{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Vector calculus with SageMath\n", "\n", "## Part 3: Using cylindrical coordinates\n", "\n", "This notebook illustrates some vector calculus capabilities of SageMath within the 3-dimensional Euclidean space. The corresponding tools have been developed within\n", "the [SageManifolds](https://sagemanifolds.obspm.fr) project.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Notebooks/SM_vector_calc_cylindrical.ipynb) to download the notebook file (ipynb format). To run it, you must start SageMath with the Jupyter interface, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 8.3 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.8.beta2, Release Date: 2019-04-14'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display math formulas using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The 3-dimensional Euclidean space\n", "\n", "We start by declaring the 3-dimensional Euclidean space $\\mathbb{E}^3$, with $(\\rho,\\phi,z)$ as cylindrical coordinates:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Euclidean space E^3\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbb{E}^{3}$ " ], "text/plain": [ "Euclidean space E^3" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E. = EuclideanSpace(coordinates='cylindrical')\n", "print(E)\n", "E" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\mathbb{E}^3$ is endowed with the chart of cylindrical coordinates:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3},({\\rho}, {\\phi}, z)\\right)\\right]$ " ], "text/plain": [ "[Chart (E^3, (rh, ph, z))]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "as well as with the associated orthonormal vector frame $(e_\\rho, e_\\phi, e_z)$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3}, \\left(\\frac{\\partial}{\\partial {\\rho} },\\frac{\\partial}{\\partial {\\phi} },\\frac{\\partial}{\\partial z }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(e_{ {\\rho} },e_{ {\\phi} },e_{ z }\\right)\\right)\\right]$ " ], "text/plain": [ "[Coordinate frame (E^3, (d/drh,d/dph,d/dz)),\n", " Vector frame (E^3, (e_rh,e_ph,e_z))]" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the above output, $\\left(\\frac{\\partial}{\\partial\\rho}, \\frac{\\partial}{\\partial\\phi}, \\frac{\\partial}{\\partial z}\\right)$ is the coordinate frame associated with $(\\rho,\\phi,z)$; it is not an orthonormal frame and will not be used below." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vector fields" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define a vector field on $\\mathbb{E}^3$ from its components in the orthonormal vector frame $(e_\\rho,e_\\phi,e_z)$:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = {\\rho} {\\left(\\sin\\left(2 \\, {\\phi}\\right) + 1\\right)} e_{ {\\rho} } + 2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} e_{ {\\phi} } + z e_{ z }$ " ], "text/plain": [ "v = rh*(sin(2*ph) + 1) e_rh + 2*rh*cos(ph)^2 e_ph + z e_z" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v = E.vector_field(rh*(1+sin(2*ph)), 2*rh*cos(ph)^2, z,\n", " name='v')\n", "v.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can access to the components of $v$ via the square bracket operator:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\rho} {\\left(\\sin\\left(2 \\, {\\phi}\\right) + 1\\right)}$ " ], "text/plain": [ "rh*(sin(2*ph) + 1)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v[1]" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[{\\rho} {\\left(\\sin\\left(2 \\, {\\phi}\\right) + 1\\right)}, 2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2}, z\\right]$ " ], "text/plain": [ "[rh*(sin(2*ph) + 1), 2*rh*cos(ph)^2, z]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A vector field can evaluated at any point of $\\mathbb{E}^3$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the Euclidean space E^3\n" ] } ], "source": [ "p = E((1, pi, 0), name='p')\n", "print(p)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1, \\pi, 0\\right)$ " ], "text/plain": [ "(1, pi, 0)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p.coordinates()" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector v at Point p on the Euclidean space E^3\n" ] } ], "source": [ "vp = v.at(p)\n", "print(vp)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = e_{ {\\rho} } + 2 e_{ {\\phi} }$ " ], "text/plain": [ "v = e_rh + 2 e_ph" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vp.