%% %% 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\