{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<script async src=\"https://www.googletagmanager.com/gtag/js?id=UA-59152712-8\"></script>\n",
"<script>\n",
" window.dataLayer = window.dataLayer || [];\n",
" function gtag(){dataLayer.push(arguments);}\n",
" gtag('js', new Date());\n",
"\n",
" gtag('config', 'UA-59152712-8');\n",
"</script>\n",
"\n",
"# Time Evolution of Maxwell's Equations & Constraints in Flat Spacetime and Cartesian Coordinates\n",
"\n",
"## Authors: Terrence Pierre Jacques, Zachariah Etienne and Ian Ruchlin\n",
"\n",
"### This module constructs the evolution equations for Maxwell's equations as symbolic (SymPy) expressions, for an electromagnetic field in vacuum, as defined in [Tutorial-VacuumMaxwell_formulation_Cartesian](Tutorial-VacuumMaxwell_formulation_Cartesian.ipynb).\n",
"\n",
"**Notebook Status:** <font color='green'><b> Validated </b></font>\n",
"\n",
"**Validation Notes:** All expressions generated in this module have been validated.\n",
"\n",
"### NRPy+ Source Code for this module: [Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py](../edit/Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py)\n",
"\n",
"[comment]: <> (Introduction: TODO)\n",
"\n",
"This tutorial takes the expressions defined in [Tutorial-VacuumMaxwell_formulation_Cartesian](Tutorial-VacuumMaxwell_formulation_Cartesian.ipynb), and constructs them using NRPy+ in SymPy expressions."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='toc'></a>\n",
"\n",
"# Table of Contents\n",
"$$\\label{toc}$$\n",
"\n",
"1. [Step 1](#initializenrpy): Initialize needed Python/NRPy+ modules\n",
"1. [Step 2](#system1): System 1\n",
" 1. [Step 2.a](#system1_rhs): Construct evolution equations\n",
" 1. [Step 2.b](#system1_val): Code Validation\n",
"1. [Step 3](#system2): System 2\n",
" 1. [Step 3.a](#system2_rhs): Construct evolution equations\n",
" 1. [Step 3.b](#system2_val): Code Validation\n",
"1. [Step 4](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='initializenrpy'></a>\n",
"\n",
"# Step 1: Initialize needed Python/NRPy+ modules \\[Back to [top](#toc)\\]\n",
"\n",
"$$\\label{initializenrpy}$$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"execution": {
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"shell.execute_reply": "2021-03-07T17:18:46.624947Z"
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"source": [
"# Import needed Python modules\n",
"import NRPy_param_funcs as par # NRPy+: Parameter interface\n",
"import indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\n",
"import grid as gri # NRPy+: Functions having to do with numerical grids\n",
"\n",
"#Step 0: Set the spatial dimension parameter to 3.\n",
"par.set_parval_from_str(\"grid::DIM\", 3)\n",
"DIM = par.parval_from_str(\"grid::DIM\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='system1'></a>\n",
"\n",
"# Step 2: System I \\[Back to [top](#toc)\\]\n",
"$$\\label{system1}$$\n",
"\n",
"<a id='system1_rhs'></a>\n",
"\n",
"## Step 2.a: Construct evolution equations \\[Back to [top](#toc)\\]\n",
"$$\\label{system1_rhs}$$\n",
"\n",
"Following our derivations in [Tutorial-VacuumMaxwell_formulation_Cartesian](Tutorial-VacuumMaxwell_formulation_Cartesian.ipynb), we construct the system of equations for System I, in flat space and cartesian coordinates:\n",
"\n",
"\\begin{align}\n",
"\\partial_t A^i &= -E^i - \\partial^i \\varphi, \\\\\n",
"\\partial_t E^i &= \\partial^i \\partial_j A^j - \\partial_j \\partial^j A^i, \\\\\n",