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may define a vector field with generic components:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) e_{ {\\rho} } + u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) e_{ {\\phi} } + u_{z}\\left({\\rho}, {\\phi}, z\\right) e_{ z }$ " ], "text/plain": [ "u = u_rho(rh, ph, z) e_rh + u_phi(rh, ph, z) e_ph + u_z(rh, ph, z) e_z" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = E.vector_field(function('u_rho')(rh,ph,z),\n", " function('u_phi')(rh,ph,z),\n", " function('u_z')(rh,ph,z),\n", " name='u')\n", "u.display()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[u_{\\rho}\\left({\\rho}, {\\phi}, z\\right), u_{\\phi}\\left({\\rho}, {\\phi}, z\\right), u_{z}\\left({\\rho}, {\\phi}, z\\right)\\right]$ " ], "text/plain": [ "[u_rho(rh, ph, z), u_phi(rh, ph, z), u_z(rh, ph, z)]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u[:]" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = u_{\\rho}\\left(1, \\pi, 0\\right) e_{ {\\rho} } + u_{\\phi}\\left(1, \\pi, 0\\right) e_{ {\\phi} } + u_{z}\\left(1, \\pi, 0\\right) e_{ z }$ " ], "text/plain": [ "u = u_rho(1, pi, 0) e_rh + u_phi(1, pi, 0) e_ph + u_z(1, pi, 0) e_z" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "up = u.at(p)\n", "up.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Algebraic operations on vector fields\n", "\n", "### Dot product\n", "\n", "The dot (or scalar) product of the vector fields $u$ and $v$ is obtained by the method `dot_product`, which admits `dot` as a shortcut alias:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{u}\\cdot{v}$ " ], "text/plain": [ "Scalar field u.v on the Euclidean space E^3" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = u.dot(v)\n", "s" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field u.v on the Euclidean space E^3\n" ] } ], "source": [ "print(s)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$s= u\\cdot v$ is a *scalar field*, i.e. a map $\\mathbb{E}^3 \\rightarrow \\mathbb{R}$:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} {u}\\cdot{v}:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & 2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) + {\\left(2 \\, \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right) + 1\\right)} {\\rho} u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) + z u_{z}\\left({\\rho}, {\\phi}, z\\right) \\end{array}$ " ], "text/plain": [ "u.v: E^3 --> R\n", " (rh, ph, z) |--> 2*rh*cos(ph)^2*u_phi(rh, ph, z) + (2*cos(ph)*sin(ph) + 1)*rh*u_rho(rh, ph, z) + z*u_z(rh, ph, z)" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It maps points of $\\mathbb{E}^3$ to real numbers:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, u_{\\phi}\\left(1, \\pi, 0\\right) + u_{\\rho}\\left(1, \\pi, 0\\right)$ " ], "text/plain": [ "2*u_phi(1, pi, 0) + u_rho(1, pi, 0)" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Its coordinate expression is" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) + {\\left(2 \\, \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right) + 1\\right)} {\\rho} u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) + z u_{z}\\left({\\rho}, {\\phi}, z\\right)$ " ], "text/plain": [ "2*rh*cos(ph)^2*u_phi(rh, ph, z) + (2*cos(ph)*sin(ph) + 1)*rh*u_rho(rh, ph, z) + z*u_z(rh, ph, z)" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Norm\n", "\n", "The norm of a vector field is " ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\|u\\right\\|$ " ], "text/plain": [ "Scalar field |u| on the Euclidean space E^3" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = norm(u)\n", "s" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\left\\|u\\right\\|:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & \\sqrt{u_{\\phi}\\left({\\rho}, {\\phi}, z\\right)^{2} + u_{\\rho}\\left({\\rho}, {\\phi}, z\\right)^{2} + u_{z}\\left({\\rho}, {\\phi}, z\\right)^{2}} \\end{array}$ " ], "text/plain": [ "|u|: E^3 --> R\n", " (rh, ph, z) |--> sqrt(u_phi(rh, ph, z)^2 + u_rho(rh, ph, z)^2 + u_z(rh, ph, z)^2)" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display()" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{u_{\\phi}\\left({\\rho}, {\\phi}, z\\right)^{2} + u_{\\rho}\\left({\\rho}, {\\phi}, z\\right)^{2} + u_{z}\\left({\\rho}, {\\phi}, z\\right)^{2}}$ " ], "text/plain": [ "sqrt(u_phi(rh, ph, z)^2 + u_rho(rh, ph, z)^2 + u_z(rh, ph, z)^2)" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The norm is related to the dot product by $\\|u\\|^2 = u\\cdot u$, as we can check:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "norm(u)^2 == u.dot(u)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $v$, we have" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{{\\left(4 \\, \\cos\\left({\\phi}\\right)^{2} + 4 \\, \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right) + 1\\right)} {\\rho}^{2} + z^{2}}$ " ], "text/plain": [ "sqrt((4*cos(ph)^2 + 4*cos(ph)*sin(ph) + 1)*rh^2 + z^2)" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "norm(v).