcfp[1]) # value of t(dx0,dx1,dx1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "### General tensors from $T^{(k,m)}_pM$ \n", "\n", "
\n", "\n", "Since the definition of $T^{(k,m)}_pM$ was given above we start from definition of tensor product." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $t_p\\in T^{(k_1,m_1)},s_p\\in T^{(k_2,m_2)}_pM$ we define the **tensor product** $t_p\\otimes s_p\\in T_p^{(k_1+k_2,m_1+m_2)}$ by

\n", "\\begin{equation}\n", "(t_p ⊗ s_p )(\\alpha_1,\\ldots\\alpha_{k_1+k_2},v_1 ,\\ldots , v_{m_1+m_2} )\\\\\n", "=t_p (\\alpha_1,\\ldots,\\alpha_{k_1},v_1 ,\\ldots, v_{m_1} ) s_p (\\alpha_{k_1+1},\\ldots,\\alpha_{k_1+k_2},v_{m_1+1},\\ldots, v_{m_1+m_2} ),\n", "\\tag{9.8}\n", "\\end{equation}

\n", "for $\\alpha_1,\\ldots,\\alpha_{k_1+k_2}\\in T^*_pM$ and $v_1,\\ldots,v_{m_1+m_2}\\in T_pM$.

\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Generalizing the formulas (9.5) and (9.7) we obtain the following expression for the general tensor $t_p\\in T^{(k,l)}_pM$ in local components\n", "\n", "\\begin{equation}\n", "t_p=\n", "t_p\\big(dx^{i_1}_p,..,dx^{i_k}_p,\\frac{\\partial}{\\partial x^{j_1}}\\big|_p,..,\\frac{\\partial}{\\partial x^{j_m}}\\big|_p\\big)\n", "\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\otimes..\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\otimes dx^{j_1}_p\\otimes..\\otimes dx^{j_m}_p.\n", "\\label{}\\tag{9.9}\n", "\\end{equation}\n", "\n", "Very often the notation\n", "\\begin{equation}\n", "t^{i_1\\ldots i_k}_{j_1\\ldots j_m}=t_p\\big(dx^{i_1}_p,..,dx^{i_k}_p,\\frac{\\partial}{\\partial x^{j_1}}\\big|_p,..,\\frac{\\partial}{\\partial x^{j_m}}\\big|_p\\big)\n", "\\tag{9.10}\n", "\\end{equation}\n", "is used and then \n", "\\begin{equation}\n", "t_p=t^{i_1\\ldots i_k}_{j_1\\ldots j_m}\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\otimes\\dots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\otimes dx^{j_1}_p\\otimes\\ldots\\otimes dx^{j_m}_p.\n", "\\tag{9.11}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To check that the elements\n", "\n", "\\begin{equation}\n", "\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\otimes\\dots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\otimes dx^{j_1}_p\\otimes\\ldots\\otimes dx^{j_m}_p\n", "\\label{} \\tag{9.12}\n", "\\end{equation}\n", "\n", "are linearly independent, assume that the linear combination\n", "\n", "$$a^{i_1\\ldots i_k}_{j_1\\ldots j_m}\\frac{\\partial}{\\partial x^{i_1}}\\big|_p\\otimes\\dots\\otimes\\frac{\\partial}{\\partial x^{i_k}}\\big|_p\\otimes dx^{j_1}_p\\otimes\\ldots\\otimes dx^{j_m}_p\n", "$$\n", "\n", "vanishes. If we apply this combination to $\\quad(dx^{q_1}_p,\\ldots,dx^{q_k}_p,\\frac{\\partial}{\\partial x^{r_1}}\\big|_p,..,\\frac{\\partial}{\\partial x^{r_m}}\\big|_p\\big)\\ $ we get $\\ \\displaystyle a^{q_1\\ldots q_k}_{r_1\\ldots r_m}=0$.
\n", "Since previously we have checked that elements of the type (9.12) span $T^{(k,l)}_pM$ we have proved, that these elements form a **basis** for $T^{(k,m)}_pM$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 9.10**\n", "\n", "To show an example of a tensor $t_p\\in T^{(2,2)}_pM$, we first define a 4-dimensional table with lower and upper indices." ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left[\\left[\\left[{t^{00}_{00}}, {t^{00}_{01}}\\right], \\left[{t^{00}_{10}}, {t^{00}_{11}}\\right]\\right], \\left[\\left[{t^{01}_{00}}, {t^{01}_{01}}\\right], \\left[{t^{01}_{10}}, {t^{01}_{11}}\\right]\\right]\\right], \\left[\\left[\\left[{t^{10}_{00}}, {t^{10}_{01}}\\right], \\left[{t^{10}_{10}}, {t^{10}_{11}}\\right]\\right], \\left[\\left[{t^{11}_{00}}, {t^{11}_{01}}\\right], \\left[{t^{11}_{10}}, {t^{11}_{11}}\\right]\\right]\\right]\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left[\\left[\\left[{t^{00}_{00}}, {t^{00}_{01}}\\right], \\left[{t^{00}_{10}}, {t^{00}_{11}}\\right]\\right], \\left[\\left[{t^{01}_{00}}, {t^{01}_{01}}\\right], \\left[{t^{01}_{10}}, {t^{01}_{11}}\\right]\\right]\\right], \\left[\\left[\\left[{t^{10}_{00}}, {t^{10}_{01}}\\right], \\left[{t^{10}_{10}}, {t^{10}_{11}}\\right]\\right], \\left[\\left[{t^{11}_{00}}, {t^{11}_{01}}\\right], \\left[{t^{11}_{10}}, {t^{11}_{11}}\\right]\\right]\\right]\\right]$" ], "text/plain": [ "[[[[t0000, t0001], [t0010, t0011]], [[t0100, t0101], [t0110, t0111]]],\n", " [[[t1000, t1001], [t1010, t1011]], [[t1100, t1101], [t1110, t1111]]]]" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 2 # dimension of manifold \n", "st=[[[[var('t'+str(i0)+str(i1)+str(i2)+str(i3),\n", " latex_name='t'+'^'+'{'+str(i0)+str(i1)+'}'+'_'+'{'+str(i2)+str(i3)+'}') \n", " for i3 in range(N)] for i2 in range(N)] # components of t with upper\n", " for i1 in range(N)] for i0 in range(N)] # and lower indices\n", "st" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now define a 2-dimensional manifold $M$, tangent space at $p$ and $\\ t_p\\in T^{(2,2)}_pM$." ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [], "source": [ "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", "p = M.point(X[:], name='p') # point of M\n", "Tp = M.tangent_space(p) # tangent space at p\n", "t = Tp.tensor((2,2), name='t') # tensor of type (2,2)\n", "t[:] = st # define tensor components" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can display a general tensor from $T_p^{(2,2)}M$ on a 2-dimensional manifold ($2^4$ components)." ] }, { "cell_type": "code", "execution_count": 40, "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": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t.disp() # show t" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [], "source": [ "#tt=t*t\n", "#tt[:] very long output" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A more concrete example can be defined as follows:" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [], "source": [ "ct=range(1,17) # [1,...,16] consecutive components\n", "for i0 in range(N):\n", " for i1 in range(N):\n", " for i2 in range(N):\n", " for i3 in range(N):\n", " t[i0,i1,i2,i3]=ct[i0*N*N*N+i1*N*N+i2*N+i3]" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle t = \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 2 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 4 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 5 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 6 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 7 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 8 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 9 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 10 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 11 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 12 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 13 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 14 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 15 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 16 \\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 = \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 2 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 3 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 4 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 5 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 6 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 7 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 8 \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 9 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 10 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 11 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 12 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{0}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}} + 13 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{0}} + 14 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{0}}\\otimes \\mathrm{d} {x^{1}} + 15 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{0}} + 16 \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\frac{\\partial}{\\partial {x^{1}} }\\otimes \\mathrm{d} {x^{1}}\\otimes \\mathrm{d} {x^{1}}$" ], "text/plain": [ "t = ∂/∂x0⊗∂/∂x0⊗dx0⊗dx0 + 2 ∂/∂x0⊗∂/∂x0⊗dx0⊗dx1 + 3 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx0 + 4 ∂/∂x0⊗∂/∂x0⊗dx1⊗dx1 + 5 ∂/∂x0⊗∂/∂x1⊗dx0⊗dx0 + 6 ∂/∂x0⊗∂/∂x1⊗dx0⊗dx1 + 7 ∂/∂x0⊗∂/∂x1⊗dx1⊗dx0 + 8 ∂/∂x0⊗∂/∂x1⊗dx1⊗dx1 + 9 ∂/∂x1⊗∂/∂x0⊗dx0⊗dx0 + 10 ∂/∂x1⊗∂/∂x0⊗dx0⊗dx1 + 11 ∂/∂x1⊗∂/∂x0⊗dx1⊗dx0 + 12 ∂/∂x1⊗∂/∂x0⊗dx1⊗dx1 + 13 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx0 + 14 ∂/∂x1⊗∂/∂x1⊗dx0⊗dx1 + 15 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx0 + 16 ∂/∂x1⊗∂/∂x1⊗dx1⊗dx1" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t.disp()" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "#(t*t).disp() - latex code\n", "#(t*t)[:] # long output" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using the bases of $T_pM$ and $T^*_pM$," ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Xfp- Basis (∂/∂x0,∂/∂x1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M\n", "Xcfp- Dual basis (dx0,dx1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M\n" ] } ], "source": [ "Xf = X.frame() # frame (d/dx0,d/dx1)\n", "Xfp = Xf.at(p); print(\"Xfp-\", Xfp) # frame (d/dx0,d/dx1) at p\n", "Xcf = X.coframe() # coframe (dx0,dx1)\n", "Xcfp = Xcf.at(p); print(\"Xcfp-\", Xcfp) # coframe (dx0,dx1) at p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "one can check that the component $\\ \\ t^{10}_{01}\\ \\ $ is equal to $\\ \\ t(dx_1,dx_0,\\frac{\\partial}{\\partial x_0},\\frac{\\partial}{\\partial x_1})$:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 10\$" ], "text/latex": [ "$\\displaystyle 10$" ], "text/plain": [ "10" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t(Xcfp[1], Xcfp[0], Xfp[0], Xfp[1]) # value of t(dx1,dx0,d/dx0,d/dx1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## What's next?\n", "\n", "Take a look at the notebook [Alternating forms on modules](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/10Manifold_AlternatingForms_onModules.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 }