"\\partial_t \\varphi &= -\\partial_i A^i,\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:46.640662Z",
"iopub.status.busy": "2021-03-07T17:18:46.639972Z",
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"shell.execute_reply": "2021-03-07T17:18:46.642537Z"
}
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"outputs": [],
"source": [
"# Register gridfunctions that are needed as input.\n",
"\n",
"# Declare the rank-1 indexed expressions E_{i}, A_{i},\n",
"# and \\partial_{i} \\psi, that are to be evolved in time.\n",
"# Derivative variables like these must have an underscore\n",
"# in them, so the finite difference module can parse\n",
"# the variable name properly.\n",
"\n",
"# E_i\n",
"EU = ixp.register_gridfunctions_for_single_rank1(\"EVOL\", \"EU\")\n",
"\n",
"# A_i, _AD is unused\n",
"_AU = ixp.register_gridfunctions_for_single_rank1(\"EVOL\", \"AU\")\n",
"\n",
"# \\psi is a scalar function that is time evolved\n",
"# _psi is unused\n",
"_psi = gri.register_gridfunctions(\"EVOL\", [\"psi\"])\n",
"\n",
"# \\partial_i \\psi\n",
"psi_dD = ixp.declarerank1(\"psi_dD\")\n",
"\n",
"# \\partial_k ( A_i ) --> rank two tensor\n",
"AU_dD = ixp.declarerank2(\"AU_dD\", \"nosym\")\n",
"\n",
"# \\partial_k partial_m ( A_i ) --> rank three tensor\n",
"AU_dDD = ixp.declarerank3(\"AU_dDD\", \"sym12\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{align}\n",
"\\partial_t A^i &= -E^i - \\partial^i \\varphi, \\\\\n",
"\\partial_t E^i &= \\partial^i \\partial_j A^j - \\partial_j \\partial^j A^i, \\\\\n",
"\\partial_t \\varphi &= -\\partial_i A^i,\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:46.652659Z",
"iopub.status.busy": "2021-03-07T17:18:46.652026Z",
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"shell.execute_reply": "2021-03-07T17:18:46.696401Z"
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"outputs": [],
"source": [
"# \\partial_t \\psi = -\\partial_i A_i\n",
"psi_rhs = 0\n",
"\n",
"# \\partial_t A_i = E_i - \\partial_i \\psi\n",
"ArhsU = ixp.zerorank1()\n",
"\n",
"# \\partial_t E_i = -\\partial_j^2 A_i + \\partial_j \\partial_i A_j\n",
"ErhsU = ixp.zerorank1()\n",
"\n",
"# RHSs - equations 10 and 11 from https://arxiv.org/abs/gr-qc/0201051\n",
"for i in range(DIM):\n",
" ArhsU[i] = -EU[i] - psi_dD[i]\n",
" psi_rhs += -AU_dD[i][i]\n",
" for j in range(DIM):\n",
" ErhsU[i] += -AU_dDD[i][j][j] + AU_dDD[j][j][i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Constraint:\n",
"\n",
"\\begin{align}\n",
"\\mathcal{C} \\equiv \\partial_i E^i\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:46.701977Z",
"iopub.status.busy": "2021-03-07T17:18:46.701330Z",
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"shell.execute_reply": "2021-03-07T17:18:46.703946Z"
}
},
"outputs": [],
"source": [
"# \\partial_k ( E^i ) --> rank two tensor\n",
"EU_dD = ixp.declarerank2(\"EU_dD\", \"nosym\")\n",
"\n",
"# C = \\partial_i E^i\n",
"C = EU_dD[0][0] + EU_dD[1][1] + EU_dD[2][2]"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:46.715265Z",
"iopub.status.busy": "2021-03-07T17:18:46.714431Z",
"iopub.status.idle": "2021-03-07T17:18:46.718545Z",
"shell.execute_reply": "2021-03-07T17:18:46.719052Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"C = EU_dD00 + EU_dD11 + EU_dD22\n",
"ArhsU[0] = -EU0 - psi_dD0\n",
"ArhsU[1] = -EU1 - psi_dD1\n",
"ArhsU[2] = -EU2 - psi_dD2\n",
"ErhsU[0] = -AU_dDD011 - AU_dDD022 + AU_dDD101 + AU_dDD202\n",
"ErhsU[1] = AU_dDD001 - AU_dDD100 - AU_dDD122 + AU_dDD212\n",