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Cross product\n", "\n", "The cross product of $u$ by $v$ is obtained by the method `cross_product`, which admits `cross` as a shortcut alias:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field u x v on the Euclidean space E^3\n" ] } ], "source": [ "s = u.cross(v)\n", "print(s)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{u}\\times{v} = \\left( -2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} u_{z}\\left({\\rho}, {\\phi}, z\\right) + z u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) \\right) e_{ {\\rho} } + \\left( {\\left(2 \\, \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right) + 1\\right)} {\\rho} u_{z}\\left({\\rho}, {\\phi}, z\\right) - z u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) \\right) e_{ {\\phi} } + \\left( 2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) - {\\left(2 \\, \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right) + 1\\right)} {\\rho} u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) \\right) e_{ z }$ " ], "text/plain": [ "u x v = (-2*rh*cos(ph)^2*u_z(rh, ph, z) + z*u_phi(rh, ph, z)) e_rh + ((2*cos(ph)*sin(ph) + 1)*rh*u_z(rh, ph, z) - z*u_rho(rh, ph, z)) e_ph + (2*rh*cos(ph)^2*u_rho(rh, ph, z) - (2*cos(ph)*sin(ph) + 1)*rh*u_phi(rh, ph, z)) e_z" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Scalar triple product\n", "\n", "Let us introduce a third vector field. As a example, we do not pass the components as arguments of `vector_field`, as we did for $u$ and $v$; instead, we set them in a second stage, via the square bracket operator, any unset component being assumed to be zero:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}w = {\\rho} e_{ {\\rho} } + z e_{ z }$ " ], "text/plain": [ "w = rh e_rh + z e_z" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w = E.vector_field(name='w')\n", "w[1] = rh\n", "w[3] = z\n", "w.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The scalar triple product of the vector fields $u$, $v$ and $w$ is obtained as follows: " ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field epsilon(u,v,w) on the Euclidean space E^3\n" ] } ], "source": [ "triple_product = E.scalar_triple_product()\n", "s = triple_product(u, v, w)\n", "print(s)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\rho}^{2} \\cos\\left({\\phi}\\right)^{2} u_{z}\\left({\\rho}, {\\phi}, z\\right) - 2 \\, {\\left({\\rho} \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right) u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) - {\\rho} \\cos\\left({\\phi}\\right)^{2} u_{\\rho}\\left({\\rho}, {\\phi}, z\\right)\\right)} z$ " ], "text/plain": [ "-2*rh^2*cos(ph)^2*u_z(rh, ph, z) - 2*(rh*cos(ph)*sin(ph)*u_phi(rh, ph, z) - rh*cos(ph)^2*u_rho(rh, ph, z))*z" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that the scalar triple product of $u$, $v$ and $w$ is $u\\cdot(v\\times w)$:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s == u.dot(v.cross(w))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Differential operators" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "While the standard operators $\\mathrm{grad}$, $\\mathrm{div}$, $\\mathrm{curl}$, etc. involved in vector calculus are accessible via the dot notation (e.g. `v.div()`), let us import functions `grad`, `div`, `curl`, etc. that allow for using standard mathematical notations\n", "(e.g. `div(v)`):" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [], "source": [ "from sage.manifolds.operators import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Gradient of a scalar field\n", "\n", "We first introduce a scalar field, via its expression in terms of Cartesian coordinates; in this example, we consider a unspecified function of $(\\rho,\\phi,z)$:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & f\\left({\\rho}, {\\phi}, z\\right) \\end{array}$ " ], "text/plain": [ "F: E^3 --> R\n", " (rh, ph, z) |--> f(rh, ph, z)" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "F = E.scalar_field(function('f')(rh,ph,z), name='F')\n", "F.