{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Strain and stress tensors in spherical coordinates" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*This worksheet illustrates some features of [SageManifolds](http://sagemanifolds.obspm.fr/) (v0.9) on computations regarding elasticity theory in spherical coordinates.*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Euclidean 3-space and spherical coordinates" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We then introduce the Euclidean space as a 3-dimensional differentiable manifold:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3-dimensional differentiable manifold M\n" ] } ], "source": [ "M = Manifold(3, 'M', start_index=1)\n", "print M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We shall make use of spherical coordinates $(r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (M, (r, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (M, (r, th, ph))" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher. = M.chart(r'r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "print spher\n", "spher" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Spherical coordinates do not form a regular coordinate system of the Euclidean space. So declaring that they span $M$ means that, strictly speaking, the manifold $M$ is not the whole Euclidean space, but the Euclidean space minus some half plane (the azimuthal origin). However, in this worksheet, this difference will not matter. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The natural vector frame of spherical coordinates is" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M ,\\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$ " ], "text/plain": [ "Coordinate frame (M, (d/dr,d/dth,d/dph))" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We shall expand vector and tensor fields on the orthonormal frame $(e_1, e_2, e_3)$ associated with spherical coordinates, which is related to the natural frame $(\\partial/\\partial r, \\partial/\\partial\\theta, \\partial/\\partial\\phi)$ displayed above by means of the following field of automorphisms: " ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} r + \\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\theta} + \\frac{1}{r \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "d/dr*dr + 1/r d/dth*dth + 1/(r*sin(th)) d/dph*dph" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "to_orthonormal = M.automorphism_field()\n", "to_orthonormal[1,1] = 1\n", "to_orthonormal[2,2] = 1/r\n", "to_orthonormal[3,3] = 1/(r*sin(th))\n", "to_orthonormal.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In other words, the change-of-basis matrix is " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "1 & 0 & 0 \\\\\n", "0 & \\frac{1}{r} & 0 \\\\\n", "0 & 0 & \\frac{1}{r \\sin\\left({\\theta}\\right)}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[ 1 0 0]\n", "[ 0 1/r 0]\n", "[ 0 0 1/(r*sin(th))]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "to_orthonormal[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We construct the orthonormal frame from the natural frame of spherical coordinates by this change of basis: " ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M, \\left(e_1,e_2,e_3\\right)\\right)$ " ], "text/plain": [ "Vector frame (M, (e_1,e_2,e_3))" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e = spher.frame().new_frame(to_orthonormal, 'e')\n", "e" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_1 = \\frac{\\partial}{\\partial r }$ " ], "text/plain": [ "e_1 = d/dr" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[1].display()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_2 = \\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }$ " ], "text/plain": [ "e_2 = 1/r d/dth" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[2].display()" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_3 = \\frac{1}{r \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "e_3 = 1/(r*sin(th)) d/dph" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[3].display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At this stage, the default vector frame on $M$ is the first one introduced, namely the natural frame of spherical coordinates: " ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M ,\\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$ " ], "text/plain": [ "Coordinate frame (M, (d/dr,d/dth,d/dph))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since we prefer the orthonormal frame, we declare" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "M.set_default_frame(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then, by default, all vector and tensor fields are displayed with respect to that frame:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_3 = e_3$ " ], "text/plain": [ "e_3 = e_3" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[3].display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get the same output as in `Out[10]`, one should specify the frame for display, since this is no longer the default one:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_3 = \\frac{1}{r \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "e_3 = 1/(r*sin(th)) d/dph" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[3].display(spher.