"ErhsU[2] = AU_dDD002 + AU_dDD112 - AU_dDD200 - AU_dDD211\n",
"psi_rhs = -AU_dD00 - AU_dD11 - AU_dD22\n"
]
}
],
"source": [
"print('C = ', C)\n",
"print('ArhsU[0] = ', ArhsU[0])\n",
"print('ArhsU[1] = ', ArhsU[1])\n",
"print('ArhsU[2] = ', ArhsU[2])\n",
"print('ErhsU[0] = ', ErhsU[0])\n",
"print('ErhsU[1] = ', ErhsU[1])\n",
"print('ErhsU[2] = ', ErhsU[2])\n",
"print('psi_rhs = ', psi_rhs)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='system1_val'></a>\n",
"\n",
"## Step 2.b: NRPy+ Module Code Validation \\[Back to [top](#toc)\\]\n",
"$$\\label{system1_val}$$\n",
"\n",
"Here, as a code validation check, we verify agreement in the SymPy expressions for the RHSs of Maxwell's equations (in System I) between\n",
"1. this tutorial and \n",
"2. the NRPy+ [Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py](../edit/Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py) module.\n"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:46.726534Z",
"iopub.status.busy": "2021-03-07T17:18:46.725819Z",
"iopub.status.idle": "2021-03-07T17:18:47.162628Z",
"shell.execute_reply": "2021-03-07T17:18:47.163113Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Currently using System_I RHSs \n",
"\n",
"Consistency check between Tutorial-VacuumMaxwell_Cartesian_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.\n",
"C - mwevol.C = 0\n",
"psi_rhs - mwevol.psi_rhs = 0\n",
"ArhsU[0] - mwevol.ArhsU[0] = 0\n",
"ErhsU[0] - mwevol.ErhsU[0] = 0\n",
"ArhsU[1] - mwevol.ArhsU[1] = 0\n",
"ErhsU[1] - mwevol.ErhsU[1] = 0\n",
"ArhsU[2] - mwevol.ArhsU[2] = 0\n",
"ErhsU[2] - mwevol.ErhsU[2] = 0\n"
]
}
],
"source": [
"# Reset the list of gridfunctions, as registering a gridfunction\n",
"# twice will spawn an error.\n",
"gri.glb_gridfcs_list = []\n",
"\n",
"# Call the VacuumMaxwellRHSs() function from within the\n",
"# Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py module,\n",
"# which should do exactly the same as the steps above.\n",
"\n",
"# Set which system to use, which are defined in Maxwell/InitialData.py\n",
"par.initialize_param(par.glb_param(\"char\", \"Maxwell.InitialData\",\"System_to_use\",\"System_I\"))\n",
"\n",
"import Maxwell.VacuumMaxwell_Flat_Evol_Cartesian as mwevol\n",
"mwevol.VacuumMaxwellRHSs()\n",
"\n",
"print(\"Consistency check between Tutorial-VacuumMaxwell_Cartesian_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.\")\n",
"\n",
"print(\"C - mwevol.C = \" + str(C - mwevol.C))\n",
"print(\"psi_rhs - mwevol.psi_rhs = \" + str(psi_rhs - mwevol.psi_rhs))\n",
"for i in range(DIM):\n",
"\n",
" print(\"ArhsU[\"+str(i)+\"] - mwevol.ArhsU[\"+str(i)+\"] = \" + str(ArhsU[i] - mwevol.ArhsU[i]))\n",
" print(\"ErhsU[\"+str(i)+\"] - mwevol.ErhsU[\"+str(i)+\"] = \" + str(ErhsU[i] - mwevol.ErhsU[i]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='system2'></a>\n",
"\n",
"# Step 3: System II \\[Back to [top](#toc)\\]\n",
"$$\\label{system2}$$\n",
"\n",
"<a id='system2_rhs'></a>\n",
"\n",
"## Step 3.a: Construct evolution equations \\[Back to [top](#toc)\\]\n",
"$$\\label{system2_rhs}$$\n",
"\n",
"Next, again following [Tutorial-VacuumMaxwell_formulation_Cartesian](Tutorial-VacuumMaxwell_formulation_Cartesian.ipynb), we construct the system of equations for System II, in flat space and cartesian coordinates:\n",
"\n",
"\\begin{align}\n",
"\\partial_t A^i &= -E^i - \\partial^i \\varphi, \\\\\n",
"\\partial_t E^i &= \\partial^i \\Gamma - \\partial_j \\partial^j A^i, \\\\\n",
"\\partial_t \\Gamma &= - \\partial_i \\partial^i \\varphi, \\\\\n",
