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%%
%% An UIT Edition example
%%
%% Example 03-07-3 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}{37}
\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}
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