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Displacement vector and strain tensor\n", "\n", "Let define the **displacement vector** $U$ in terms of its components w.r.t. the orthonormal spherical frame:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}U = U_{1}\\left(r, {\\theta}, {\\phi}\\right) e_1 + U_{2}\\left(r, {\\theta}, {\\phi}\\right) e_2 + U_{3}\\left(r, {\\theta}, {\\phi}\\right) e_3$ " ], "text/plain": [ "U = U_1(r, th, ph) e_1 + U_2(r, th, ph) e_2 + U_3(r, th, ph) e_3" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U = M.vector_field(name='U')\n", "U[:] = [function('U_1')(r,th,ph), function('U_2')(r,th,ph), function('U_3')(r,th,ph)]\n", "U.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following computations will involve the metric $g$ of the Euclidean space. At the current stage of SageManifolds, we need to introduce it explicitly, as a Riemannian metric on the manifold $M$ (in a future version of SageManifolds, one shall to declare $M$ as an Euclidean space, and not merely as a manifold, so that it will come equipped with $g$):" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Riemannian metric g on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "g = M.riemannian_metric('g')\n", "print g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $e$ is supposed to be an orthonormal frame, we declare that the components of $g$ with respect to it are\n", "$\\mathrm{diag}(1,1,1)$:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = e^1\\otimes e^1+e^2\\otimes e^2+e^3\\otimes e^3$ " ], "text/plain": [ "g = e^1*e^1 + e^2*e^2 + e^3*e^3" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[1,1], g[2,2], g[3,3] = 1, 1, 1\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of $g$ with respect to the natural frame of spherical coordinates is then " ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(spher.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The covariant derivative operator $\\nabla$ is introduced as the (Levi-Civita) connection associated with $g$: " ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Riemannian metric g on the 3-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g}$ " ], "text/plain": [ "Levi-Civita connection nabla_g associated with the Riemannian metric g on the 3-dimensional differentiable manifold M" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla = g.connection()\n", "print nabla\n", "nabla" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The connection coefficients with respect to the spherical orthonormal frame $e$ are" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{\\phantom{\\, 1}\\,2\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}} & = & -\\frac{1}{r} \\\\ \\Gamma_{\\phantom{\\, 1}\\,3\\,3}^{\\,1\\phantom{\\, 3}\\phantom{\\, 3}} & = & -\\frac{1}{r} \\\\ \\Gamma_{\\phantom{\\, 2}\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{r} \\\\ \\Gamma_{\\phantom{\\, 2}\\,3\\,3}^{\\,2\\phantom{\\, 3}\\phantom{\\, 3}} & = & -\\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{\\phantom{\\, 3}\\,1\\,3}^{\\,3\\phantom{\\, 1}\\phantom{\\, 3}} & = & \\frac{1}{r} \\\\ \\Gamma_{\\phantom{\\, 3}\\,2\\,3}^{\\,3\\phantom{\\, 2}\\phantom{\\, 3}} & = & \\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} \\end{array}$ " ], "text/plain": [ "Gam^1_22 = -1/r \n", "Gam^1_33 = -1/r \n", "Gam^2_12 = 1/r \n", "Gam^2_33 = -cos(th)/(r*sin(th)) \n", "Gam^3_13 = 1/r \n", "Gam^3_23 = cos(th)/(r*sin(th)) " ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "while those with respect to the natural frame of spherical coordinates (Christoffel symbols) are:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r } \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta} } \\phantom{\\, {\\theta} } } & = & -r \\\\ \\Gamma_{ \\phantom{\\, r } \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & -r \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta} } \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r } \\phantom{\\, {\\theta} } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta} } \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta} } \\phantom{\\, r } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta} } \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r } \\phantom{\\, {\\phi} } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta} } \\phantom{\\, {\\phi} } } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi} } \\phantom{\\, r } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi} } \\phantom{\\, {\\theta} } } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$ " ], "text/plain": [ "Gam^r_th,th = -r \n", "Gam^r_ph,ph = -r*sin(th)^2 \n", "Gam^th_r,th = 1/r \n", "Gam^th_th,r = 1/r \n", "Gam^th_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_r,ph = 1/r \n", "Gam^ph_th,ph = cos(th)/sin(th) \n", "Gam^ph_ph,r = 1/r \n", "Gam^ph_ph,th = cos(th)/sin(th) " ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla.display(spher.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The covariant derivative of the displacement vector $U$ is" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(U) of type (1,1) on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "nabU = nabla(U)\n", "print nabU" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} U = \\frac{\\partial\\,U_{1}}{\\partial r} e_1\\otimes e^1 + \\left( -\\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) - \\frac{\\partial\\,U_{1}}{\\partial th}}{r} \\right) e_1\\otimes e^2 + \\left( -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\sin\\left({\\theta}\\right) - \\frac{\\partial\\,U_{1}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e_1\\otimes e^3 + \\frac{\\partial\\,U_{2}}{\\partial r} e_2\\otimes e^1 + \\left( \\frac{U_{1}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{2}}{\\partial th}}{r} \\right) e_2\\otimes e^2 + \\left( -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - \\frac{\\partial\\,U_{2}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e_2\\otimes e^3 + \\frac{\\partial\\,U_{3}}{\\partial r} e_3\\otimes e^1 + \\frac{\\frac{\\partial\\,U_{3}}{\\partial th}}{r} e_3\\otimes e^2 + \\left( \\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + U_{1}\\left(r, {\\theta}, {\\phi}\\right) \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e_3\\otimes e^3$ " ], "text/plain": [ "nabla_g(U) = d(U_1)/dr e_1*e^1 - (U_2(r, th, ph) - d(U_1)/dth)/r e_1*e^2 - (U_3(r, th, ph)*sin(th) - d(U_1)/dph)/(r*sin(th)) e_1*e^3 + d(U_2)/dr e_2*e^1 + (U_1(r, th, ph) + d(U_2)/dth)/r e_2*e^2 - (U_3(r, th, ph)*cos(th) - d(U_2)/dph)/(r*sin(th)) e_2*e^3 + d(U_3)/dr e_3*e^1 + d(U_3)/dth/r e_3*e^2 + (U_2(r, th, ph)*cos(th) + U_1(r, th, ph)*sin(th) + d(U_3)/dph)/(r*sin(th)) e_3*e^3" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabU.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We convert it to a tensor field of type (0,2) (i.e. a bilinear form) by lowering the upper index with $g$:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (0,2) on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "nabU_form = nabU.down(g)\n", "print nabU_form" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial\\,U_{1}}{\\partial r} e^1\\otimes e^1 + \\left( -\\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) - \\frac{\\partial\\,U_{1}}{\\partial th}}{r} \\right) e^1\\otimes e^2 + \\left( -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\sin\\left({\\theta}\\right) - \\frac{\\partial\\,U_{1}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^1\\otimes e^3 + \\frac{\\partial\\,U_{2}}{\\partial r} e^2\\otimes e^1 + \\left( \\frac{U_{1}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{2}}{\\partial th}}{r} \\right) e^2\\otimes e^2 + \\left( -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - \\frac{\\partial\\,U_{2}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^2\\otimes e^3 + \\frac{\\partial\\,U_{3}}{\\partial r} e^3\\otimes e^1 + \\frac{\\frac{\\partial\\,U_{3}}{\\partial th}}{r} e^3\\otimes e^2 + \\left( \\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + U_{1}\\left(r, {\\theta}, {\\phi}\\right) \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^3$ " ], "text/plain": [ "d(U_1)/dr e^1*e^1 - (U_2(r, th, ph) - d(U_1)/dth)/r e^1*e^2 - (U_3(r, th, ph)*sin(th) - d(U_1)/dph)/(r*sin(th)) e^1*e^3 + d(U_2)/dr e^2*e^1 + (U_1(r, th, ph) + d(U_2)/dth)/r e^2*e^2 - (U_3(r, th, ph)*cos(th) - d(U_2)/dph)/(r*sin(th)) e^2*e^3 + d(U_3)/dr e^3*e^1 + d(U_3)/dth/r e^3*e^2 + (U_2(r, th, ph)*cos(th) + U_1(r, th, ph)*sin(th) + d(U_3)/dph)/(r*sin(th)) e^3*e^3" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabU_form.