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The value of $F$ at a point:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}f\\left(1, \\pi, 0\\right)$ " ], "text/plain": [ "f(1, pi, 0)" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "F(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The gradient of $F$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field grad(F) on the Euclidean space E^3\n" ] } ], "source": [ "print(grad(F))" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{grad}\\left(F\\right) = \\frac{\\partial\\,f}{\\partial {\\rho}} e_{ {\\rho} } + \\frac{\\frac{\\partial\\,f}{\\partial {\\phi}}}{{\\rho}} e_{ {\\phi} } + \\frac{\\partial\\,f}{\\partial z} e_{ z }$ " ], "text/plain": [ "grad(F) = d(f)/drh e_rh + d(f)/dph/rh e_ph + d(f)/dz e_z" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "grad(F).display()" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\left\\|\\mathrm{grad}\\left(F\\right)\\right\\|:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & \\frac{\\sqrt{{\\rho}^{2} \\left(\\frac{\\partial\\,f}{\\partial {\\rho}}\\right)^{2} + {\\rho}^{2} \\left(\\frac{\\partial\\,f}{\\partial z}\\right)^{2} + \\left(\\frac{\\partial\\,f}{\\partial {\\phi}}\\right)^{2}}}{{\\rho}} \\end{array}$ " ], "text/plain": [ "|grad(F)|: E^3 --> R\n", " (rh, ph, z) |--> sqrt(rh^2*(d(f)/drh)^2 + rh^2*(d(f)/dz)^2 + (d(f)/dph)^2)/rh" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "norm(grad(F)).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Divergence \n", "\n", "The divergence of a vector field:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{div}\\left(u\\right):& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & \\frac{{\\rho} \\frac{\\partial\\,u_{\\rho}}{\\partial {\\rho}} + {\\rho} \\frac{\\partial\\,u_{z}}{\\partial z} + u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) + \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}}}{{\\rho}} \\end{array}$ " ], "text/plain": [ "div(u): E^3 --> R\n", " (rh, ph, z) |--> (rh*d(u_rho)/drh + rh*d(u_z)/dz + u_rho(rh, ph, z) + d(u_phi)/dph)/rh" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = div(u)\n", "s.display()" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{u_{\\rho}\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}} + \\frac{\\frac{\\partial}{\\partial {\\phi}}u_{\\phi}\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}} + \\frac{\\partial}{\\partial {\\rho}}u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) + \\frac{\\partial}{\\partial z}u_{z}\\left({\\rho}, {\\phi}, z\\right)$ " ], "text/plain": [ "u_rho(rh, ph, z)/rh + diff(u_phi(rh, ph, z), ph)/rh + diff(u_rho(rh, ph, z), rh) + diff(u_z(rh, ph, z), z)" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr().expand()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $v$ and $w$, we have" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3$ " ], "text/plain": [ "3" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "div(v).expr()" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3$ " ], "text/plain": [ "3" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "div(w).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An identity valid for any scalar field $F$ and any vector field $u$:" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "div(F*u) == F*div(u) + u.dot(grad(F))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Curl\n", "\n", "The curl of a vector field:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field curl(u) on the Euclidean space E^3\n" ] } ], "source": [ "s = curl(u)\n", "print(s)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{curl}\\left(u\\right) = \\left( -\\frac{{\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial z} - \\frac{\\partial\\,u_{z}}{\\partial {\\phi}}}{{\\rho}} \\right) e_{ {\\rho} } + \\left( \\frac{\\partial\\,u_{\\rho}}{\\partial z} - \\frac{\\partial\\,u_{z}}{\\partial {\\rho}} \\right) e_{ {\\phi} } + \\left( \\frac{{\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}} + u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) - \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}} \\right) e_{ z }$ " ], "text/plain": [ "curl(u) = -(rh*d(u_phi)/dz - d(u_z)/dph)/rh e_rh + (d(u_rho)/dz - d(u_z)/drh) e_ph + (rh*d(u_phi)/drh + u_phi(rh, ph, z) - d(u_rho)/dph)/rh e_z" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To use the notation `rot` instead of `curl`, simply do" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [], "source": [ "rot = curl" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An alternative is" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "from sage.manifolds.operators import curl as rot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have then" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{curl}\\left(u\\right) = \\left( -\\frac{{\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial z} - \\frac{\\partial\\,u_{z}}{\\partial {\\phi}}}{{\\rho}} \\right) e_{ {\\rho} } + \\left( \\frac{\\partial\\,u_{\\rho}}{\\partial z} - \\frac{\\partial\\,u_{z}}{\\partial {\\rho}} \\right) e_{ {\\phi} } + \\left( \\frac{{\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}} + u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) - \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}} \\right) e_{ z }$ " ], "text/plain": [ "curl(u) = -(rh*d(u_phi)/dz - d(u_z)/dph)/rh e_rh + (d(u_rho)/dz - d(u_z)/drh) e_ph + (rh*d(u_phi)/drh + u_phi(rh, ph, z) - d(u_rho)/dph)/rh e_z" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rot(u).display()" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rot(u) == curl(u)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $v$ and $w$, we have" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{curl}\\left(v\\right) = 2 e_{ z }$ " ], "text/plain": [ "curl(v) = 2 e_z" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "curl(v).display()" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{curl}\\left(w\\right) = 0$ " ], "text/plain": [ "curl(w) = 0" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "curl(w).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The curl of a gradient is always zero:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{curl}\\left(\\mathrm{grad}\\left(F\\right)\\right) = 0$ " ], "text/plain": [ "curl(grad(F)) = 0" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "curl(grad(F)).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The divergence of a curl is always zero:" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{div}\\left(\\mathrm{curl}\\left(u\\right)\\right):& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & 0 \\end{array}$ " ], "text/plain": [ "div(curl(u)): E^3 --> R\n", " (rh, ph, z) |--> 0" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "div(curl(u)).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An identity valid for any scalar field $F$ and any vector field $u$:" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "curl(F*u) == grad(F).cross(u) + F*curl(u)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Laplacian" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Laplacian of a scalar field:" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Delta\\left(F\\right):& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & \\frac{{\\rho}^{2} \\frac{\\partial^2\\,f}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,f}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,f}{\\partial {\\rho}} + \\frac{\\partial^2\\,f}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\end{array}$ " ], "text/plain": [ "Delta(F): E^3 --> R\n", " (rh, ph, z) |--> (rh^2*d^2(f)/drh^2 + rh^2*d^2(f)/dz^2 + rh*d(f)/drh + d^2(f)/dph^2)/rh^2" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = laplacian(F)\n", "s.display()" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\frac{\\partial}{\\partial {\\rho}}f\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}} + \\frac{\\frac{\\partial^{2}}{(\\partial {\\phi})^{2}}f\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}^{2}} + \\frac{\\partial^{2}}{(\\partial {\\rho})^{2}}f\\left({\\rho}, {\\phi}, z\\right) + \\frac{\\partial^{2}}{(\\partial z)^{2}}f\\left({\\rho}, {\\phi}, z\\right)$ " ], "text/plain": [ "diff(f(rh, ph, z), rh)/rh + diff(f(rh, ph, z), ph, ph)/rh^2 + diff(f(rh, ph, z), rh, rh) + diff(f(rh, ph, z), z, z)" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr().expand()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a scalar field, the Laplacian is nothing but the divergence of the gradient:" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "laplacian(F) == div(grad(F))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Laplacian of a vector field:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Delta\\left(u\\right) = \\left( \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{\\rho}}{\\partial {\\rho}} - u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) - 2 \\, \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\right) e_{ {\\rho} } + \\left( \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{\\phi}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}} - u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) + \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi} ^ 2} + 2 \\, \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}^{2}} \\right) e_{ {\\phi} } + \\left( \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{z}}{\\partial {\\rho}} + \\frac{\\partial^2\\,u_{z}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\right) e_{ z }$ " ], "text/plain": [ "Delta(u) = (rh^2*d^2(u_rho)/drh^2 + rh^2*d^2(u_rho)/dz^2 + rh*d(u_rho)/drh - u_rho(rh, ph, z) - 2*d(u_phi)/dph + d^2(u_rho)/dph^2)/rh^2 e_rh + (rh^2*d^2(u_phi)/drh^2 + rh^2*d^2(u_phi)/dz^2 + rh*d(u_phi)/drh - u_phi(rh, ph, z) + d^2(u_phi)/dph^2 + 2*d(u_rho)/dph)/rh^2 e_ph + (rh^2*d^2(u_z)/drh^2 + rh^2*d^2(u_z)/dz^2 + rh*d(u_z)/drh + d^2(u_z)/dph^2)/rh^2 e_z" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Du = laplacian(u)\n", "Du.