"\\partial_t \\varphi &= -\\Gamma,\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:47.173619Z",
"iopub.status.busy": "2021-03-07T17:18:47.172518Z",
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"shell.execute_reply": "2021-03-07T17:18:47.177083Z"
}
},
"outputs": [],
"source": [
"# We inherit here all of the definitions from System I, above\n",
"\n",
"# Register the scalar auxiliary variable \\Gamma\n",
"Gamma = gri.register_gridfunctions(\"EVOL\", [\"Gamma\"])\n",
"\n",
"# Declare the ordinary gradient \\partial_{i} \\Gamma\n",
"Gamma_dD = ixp.declarerank1(\"Gamma_dD\")\n",
"\n",
"# partial_i \\partial_j \\psi\n",
"psi_dDD = ixp.declarerank2(\"psi_dDD\", \"sym01\")\n",
"\n",
"psi_rhs = -Gamma"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{align}\n",
"\\partial_t A^i &= -E^i - \\partial^i \\varphi, \\\\\n",
"\\partial_t E^i &= \\partial^i \\Gamma - \\partial_j \\partial^j A^i, \\\\\n",
"\\partial_t \\Gamma &= - \\partial_i \\partial^i \\varphi, \\\\\n",
"\\partial_t \\varphi &= -\\Gamma,\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:47.190722Z",
"iopub.status.busy": "2021-03-07T17:18:47.189928Z",
"iopub.status.idle": "2021-03-07T17:18:47.192137Z",
"shell.execute_reply": "2021-03-07T17:18:47.192609Z"
}
},
"outputs": [],
"source": [
"# Equation 15 https://arxiv.org/abs/gr-qc/0201051\n",
"# \\partial_t \\Gamma = -\\partial_i^2 \\psi\n",
"Gamma_rhs = 0\n",
"\n",
"# \\partial_t A_i = -E_i - \\partial_i \\psi\n",
"ArhsU = ixp.zerorank1()\n",
"\n",
"# \\partial_t E_i = -\\partial_j^2 A_i + \\partial_i \\Gamma\n",
"ErhsU = ixp.zerorank1()\n",
"\n",
"# RHSs - equations 10 and 14 https://arxiv.org/abs/gr-qc/0201051\n",
"for i in range(DIM):\n",
" ArhsU[i] = -EU[i] - psi_dD[i]\n",
" Gamma_rhs -= psi_dDD[i][i]\n",
" ErhsU[i] += Gamma_dD[i]\n",
" for j in range(DIM):\n",
" ErhsU[i] -= AU_dDD[i][j][j]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Constraints:\n",
"\n",
"\\begin{align}\n",
"\\mathcal{G} &\\equiv \\Gamma - \\partial_i A^i, \\\\\n",
"\\mathcal{C} &\\equiv \\partial_i E^i.\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:47.200830Z",
"iopub.status.busy": "2021-03-07T17:18:47.199752Z",
"iopub.status.idle": "2021-03-07T17:18:47.204835Z",
"shell.execute_reply": "2021-03-07T17:18:47.205386Z"
}
},
"outputs": [],
"source": [
"# \\partial_k ( E^i ) --> rank two tensor\n",
"EU_dD = ixp.declarerank2(\"EU_dD\", \"nosym\")\n",
"\n",
"# C = \\partial_i E^i\n",
"C = EU_dD[0][0] + EU_dD[1][1] + EU_dD[2][2]\n",
"\n",
"# G = \\Gamma - \\partial_i A^i\n",
"G = Gamma - AU_dD[0][0] - AU_dD[1][1] - AU_dD[2][2]"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:47.216491Z",
"iopub.status.busy": "2021-03-07T17:18:47.215710Z",
"iopub.status.idle": "2021-03-07T17:18:47.221227Z",
"shell.execute_reply": "2021-03-07T17:18:47.220653Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"C = EU_dD00 + EU_dD11 + EU_dD22\n",
"G = -AU_dD00 - AU_dD11 - AU_dD22 + Gamma\n",
"ArhsU[0] = -EU0 - psi_dD0\n",
"ArhsU[1] = -EU1 - psi_dD1\n",
"ArhsU[2] = -EU2 - psi_dD2\n",
"ErhsU[0] = -AU_dDD000 - AU_dDD011 - AU_dDD022 + Gamma_dD0\n",
"ErhsU[1] = -AU_dDD100 - AU_dDD111 - AU_dDD122 + Gamma_dD1\n",
"ErhsU[2] = -AU_dDD200 - AU_dDD211 - AU_dDD222 + Gamma_dD2\n",
"psi_rhs = -Gamma\n",
"Gamma_rhs = -psi_dDD00 - psi_dDD11 - psi_dDD22\n"
]
}
],
"source": [
"print('C = ', C)\n",
"print('G = ', G)\n",
"print('ArhsU[0] = ', ArhsU[0])\n",
"print('ArhsU[1] = ', ArhsU[1])\n",
"print('ArhsU[2] = ', ArhsU[2])\n",
"print('ErhsU[0] = ', ErhsU[0])\n",