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **strain tensor** $\\varepsilon$ is defined as the symmetrized part of this tensor:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "E = nabU_form.symmetrize()\n", "print E" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon = \\frac{\\partial\\,U_{1}}{\\partial r} e^1\\otimes e^1 + \\left( \\frac{r \\frac{\\partial\\,U_{2}}{\\partial r} - U_{2}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{1}}{\\partial th}}{2 \\, r} \\right) e^1\\otimes e^2 + \\left( \\frac{{\\left(r \\frac{\\partial\\,U_{3}}{\\partial r} - U_{3}\\left(r, {\\theta}, {\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{1}}{\\partial ph}}{2 \\, r \\sin\\left({\\theta}\\right)} \\right) e^1\\otimes e^3 + \\left( \\frac{r \\frac{\\partial\\,U_{2}}{\\partial r} - U_{2}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{1}}{\\partial th}}{2 \\, r} \\right) e^2\\otimes e^1 + \\left( \\frac{U_{1}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{2}}{\\partial th}}{r} \\right) e^2\\otimes e^2 + \\left( -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - \\sin\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} - \\frac{\\partial\\,U_{2}}{\\partial ph}}{2 \\, r \\sin\\left({\\theta}\\right)} \\right) e^2\\otimes e^3 + \\left( \\frac{{\\left(r \\frac{\\partial\\,U_{3}}{\\partial r} - U_{3}\\left(r, {\\theta}, {\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{1}}{\\partial ph}}{2 \\, r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^1 + \\left( -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - \\sin\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} - \\frac{\\partial\\,U_{2}}{\\partial ph}}{2 \\, r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^2 + \\left( \\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + U_{1}\\left(r, {\\theta}, {\\phi}\\right) \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^3$ " ], "text/plain": [ "E = d(U_1)/dr e^1*e^1 + 1/2*(r*d(U_2)/dr - U_2(r, th, ph) + d(U_1)/dth)/r e^1*e^2 + 1/2*((r*d(U_3)/dr - U_3(r, th, ph))*sin(th) + d(U_1)/dph)/(r*sin(th)) e^1*e^3 + 1/2*(r*d(U_2)/dr - U_2(r, th, ph) + d(U_1)/dth)/r e^2*e^1 + (U_1(r, th, ph) + d(U_2)/dth)/r e^2*e^2 - 1/2*(U_3(r, th, ph)*cos(th) - sin(th)*d(U_3)/dth - d(U_2)/dph)/(r*sin(th)) e^2*e^3 + 1/2*((r*d(U_3)/dr - U_3(r, th, ph))*sin(th) + d(U_1)/dph)/(r*sin(th)) e^3*e^1 - 1/2*(U_3(r, th, ph)*cos(th) - sin(th)*d(U_3)/dth - d(U_2)/dph)/(r*sin(th)) e^3*e^2 + (U_2(r, th, ph)*cos(th) + U_1(r, th, ph)*sin(th) + d(U_3)/dph)/(r*sin(th)) e^3*e^3" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.set_name('E', latex_name=r'\\varepsilon')\n", "E.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us display the components of $\\varepsilon$, skipping those that can be deduced by symmetry:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\varepsilon_{\\,1\\,1}^{\\phantom{\\, 1}\\phantom{\\, 1}} & = & \\frac{\\partial\\,U_{1}}{\\partial r} \\\\ \\varepsilon_{\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{r \\frac{\\partial\\,U_{2}}{\\partial r} - U_{2}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{1}}{\\partial th}}{2 \\, r} \\\\ \\varepsilon_{\\,1\\,3}^{\\phantom{\\, 1}\\phantom{\\, 3}} & = & \\frac{{\\left(r \\frac{\\partial\\,U_{3}}{\\partial r} - U_{3}\\left(r, {\\theta}, {\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{1}}{\\partial ph}}{2 \\, r \\sin\\left({\\theta}\\right)} \\\\ \\varepsilon_{\\,2\\,2}^{\\phantom{\\, 2}\\phantom{\\, 2}} & = & \\frac{U_{1}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{2}}{\\partial th}}{r} \\\\ \\varepsilon_{\\,2\\,3}^{\\phantom{\\, 2}\\phantom{\\, 3}} & = & -\\frac{U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - \\sin\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} - \\frac{\\partial\\,U_{2}}{\\partial ph}}{2 \\, r \\sin\\left({\\theta}\\right)} \\\\ \\varepsilon_{\\,3\\,3}^{\\phantom{\\, 3}\\phantom{\\, 3}} & = & \\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + U_{1}\\left(r, {\\theta}, {\\phi}\\right) \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\end{array}$ " ], "text/plain": [ "E_11 = d(U_1)/dr \n", "E_12 = 1/2*(r*d(U_2)/dr - U_2(r, th, ph) + d(U_1)/dth)/r \n", "E_13 = 1/2*((r*d(U_3)/dr - U_3(r, th, ph))*sin(th) + d(U_1)/dph)/(r*sin(th)) \n", "E_22 = (U_1(r, th, ph) + d(U_2)/dth)/r \n", "E_23 = -1/2*(U_3(r, th, ph)*cos(th) - sin(th)*d(U_3)/dth - d(U_2)/dph)/(r*sin(th)) \n", "E_33 = (U_2(r, th, ph)*cos(th) + U_1(r, th, ph)*sin(th) + d(U_3)/dph)/(r*sin(th)) " ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Stress tensor and Hooke's law\n", "\n", "To form the stress tensor according to Hooke's law, we introduce first the Lamé constants:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\lambda}$ " ], "text/plain": [ "ll" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('ll', latex_name=r'\\lambda')" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu}$ " ], "text/plain": [ "mu" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('mu', latex_name=r'\\mu')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The trace (with respect to $g$) of the bilinear form $\\varepsilon$ is obtained by (i) raising the first index (`pos=0`) by means of $g$ and (ii) by taking the trace of the resulting endomorphism:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "trE = E.up(g, pos=0).trace()\n", "print trE" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left(r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, U_{1}\\left(r, {\\theta}, {\\phi}\\right) + \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\end{array}$ " ], "text/plain": [ "M --> R\n", "(r, th, ph) |--> (U_2(r, th, ph)*cos(th) + (r*d(U_1)/dr + 2*U_1(r, th, ph) + d(U_2)/dth)*sin(th) + d(U_3)/dph)/(r*sin(th))" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "trE.