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since this expression is quite lengthy, we may ask for a display component by component:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Delta\\left(u\\right)_{\\phantom{\\, 1}}^{\\,1} & = & \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{\\rho}}{\\partial {\\rho}} - u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) - 2 \\, \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\\\ \\Delta\\left(u\\right)_{\\phantom{\\, 2}}^{\\,2} & = & \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{\\phi}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}} - u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) + \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi} ^ 2} + 2 \\, \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}^{2}} \\\\ \\Delta\\left(u\\right)_{\\phantom{\\, 3}}^{\\,3} & = & \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{z}}{\\partial {\\rho}} + \\frac{\\partial^2\\,u_{z}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\end{array}$ " ], "text/plain": [ "Delta(u)^1 = (rh^2*d^2(u_rho)/drh^2 + rh^2*d^2(u_rho)/dz^2 + rh*d(u_rho)/drh - u_rho(rh, ph, z) - 2*d(u_phi)/dph + d^2(u_rho)/dph^2)/rh^2 \n", "Delta(u)^2 = (rh^2*d^2(u_phi)/drh^2 + rh^2*d^2(u_phi)/dz^2 + rh*d(u_phi)/drh - u_phi(rh, ph, z) + d^2(u_phi)/dph^2 + 2*d(u_rho)/dph)/rh^2 \n", "Delta(u)^3 = (rh^2*d^2(u_z)/drh^2 + rh^2*d^2(u_z)/dz^2 + rh*d(u_z)/drh + d^2(u_z)/dph^2)/rh^2 " ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Du.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may expand each component:" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Delta\\left(u\\right)_{\\phantom{\\, 1}}^{\\,1} & = & \\frac{\\frac{\\partial\\,u_{\\rho}}{\\partial {\\rho}}}{{\\rho}} - \\frac{u_{\\rho}\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}^{2}} - \\frac{2 \\, \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}}}{{\\rho}^{2}} + \\frac{\\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho} ^ 2} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial z ^ 2} \\\\ \\Delta\\left(u\\right)_{\\phantom{\\, 2}}^{\\,2} & = & \\frac{\\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}}}{{\\rho}} - \\frac{u_{\\phi}\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}^{2}} + \\frac{\\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} + \\frac{2 \\, \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}^{2}} + \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho} ^ 2} + \\frac{\\partial^2\\,u_{\\phi}}{\\partial z ^ 2} \\\\ \\Delta\\left(u\\right)_{\\phantom{\\, 3}}^{\\,3} & = & \\frac{\\frac{\\partial\\,u_{z}}{\\partial {\\rho}}}{{\\rho}} + \\frac{\\frac{\\partial^2\\,u_{z}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} + \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho} ^ 2} + \\frac{\\partial^2\\,u_{z}}{\\partial z ^ 2} \\end{array}$ " ], "text/plain": [ "Delta(u)^1 = d(u_rho)/drh/rh - u_rho(rh, ph, z)/rh^2 - 2*d(u_phi)/dph/rh^2 + d^2(u_rho)/dph^2/rh^2 + d^2(u_rho)/drh^2 + d^2(u_rho)/dz^2 \n", "Delta(u)^2 = d(u_phi)/drh/rh - u_phi(rh, ph, z)/rh^2 + d^2(u_phi)/dph^2/rh^2 + 2*d(u_rho)/dph/rh^2 + d^2(u_phi)/drh^2 + d^2(u_phi)/dz^2 \n", "Delta(u)^3 = d(u_z)/drh/rh + d^2(u_z)/dph^2/rh^2 + d^2(u_z)/drh^2 + d^2(u_z)/dz^2 " ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "for i in E.irange():\n", " Du[i].expand()\n", "Du.display_comp()" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\frac{\\partial\\,u_{\\rho}}{\\partial {\\rho}}}{{\\rho}} - \\frac{u_{\\rho}\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}^{2}} - \\frac{2 \\, \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}}}{{\\rho}^{2}} + \\frac{\\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho} ^ 2} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial z ^ 2}$ " ], "text/plain": [ "d(u_rho)/drh/rh - u_rho(rh, ph, z)/rh^2 - 2*d(u_phi)/dph/rh^2 + d^2(u_rho)/dph^2/rh^2 + d^2(u_rho)/drh^2 + d^2(u_rho)/dz^2" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Du[1]" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}}}{{\\rho}} - \\frac{u_{\\phi}\\left({\\rho}, {\\phi}, z\\right)}{{\\rho}^{2}} + \\frac{\\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} + \\frac{2 \\, \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}^{2}} + \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho} ^ 2} + \\frac{\\partial^2\\,u_{\\phi}}{\\partial z ^ 2}$ " ], "text/plain": [ "d(u_phi)/drh/rh - u_phi(rh, ph, z)/rh^2 + d^2(u_phi)/dph^2/rh^2 + 2*d(u_rho)/dph/rh^2 + d^2(u_phi)/drh^2 + d^2(u_phi)/dz^2" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Du[2]" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\frac{\\partial\\,u_{z}}{\\partial {\\rho}}}{{\\rho}} + \\frac{\\frac{\\partial^2\\,u_{z}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} + \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho} ^ 2} + \\frac{\\partial^2\\,u_{z}}{\\partial z ^ 2}$ " ], "text/plain": [ "d(u_z)/drh/rh + d^2(u_z)/dph^2/rh^2 + d^2(u_z)/drh^2 + d^2(u_z)/dz^2" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Du[3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a test, we may check that these formulas coincide with those of\n", "Wikipedia's article [*Del in cylindrical and spherical coordinates*](https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Del_formula)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For $v$ and $w$, we have " ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Delta\\left(v\\right) = 0$ " ], "text/plain": [ "Delta(v) = 0" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "laplacian(v).display()" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Delta\\left(w\\right) = 0$ " ], "text/plain": [ "Delta(w) = 0" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "laplacian(w).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{curl}\\left(\\mathrm{curl}\\left(u\\right)\\right) = \\left( -\\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial z ^ 2} - {\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho}\\partial z} - {\\rho} \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho}\\partial {\\phi}} - \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}} + \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\right) e_{ {\\rho} } + \\left( -\\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{\\phi}}{\\partial z ^ 2} + {\\rho} \\frac{\\partial\\,u_{\\phi}}{\\partial {\\rho}} - {\\rho} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho}\\partial {\\phi}} - {\\rho} \\frac{\\partial^2\\,u_{z}}{\\partial {\\phi}\\partial z} - u_{\\phi}\\left({\\rho}, {\\phi}, z\\right) + \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}^{2}} \\right) e_{ {\\phi} } + \\left( \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho}\\partial z} - {\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho} ^ 2} + {\\rho} \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi}\\partial z} + {\\rho} \\frac{\\partial\\,u_{\\rho}}{\\partial z} - {\\rho} \\frac{\\partial\\,u_{z}}{\\partial {\\rho}} - \\frac{\\partial^2\\,u_{z}}{\\partial {\\phi} ^ 2}}{{\\rho}^{2}} \\right) e_{ z }$ " ], "text/plain": [ "curl(curl(u)) = -(rh^2*d^2(u_rho)/dz^2 - rh^2*d^2(u_z)/drhdz - rh*d^2(u_phi)/drhdph - d(u_phi)/dph + d^2(u_rho)/dph^2)/rh^2 e_rh - (rh^2*d^2(u_phi)/drh^2 + rh^2*d^2(u_phi)/dz^2 + rh*d(u_phi)/drh - rh*d^2(u_rho)/drhdph - rh*d^2(u_z)/dphdz - u_phi(rh, ph, z) + d(u_rho)/dph)/rh^2 e_ph + (rh^2*d^2(u_rho)/drhdz - rh^2*d^2(u_z)/drh^2 + rh*d^2(u_phi)/dphdz + rh*d(u_rho)/dz - rh*d(u_z)/drh - d^2(u_z)/dph^2)/rh^2 e_z" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "curl(curl(u)).display()" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{grad}\\left(\\mathrm{div}\\left(u\\right)\\right) = \\left( \\frac{{\\rho}^{2} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho} ^ 2} + {\\rho}^{2} \\frac{\\partial^2\\,u_{z}}{\\partial {\\rho}\\partial z} + {\\rho} \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\rho}\\partial {\\phi}} + {\\rho} \\frac{\\partial\\,u_{\\rho}}{\\partial {\\rho}} - u_{\\rho}\\left({\\rho}, {\\phi}, z\\right) - \\frac{\\partial\\,u_{\\phi}}{\\partial {\\phi}}}{{\\rho}^{2}} \\right) e_{ {\\rho} } + \\left( \\frac{{\\rho} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho}\\partial {\\phi}} + {\\rho} \\frac{\\partial^2\\,u_{z}}{\\partial {\\phi}\\partial z} + \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi} ^ 2} + \\frac{\\partial\\,u_{\\rho}}{\\partial {\\phi}}}{{\\rho}^{2}} \\right) e_{ {\\phi} } + \\left( \\frac{{\\rho} \\frac{\\partial^2\\,u_{\\rho}}{\\partial {\\rho}\\partial z} + {\\rho} \\frac{\\partial^2\\,u_{z}}{\\partial z ^ 2} + \\frac{\\partial^2\\,u_{\\phi}}{\\partial {\\phi}\\partial z} + \\frac{\\partial\\,u_{\\rho}}{\\partial z}}{{\\rho}} \\right) e_{ z }$ " ], "text/plain": [ "grad(div(u)) = (rh^2*d^2(u_rho)/drh^2 + rh^2*d^2(u_z)/drhdz + rh*d^2(u_phi)/drhdph + rh*d(u_rho)/drh - u_rho(rh, ph, z) - d(u_phi)/dph)/rh^2 e_rh + (rh*d^2(u_rho)/drhdph + rh*d^2(u_z)/dphdz + d^2(u_phi)/dph^2 + d(u_rho)/dph)/rh^2 e_ph + (rh*d^2(u_rho)/drhdz + rh*d^2(u_z)/dz^2 + d^2(u_phi)/dphdz + d(u_rho)/dz)/rh e_z" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "grad(div(u)).