"print('ErhsU[1] = ', ErhsU[1])\n",
"print('ErhsU[2] = ', ErhsU[2])\n",
"print('psi_rhs = ', psi_rhs)\n",
"print('Gamma_rhs =', Gamma_rhs)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='system2_val'></a>\n",
"\n",
"## Step 3.b: NRPy+ Module Code Validation \\[Back to [top](#toc)\\]\n",
"$$\\label{system2_val}$$\n",
"\n",
"Here, as a code validation check, we verify agreement in the SymPy expressions for the RHSs of Maxwell's equations (in System II) between\n",
"1. this tutorial and \n",
"2. the NRPy+ [Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py](../edit/Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py) module."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:47.290416Z",
"iopub.status.busy": "2021-03-07T17:18:47.254631Z",
"iopub.status.idle": "2021-03-07T17:18:47.299468Z",
"shell.execute_reply": "2021-03-07T17:18:47.299993Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Currently using System_II RHSs \n",
"\n",
"Consistency check between Tutorial-VacuumMaxwell_Cartesian_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.\n",
"C - mwevol.C = 0\n",
"G - mwevol.G = 0\n",
"psi_rhs - mwevol.psi_rhs = 0\n",
"Gamma_rhs - mwevol.Gamma_rhs = 0\n",
"ArhsU[0] - mwevol.ArhsU[0] = 0\n",
"ErhsU[0] - mwevol.ErhsU[0] = 0\n",
"ArhsU[1] - mwevol.ArhsU[1] = 0\n",
"ErhsU[1] - mwevol.ErhsU[1] = 0\n",
"ArhsU[2] - mwevol.ArhsU[2] = 0\n",
"ErhsU[2] - mwevol.ErhsU[2] = 0\n"
]
}
],
"source": [
"# Reset the list of gridfunctions, as registering a gridfunction\n",
"# twice will spawn an error.\n",
"gri.glb_gridfcs_list = []\n",
"\n",
"# Call the VacuumMaxwellRHSs() function from within the\n",
"# Maxwell/VacuumMaxwell_Flat_Evol_Cartesian.py module,\n",
"# which should do exactly the same as the steps above.\n",
"\n",
"par.set_paramsvals_value(\"Maxwell.InitialData::System_to_use=System_II\")\n",
"mwevol.VacuumMaxwellRHSs()\n",
"\n",
"print(\"Consistency check between Tutorial-VacuumMaxwell_Cartesian_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.\")\n",
"\n",
"print(\"C - mwevol.C = \" + str(C - mwevol.C))\n",
"print(\"G - mwevol.G = \" + str(G - mwevol.G))\n",
"print(\"psi_rhs - mwevol.psi_rhs = \" + str(psi_rhs - mwevol.psi_rhs))\n",
"print(\"Gamma_rhs - mwevol.Gamma_rhs = \" + str(Gamma_rhs - mwevol.Gamma_rhs))\n",
"for i in range(DIM):\n",
"\n",
" print(\"ArhsU[\"+str(i)+\"] - mwevol.ArhsU[\"+str(i)+\"] = \" + str(ArhsU[i] - mwevol.ArhsU[i]))\n",
" print(\"ErhsU[\"+str(i)+\"] - mwevol.ErhsU[\"+str(i)+\"] = \" + str(ErhsU[i] - mwevol.ErhsU[i]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id='latex_pdf_output'></a>\n",
"\n",
"# Step 4: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n",
"$$\\label{latex_pdf_output}$$\n",
"\n",
"The following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n",
"[Tutorial-VacuumMaxwell_Cartesian_RHSs.pdf](Tutorial-VacuumMaxwell_Cartesian_RHSs.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:18:47.304599Z",
"iopub.status.busy": "2021-03-07T17:18:47.303930Z",
"iopub.status.idle": "2021-03-07T17:18:51.178598Z",
"shell.execute_reply": "2021-03-07T17:18:51.179321Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Created Tutorial-VacuumMaxwell_Cartesian_RHSs.tex, and compiled LaTeX file\n",
" to PDF file Tutorial-VacuumMaxwell_Cartesian_RHSs.pdf\n"
]
}
],
"source": [
"import cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\n",
"cmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-VacuumMaxwell_Cartesian_RHSs\")"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}