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authorKarl Berry <karl@freefriends.org>2020-06-06 21:22:16 +0000
committerKarl Berry <karl@freefriends.org>2020-06-06 21:22:16 +0000
commit776d1f6812d6d62ea7bb8860b199b9f99dd5e170 (patch)
treef1a6c0558caa25137dc2cec1fd7a849ca1a49dc9 /Master/texmf-dist/doc/latex/math-e/03-07-1.ltx
parent32844d45f401c937044cdf7b9f1f6bf234db007a (diff)
rm math-e, author request
git-svn-id: svn://tug.org/texlive/trunk@55450 c570f23f-e606-0410-a88d-b1316a301751
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-%%
-%% An UIT Edition example
-%%
-%% Example 03-07-1 on page 30.
-%%
-%% Copyright (C) 2010 Herbert Voss
-%%
-%% It may be distributed and/or modified under the conditions
-%% of the LaTeX Project Public License, either version 1.3
-%% of this license or (at your option) any later version.
-%%
-%% See http://www.latex-project.org/lppl.txt for details.
-%%
-%%
-%% ====
-% Show page(s) 1
-%%
-\documentclass[]{exaarticle}
-\pagestyle{empty}
-\setlength\textwidth{375.57637pt}
-\usepackage[utf8]{inputenc}
-\usepackage[english]{babel}
-\setcounter{equation}{35}
-\renewcommand\theequation{3.\arabic{equation}}
-\usepackage{amsmath,esint,array,esvect}
-\setlength\parindent{0pt}
-\StartShownPreambleCommands
-\def\Q#1#2{\frac{\partial #1}{\partial #2}}
-\def\half{\frac{1}{2}}
-\def\vvec#1{\vv{#1}}
-\newcommand*\diff{\mathop{}\!\mathrm{d}}
-\newcommand*\<{\negthickspace}
-\newcommand*\TT{\boldsymbol{\mathsf{T}}}
-\def\DD{\boldsymbol{\mathsf{D}}}
-\StopShownPreambleCommands
-\begin{document}
-The conservation principles for mass, torque, and energy can be given in
-differential or integral form:
-\setlength\jot{15pt}
-
-\begin{description}
-\item[Differential form]
-\begin{align}
-\begin{aligned}
- \Q{\varrho}{t}+\mathrm{div}(\varrho\vv{v}) &= 0 \\
- \varrho\Q{\vv{v}}{t}+(\varrho\vv{v}\times\nabla)\vv{v} &= \vv{f}_0+\mathrm{div}\TT=\vv{f}_0
- -\mathrm{grad}p+\mathrm{div}\TT' \\
- \varrho T\frac{\diff s}{\diff t} &= \varrho\frac{\diff e}{\diff t}
- -\frac{p}{\varrho}\frac{\diff\varrho}{\diff t}=-\mathrm{div}\vv{q}+\TT':\DD
-\end{aligned}
-\end{align}
-
-\item[Integral form]
-\begin{align}
- \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vv{v}\times\vv{v}ec{n})\diff^2A &= 0\\
- \Q{}{t}\iiint\<\varrho\vv{v}\diff^3V+\oiint\varrho\vv{v}(\vv{v}\times\vv{n}\,)\diff^2A &=
- \iiint\<f_0\diff^3V+\oiint\vv{n}\times T\diff^2A \\
- \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
- \varrho\left(\vv{v}\times\vv{n}\,\right)\diff^2A & =\\
- \multispan2{\hfill${\displaystyle-\oiint\left(\vv{q}\times\vv{v}ec{n}\right)\diff^2A+
- \iiint\<\left(\vv{v}\times\vv{f}_0\right)\diff^3V+\oiint\left(\vv{v}
- \times\vv{n}~\TT\right)\diff^2A}$}.\nonumber
-\end{align}
-\end{description}
-
-\def\QQ#1#2{\frac{\partial^2 #1}{\partial #2^2}}
-\def\ee#1{\vv{e}_{#1}}
-The $\nabla$ operator in Cartesian coordinates:
-%
-\[
- \vv{\nabla}=\Q{}{x}\ee{x}+\Q{}{y}\ee{y}+\Q{}{z}\ee{z}~~,~~
- \mathrm{grad}f=\vv{\nabla}f=\Q{f}{x}\ee{x}+\Q{f}{y}\ee{y}+\Q{f}{z}\ee{z}
-\]
-%
-\[
- \mathrm{div}~\vv{a}=\vv{\nabla}\times\vv{a}=\Q{a_x}{x}+\Q{a_y}{y}+\Q{a_z}{z}~~,~~\nabla^2 f=\QQ{f}{x}+\QQ{f}{y}+\QQ{f}{z}
-\]
-%
-\[
-\mathrm{rot}~\vv{a}=\vv{\nabla}\times\vv{a}=
- \left(\Q{a_z}{y}-\Q{a_y}{z}\right)\ee{x}+\left(\Q{a_x}{z}-\Q{a_z}{x}\right)\ee{y}+
- \left(\Q{a_y}{x}-\Q{a_x}{y}\right)\ee{z}
-\]
-\end{document}