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we may check a famous identity:" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "curl(curl(u)) == grad(div(u)) - laplacian(u)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Customizations\n", "\n", "### Customizing the symbols of the orthonormal frame vectors\n", "\n", "By default, the vectors of the orthonormal frame associated with cylindrical coordinates are denoted $(e_\\rho,e_\\phi,z)$:" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(e_{ {\\rho} },e_{ {\\phi} },e_{ z }\\right)\\right)$ " ], "text/plain": [ "Vector frame (E^3, (e_rh,e_ph,e_z))" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "frame = E.cylindrical_frame()\n", "frame" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "But this can be changed, thanks to the method `set_name`:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(a_{\\rho},a_{\\phi},a_{z}\\right)\\right)$ " ], "text/plain": [ "Vector frame (E^3, (a_rh,a_ph,a_z))" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "frame.set_name('a', indices=('rh', 'ph', 'z'),\n", " latex_indices=(r'\\rho', r'\\phi', 'z'))\n", "frame" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = {\\rho} {\\left(\\sin\\left(2 \\, {\\phi}\\right) + 1\\right)} a_{\\rho} + 2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} a_{\\phi} + z a_{z}$ " ], "text/plain": [ "v = rh*(sin(2*ph) + 1) a_rh + 2*rh*cos(ph)^2 a_ph + z a_z" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display()" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(\\hat{\\rho},\\hat{\\phi},\\hat{z}\\right)\\right)$ " ], "text/plain": [ "Vector frame (E^3, (hrh,hph,hz))" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "frame.set_name(('hrh', 'hph', 'hz'), \n", " latex_symbol=(r'\\hat{\\rho}', r'\\hat{\\phi}', r'\\hat{z}'))\n", "frame" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = {\\rho} {\\left(\\sin\\left(2 \\, {\\phi}\\right) + 1\\right)} \\hat{\\rho} + 2 \\, {\\rho} \\cos\\left({\\phi}\\right)^{2} \\hat{\\phi} + z \\hat{z}$ " ], "text/plain": [ "v = rh*(sin(2*ph) + 1) hrh + 2*rh*cos(ph)^2 hph + z hz" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Customizing the coordinate symbols\n", "\n", "The coordinates symbols are defined within the angle brackets `<...>` at the construction of the Euclidean space. Above we did " ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [], "source": [ "E. = EuclideanSpace(coordinates='cylindrical')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which resulted in the coordinate symbols $(\\rho,\\phi,z)$ and in the corresponding Python variables `rh`, `ph` and `z` (SageMath symbolic expressions). Using other symbols, for instance $(R,\\Phi,Z)$, is possible through the optional argument `symbols` of the function `EuclideanSpace`. It has to be a string, usually prefixed by `r` (for *raw* string, in order to allow for the backslash character of LaTeX expressions). This string contains the coordinate fields separated by a blank space; each field contains the coordinate’s text symbol and possibly the coordinate’s LaTeX symbol (when the latter is different from the text symbol), both symbols being separated by a colon (`:`):" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [], "source": [ "E. = EuclideanSpace(coordinates='cylindrical', symbols=r'R Ph:\\Phi Z')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have then" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3},(R, {\\Phi}, Z)\\right)\\right]$ " ], "text/plain": [ "[Chart (E^3, (R, Ph, Z))]" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.atlas()" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3}, \\left(\\frac{\\partial}{\\partial R },\\frac{\\partial}{\\partial {\\Phi} },\\frac{\\partial}{\\partial Z }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(e_{ R },e_{ {\\Phi} },e_{ Z }\\right)\\right)\\right]$ " ], "text/plain": [ "[Coordinate frame (E^3, (d/dR,d/dPh,d/dZ)), Vector frame (E^3, (e_R,e_Ph,e_Z))]" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.frames()" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(e_{ R },e_{ {\\Phi} },e_{ Z }\\right)\\right)$ " ], "text/plain": [ "Vector frame (E^3, (e_R,e_Ph,e_Z))" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.cylindrical_frame()" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = R {\\left(\\sin\\left(2 \\, {\\Phi}\\right) + 1\\right)} e_{ R } + 2 \\, R \\cos\\left({\\Phi}\\right)^{2} e_{ {\\Phi} } + Z e_{ Z }$ " ], "text/plain": [ "v = R*(sin(2*Ph) + 1) e_R + 2*R*cos(Ph)^2 e_Ph + Z e_Z" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v = E.vector_field(R*(1+sin(2*Ph)), 2*R*cos(Ph)^2, Z,\n", " name='v')\n", "v.display()" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.8.beta2", "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": "2.7.15" } }, "nbformat": 4, "nbformat_minor": 2 }