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **stress tensor** $S$ is obtained via Hooke's law for isotropic material:\n", "$$ S = \\lambda \\, \\mathrm{tr}\\varepsilon \\; g + 2\\mu \\, \\varepsilon$$" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "S = ll*trE*g + 2*mu*E\n", "print S" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}S = \\left( \\frac{{\\lambda} U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, {\\lambda} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\lambda} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + {\\lambda} \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^1\\otimes e^1 + \\left( \\frac{{\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\mu} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}}{r} \\right) e^1\\otimes e^2 + \\left( \\frac{{\\left({\\mu} r \\frac{\\partial\\,U_{3}}{\\partial r} - {\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^1\\otimes e^3 + \\left( \\frac{{\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\mu} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}}{r} \\right) e^2\\otimes e^1 + \\left( \\frac{{\\lambda} U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left({\\lambda} r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, {\\left({\\lambda} + {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + {\\lambda} \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^2\\otimes e^2 + \\left( -\\frac{{\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - {\\mu} \\sin\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} - {\\mu} \\frac{\\partial\\,U_{2}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^2\\otimes e^3 + \\left( \\frac{{\\left({\\mu} r \\frac{\\partial\\,U_{3}}{\\partial r} - {\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^1 + \\left( -\\frac{{\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - {\\mu} \\sin\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} - {\\mu} \\frac{\\partial\\,U_{2}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^2 + \\left( \\frac{{\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left({\\lambda} r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, {\\left({\\lambda} + {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\lambda} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\right) e^3\\otimes e^3$ " ], "text/plain": [ "S = (ll*U_2(r, th, ph)*cos(th) + ((ll + 2*mu)*r*d(U_1)/dr + 2*ll*U_1(r, th, ph) + ll*d(U_2)/dth)*sin(th) + ll*d(U_3)/dph)/(r*sin(th)) e^1*e^1 + (mu*r*d(U_2)/dr - mu*U_2(r, th, ph) + mu*d(U_1)/dth)/r e^1*e^2 + ((mu*r*d(U_3)/dr - mu*U_3(r, th, ph))*sin(th) + mu*d(U_1)/dph)/(r*sin(th)) e^1*e^3 + (mu*r*d(U_2)/dr - mu*U_2(r, th, ph) + mu*d(U_1)/dth)/r e^2*e^1 + (ll*U_2(r, th, ph)*cos(th) + (ll*r*d(U_1)/dr + 2*(ll + mu)*U_1(r, th, ph) + (ll + 2*mu)*d(U_2)/dth)*sin(th) + ll*d(U_3)/dph)/(r*sin(th)) e^2*e^2 - (mu*U_3(r, th, ph)*cos(th) - mu*sin(th)*d(U_3)/dth - mu*d(U_2)/dph)/(r*sin(th)) e^2*e^3 + ((mu*r*d(U_3)/dr - mu*U_3(r, th, ph))*sin(th) + mu*d(U_1)/dph)/(r*sin(th)) e^3*e^1 - (mu*U_3(r, th, ph)*cos(th) - mu*sin(th)*d(U_3)/dth - mu*d(U_2)/dph)/(r*sin(th)) e^3*e^2 + ((ll + 2*mu)*U_2(r, th, ph)*cos(th) + (ll*r*d(U_1)/dr + 2*(ll + mu)*U_1(r, th, ph) + ll*d(U_2)/dth)*sin(th) + (ll + 2*mu)*d(U_3)/dph)/(r*sin(th)) e^3*e^3" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.set_name('S')\n", "S.display()" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} S_{\\,1\\,1}^{\\phantom{\\, 1}\\phantom{\\, 1}} & = & \\frac{{\\lambda} U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, {\\lambda} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\lambda} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + {\\lambda} \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\\\ S_{\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{{\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\mu} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}}{r} \\\\ S_{\\,1\\,3}^{\\phantom{\\, 1}\\phantom{\\, 3}} & = & \\frac{{\\left({\\mu} r \\frac{\\partial\\,U_{3}}{\\partial r} - {\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\\\ S_{\\,2\\,2}^{\\phantom{\\, 2}\\phantom{\\, 2}} & = & \\frac{{\\lambda} U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left({\\lambda} r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, {\\left({\\lambda} + {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + {\\lambda} \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\\\ S_{\\,2\\,3}^{\\phantom{\\, 2}\\phantom{\\, 3}} & = & -\\frac{{\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) - {\\mu} \\sin\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} - {\\mu} \\frac{\\partial\\,U_{2}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\\\ S_{\\,3\\,3}^{\\phantom{\\, 3}\\phantom{\\, 3}} & = & \\frac{{\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) \\cos\\left({\\theta}\\right) + {\\left({\\lambda} r \\frac{\\partial\\,U_{1}}{\\partial r} + 2 \\, {\\left({\\lambda} + {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\lambda} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{3}}{\\partial ph}}{r \\sin\\left({\\theta}\\right)} \\end{array}$ " ], "text/plain": [ "S_11 = (ll*U_2(r, th, ph)*cos(th) + ((ll + 2*mu)*r*d(U_1)/dr + 2*ll*U_1(r, th, ph) + ll*d(U_2)/dth)*sin(th) + ll*d(U_3)/dph)/(r*sin(th)) \n", "S_12 = (mu*r*d(U_2)/dr - mu*U_2(r, th, ph) + mu*d(U_1)/dth)/r \n", "S_13 = ((mu*r*d(U_3)/dr - mu*U_3(r, th, ph))*sin(th) + mu*d(U_1)/dph)/(r*sin(th)) \n", "S_22 = (ll*U_2(r, th, ph)*cos(th) + (ll*r*d(U_1)/dr + 2*(ll + mu)*U_1(r, th, ph) + (ll + 2*mu)*d(U_2)/dth)*sin(th) + ll*d(U_3)/dph)/(r*sin(th)) \n", "S_23 = -(mu*U_3(r, th, ph)*cos(th) - mu*sin(th)*d(U_3)/dth - mu*d(U_2)/dph)/(r*sin(th)) \n", "S_33 = ((ll + 2*mu)*U_2(r, th, ph)*cos(th) + (ll*r*d(U_1)/dr + 2*(ll + mu)*U_1(r, th, ph) + ll*d(U_2)/dth)*sin(th) + (ll + 2*mu)*d(U_3)/dph)/(r*sin(th)) " ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Each component can be accessed individually:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\mu} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}}{r}$ " ], "text/plain": [ "(mu*r*d(U_2)/dr - mu*U_2(r, th, ph) + mu*d(U_1)/dth)/r" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S[1,2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Divergence of the stress tensor\n", "\n", "The divergence of the stress tensor (with respect to $g$) is the 1-form:\n", "$$ f_i = \\nabla_j S^j_{\\ \\, i} $$\n", "In a next version of SageManifolds, there will be a function `divergence()`. For the moment, to evaluate $f$, \n", "we first form the tensor $S^j_{\\ \\, i}$ by raising the first index (`pos=0`) of $S$ with $g$:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (1,1) on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "SU = S.up(g, pos=0)\n", "print SU" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The divergence is obtained by taking the trace on the first index (`0`) and the third one (`2`) of the tensor\n", "$(\\nabla S)^j_{\\ \\, ik} = \\nabla_k S^j_{\\ \\, i}$:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form on the 3-dimensional differentiable manifold M\n" ] } ], "source": [ "divS = nabla(SU).trace(0,2)\n", "print divS" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}f = \\left( \\frac{{\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} r^{2} \\frac{\\partial^2\\,U_{1}}{\\partial r^2} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} r \\frac{\\partial\\,U_{1}}{\\partial r} + {\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{2}}{\\partial r\\partial th} - 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial^2\\,U_{1}}{\\partial th^2} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right)^{2} + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{3}}{\\partial r\\partial ph} + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}\\right)} \\cos\\left({\\theta}\\right) - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{3}}{\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial^2\\,U_{1}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}} \\right) e^1 + \\left( \\frac{{\\left({\\mu} r^{2} \\frac{\\partial^2\\,U_{2}}{\\partial r^2} + {\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{1}}{\\partial r\\partial th} + 2 \\, {\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{1}}{\\partial th} + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial^2\\,U_{2}}{\\partial th^2}\\right)} \\sin\\left({\\theta}\\right)^{2} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial ph} - {\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{2}}{\\partial th} + {\\left({\\lambda} + {\\mu}\\right)} \\frac{\\partial^2\\,U_{3}}{\\partial th\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial^2\\,U_{2}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}} \\right) e^2 + \\left( \\frac{{\\left({\\mu} r^{2} \\frac{\\partial^2\\,U_{3}}{\\partial r^2} + 2 \\, {\\mu} r \\frac{\\partial\\,U_{3}}{\\partial r} + {\\mu} \\frac{\\partial^2\\,U_{3}}{\\partial th^2}\\right)} \\sin\\left({\\theta}\\right)^{2} + {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{2}}{\\partial ph} - {\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{1}}{\\partial r\\partial ph} + {\\mu} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{1}}{\\partial ph} + {\\left({\\lambda} + {\\mu}\\right)} \\frac{\\partial^2\\,U_{2}}{\\partial th\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial^2\\,U_{3}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}} \\right) e^3$ " ], "text/plain": [ "f = (((ll + 2*mu)*r^2*d^2(U_1)/dr^2 + 2*(ll + 2*mu)*r*d(U_1)/dr + (ll + mu)*r*d^2(U_2)/drdth - 2*(ll + 2*mu)*U_1(r, th, ph) + mu*d^2(U_1)/dth^2 - (ll + 3*mu)*d(U_2)/dth)*sin(th)^2 + ((ll + mu)*r*d^2(U_3)/drdph + ((ll + mu)*r*d(U_2)/dr - (ll + 3*mu)*U_2(r, th, ph) + mu*d(U_1)/dth)*cos(th) - (ll + 3*mu)*d(U_3)/dph)*sin(th) + mu*d^2(U_1)/dph^2)/(r^2*sin(th)^2) e^1 + ((mu*r^2*d^2(U_2)/dr^2 + (ll + mu)*r*d^2(U_1)/drdth + 2*mu*r*d(U_2)/dr + 2*(ll + 2*mu)*d(U_1)/dth + (ll + 2*mu)*d^2(U_2)/dth^2)*sin(th)^2 - (ll + 3*mu)*cos(th)*d(U_3)/dph - (ll + 2*mu)*U_2(r, th, ph) + ((ll + 2*mu)*cos(th)*d(U_2)/dth + (ll + mu)*d^2(U_3)/dthdph)*sin(th) + mu*d^2(U_2)/dph^2)/(r^2*sin(th)^2) e^2 + ((mu*r^2*d^2(U_3)/dr^2 + 2*mu*r*d(U_3)/dr + mu*d^2(U_3)/dth^2)*sin(th)^2 + (ll + 3*mu)*cos(th)*d(U_2)/dph - mu*U_3(r, th, ph) + ((ll + mu)*r*d^2(U_1)/drdph + mu*cos(th)*d(U_3)/dth + 2*(ll + 2*mu)*d(U_1)/dph + (ll + mu)*d^2(U_2)/dthdph)*sin(th) + (ll + 2*mu)*d^2(U_3)/dph^2)/(r^2*sin(th)^2) e^3" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divS.set_name('f')\n", "divS.display()" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} f_{\\,1}^{\\phantom{\\, 1}} & = & \\frac{{\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} r^{2} \\frac{\\partial^2\\,U_{1}}{\\partial r^2} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} r \\frac{\\partial\\,U_{1}}{\\partial r} + {\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{2}}{\\partial r\\partial th} - 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial^2\\,U_{1}}{\\partial th^2} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right)^{2} + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{3}}{\\partial r\\partial ph} + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}\\right)} \\cos\\left({\\theta}\\right) - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{3}}{\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial^2\\,U_{1}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}} \\\\ f_{\\,2}^{\\phantom{\\, 2}} & = & \\frac{{\\left({\\mu} r^{2} \\frac{\\partial^2\\,U_{2}}{\\partial r^2} + {\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{1}}{\\partial r\\partial th} + 2 \\, {\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{1}}{\\partial th} + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial^2\\,U_{2}}{\\partial th^2}\\right)} \\sin\\left({\\theta}\\right)^{2} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial ph} - {\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{2}}{\\partial th} + {\\left({\\lambda} + {\\mu}\\right)} \\frac{\\partial^2\\,U_{3}}{\\partial th\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial^2\\,U_{2}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}} \\\\ f_{\\,3}^{\\phantom{\\, 3}} & = & \\frac{{\\left({\\mu} r^{2} \\frac{\\partial^2\\,U_{3}}{\\partial r^2} + 2 \\, {\\mu} r \\frac{\\partial\\,U_{3}}{\\partial r} + {\\mu} \\frac{\\partial^2\\,U_{3}}{\\partial th^2}\\right)} \\sin\\left({\\theta}\\right)^{2} + {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{2}}{\\partial ph} - {\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{1}}{\\partial r\\partial ph} + {\\mu} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{1}}{\\partial ph} + {\\left({\\lambda} + {\\mu}\\right)} \\frac{\\partial^2\\,U_{2}}{\\partial th\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial^2\\,U_{3}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}} \\end{array}$ " ], "text/plain": [ "f_1 = (((ll + 2*mu)*r^2*d^2(U_1)/dr^2 + 2*(ll + 2*mu)*r*d(U_1)/dr + (ll + mu)*r*d^2(U_2)/drdth - 2*(ll + 2*mu)*U_1(r, th, ph) + mu*d^2(U_1)/dth^2 - (ll + 3*mu)*d(U_2)/dth)*sin(th)^2 + ((ll + mu)*r*d^2(U_3)/drdph + ((ll + mu)*r*d(U_2)/dr - (ll + 3*mu)*U_2(r, th, ph) + mu*d(U_1)/dth)*cos(th) - (ll + 3*mu)*d(U_3)/dph)*sin(th) + mu*d^2(U_1)/dph^2)/(r^2*sin(th)^2) \n", "f_2 = ((mu*r^2*d^2(U_2)/dr^2 + (ll + mu)*r*d^2(U_1)/drdth + 2*mu*r*d(U_2)/dr + 2*(ll + 2*mu)*d(U_1)/dth + (ll + 2*mu)*d^2(U_2)/dth^2)*sin(th)^2 - (ll + 3*mu)*cos(th)*d(U_3)/dph - (ll + 2*mu)*U_2(r, th, ph) + ((ll + 2*mu)*cos(th)*d(U_2)/dth + (ll + mu)*d^2(U_3)/dthdph)*sin(th) + mu*d^2(U_2)/dph^2)/(r^2*sin(th)^2) \n", "f_3 = ((mu*r^2*d^2(U_3)/dr^2 + 2*mu*r*d(U_3)/dr + mu*d^2(U_3)/dth^2)*sin(th)^2 + (ll + 3*mu)*cos(th)*d(U_2)/dph - mu*U_3(r, th, ph) + ((ll + mu)*r*d^2(U_1)/drdph + mu*cos(th)*d(U_3)/dth + 2*(ll + 2*mu)*d(U_1)/dph + (ll + mu)*d^2(U_2)/dthdph)*sin(th) + (ll + 2*mu)*d^2(U_3)/dph^2)/(r^2*sin(th)^2) " ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divS.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that $f_1$ is quite badly displayed. We get a better view by displaying the components one by one:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} r^{2} \\frac{\\partial^2\\,U_{1}}{\\partial r^2} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} r \\frac{\\partial\\,U_{1}}{\\partial r} + {\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{2}}{\\partial r\\partial th} - 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{1}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial^2\\,U_{1}}{\\partial th^2} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{2}}{\\partial th}\\right)} \\sin\\left({\\theta}\\right)^{2} + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{3}}{\\partial r\\partial ph} + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial\\,U_{2}}{\\partial r} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\mu} \\frac{\\partial\\,U_{1}}{\\partial th}\\right)} \\cos\\left({\\theta}\\right) - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{3}}{\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial^2\\,U_{1}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}}$ " ], "text/plain": [ "(((ll + 2*mu)*r^2*d^2(U_1)/dr^2 + 2*(ll + 2*mu)*r*d(U_1)/dr + (ll + mu)*r*d^2(U_2)/drdth - 2*(ll + 2*mu)*U_1(r, th, ph) + mu*d^2(U_1)/dth^2 - (ll + 3*mu)*d(U_2)/dth)*sin(th)^2 + ((ll + mu)*r*d^2(U_3)/drdph + ((ll + mu)*r*d(U_2)/dr - (ll + 3*mu)*U_2(r, th, ph) + mu*d(U_1)/dth)*cos(th) - (ll + 3*mu)*d(U_3)/dph)*sin(th) + mu*d^2(U_1)/dph^2)/(r^2*sin(th)^2)" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divS[1]" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\mu} r^{2} \\frac{\\partial^2\\,U_{2}}{\\partial r^2} + {\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{1}}{\\partial r\\partial th} + 2 \\, {\\mu} r \\frac{\\partial\\,U_{2}}{\\partial r} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{1}}{\\partial th} + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial^2\\,U_{2}}{\\partial th^2}\\right)} \\sin\\left({\\theta}\\right)^{2} - {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial ph} - {\\left({\\lambda} + 2 \\, {\\mu}\\right)} U_{2}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{2}}{\\partial th} + {\\left({\\lambda} + {\\mu}\\right)} \\frac{\\partial^2\\,U_{3}}{\\partial th\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\mu} \\frac{\\partial^2\\,U_{2}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}}$ " ], "text/plain": [ "((mu*r^2*d^2(U_2)/dr^2 + (ll + mu)*r*d^2(U_1)/drdth + 2*mu*r*d(U_2)/dr + 2*(ll + 2*mu)*d(U_1)/dth + (ll + 2*mu)*d^2(U_2)/dth^2)*sin(th)^2 - (ll + 3*mu)*cos(th)*d(U_3)/dph - (ll + 2*mu)*U_2(r, th, ph) + ((ll + 2*mu)*cos(th)*d(U_2)/dth + (ll + mu)*d^2(U_3)/dthdph)*sin(th) + mu*d^2(U_2)/dph^2)/(r^2*sin(th)^2)" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divS[2]" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\mu} r^{2} \\frac{\\partial^2\\,U_{3}}{\\partial r^2} + 2 \\, {\\mu} r \\frac{\\partial\\,U_{3}}{\\partial r} + {\\mu} \\frac{\\partial^2\\,U_{3}}{\\partial th^2}\\right)} \\sin\\left({\\theta}\\right)^{2} + {\\left({\\lambda} + 3 \\, {\\mu}\\right)} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{2}}{\\partial ph} - {\\mu} U_{3}\\left(r, {\\theta}, {\\phi}\\right) + {\\left({\\left({\\lambda} + {\\mu}\\right)} r \\frac{\\partial^2\\,U_{1}}{\\partial r\\partial ph} + {\\mu} \\cos\\left({\\theta}\\right) \\frac{\\partial\\,U_{3}}{\\partial th} + 2 \\, {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial\\,U_{1}}{\\partial ph} + {\\left({\\lambda} + {\\mu}\\right)} \\frac{\\partial^2\\,U_{2}}{\\partial th\\partial ph}\\right)} \\sin\\left({\\theta}\\right) + {\\left({\\lambda} + 2 \\, {\\mu}\\right)} \\frac{\\partial^2\\,U_{3}}{\\partial ph^2}}{r^{2} \\sin\\left({\\theta}\\right)^{2}}$ " ], "text/plain": [ "((mu*r^2*d^2(U_3)/dr^2 + 2*mu*r*d(U_3)/dr + mu*d^2(U_3)/dth^2)*sin(th)^2 + (ll + 3*mu)*cos(th)*d(U_2)/dph - mu*U_3(r, th, ph) + ((ll + mu)*r*d^2(U_1)/drdph + mu*cos(th)*d(U_3)/dth + 2*(ll + 2*mu)*d(U_1)/dph + (ll + mu)*d^2(U_2)/dthdph)*sin(th) + (ll + 2*mu)*d^2(U_3)/dph^2)/(r^2*sin(th)^2)" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divS[3]" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 6.10.beta7", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }