%% %% This is file `mandi.sty', %% generated with the docstrip utility. %% %% The original source files were: %% %% mandi.dtx (with options: `package') %% %% Copyright (C) 2011,2012 by Paul J. Heafner %% --------------------------------------------------------------------------- %% This work 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. The latest version of this license is in %% http://www.latex-project.org/lppl.txt %% and version 1.3 or later is part of all distributions of LaTeX version %% 2005/12/01 or later. %% %% This work has the LPPL maintenance status `maintained'. %% %% The Current Maintainer of this work is Paul J. Heafner. %% %% This work consists of the files mandi.dtx %% %% and includes the derived files README %% mandi.ins %% mandi.sty %% mandi.pdf and %% vdemo.py. %% --------------------------------------------------------------------------- %% \ProvidesPackage{mandi}[2013/04/10 2.1.0 Macros for intro physics and astronomy] \NeedsTeXFormat{LaTeX2e}[1999/12/01] \RequirePackage{amsmath} \RequirePackage{amssymb} \RequirePackage{bigints} \RequirePackage{cancel} \RequirePackage[leftbars,color]{changebar} \RequirePackage[dvipsnames]{xcolor} \RequirePackage{environ} \RequirePackage{etoolbox} \RequirePackage{filehook} \RequirePackage{extarrows} \RequirePackage[T1]{fontenc} \RequirePackage{graphicx} \RequirePackage{epstopdf} \RequirePackage{textcomp} \RequirePackage{letltxmacro} \RequirePackage{listings} \RequirePackage[framemethod=TikZ]{mdframed} \RequirePackage{suffix} \RequirePackage{xargs} \RequirePackage{xparse} \RequirePackage{xspace} \RequirePackage{ifthen} \RequirePackage{calligra} \DeclareMathAlphabet{\mathcalligra}{T1}{calligra}{m}{n} \DeclareFontShape{T1}{calligra}{m}{n}{<->s*[2.2]callig15}{} \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `basename #1 .tif`.png} \DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} \usetikzlibrary{shadows} \definecolor{vpythoncolor}{rgb}{0.95,0.95,0.95} \newcommand{\lstvpython}{\lstset{language=Python,numbers=left,numberstyle=\tiny, backgroundcolor=\color{vpythoncolor},upquote=true,breaklines}} \newboolean{@optitalicvectors} \newboolean{@optdoubleabsbars} \newboolean{@optbaseunits} \newboolean{@optdrvdunits} \setboolean{@optitalicvectors}{false} \setboolean{@optdoubleabsbars}{false} \setboolean{@optbaseunits}{false} \setboolean{@optdrvdunits}{false} \DeclareOption{italicvectors}{\setboolean{@optitalicvectors}{true}} \DeclareOption{doubleabsbars}{\setboolean{@optdoubleabsbars}{true}} \DeclareOption{baseunits}{\setboolean{@optbaseunits}{true}} \DeclareOption{drvdunits}{\setboolean{@optdrvdunits}{true}} \ProcessOptions\relax \@ifpackageloaded{amssymb}{% \csundef{square} \typeout{mandi: Package amssymb detected. Its \protect\square\space has been redefined.} }{% \typeout{mandi: Package amssymb not detected.} }% \newcommand{\per}{\ensuremath{/}} \newcommand{\usk}{\ensuremath{\cdot}} \newcommand{\unit}[2]{\ensuremath{{#1}\,{#2}}} \newcommand{\ampere}{\ensuremath{\mathrm{A}}} \newcommand{\arcminute}{\ensuremath{'}} \newcommand{\arcsecond}{\ensuremath{''}} \newcommand{\atomicmassunit}{\ensuremath{\mathrm{u}}} \newcommand{\candela}{\ensuremath{\mathrm{cd}}} \newcommand{\coulomb}{\ensuremath{\mathrm{C}}} \newcommand{\degree}{\ensuremath{^{\circ}}} \newcommand{\electronvolt}{\ensuremath{\mathrm{eV}}} \newcommand{\farad}{\ensuremath{\mathrm{F}}} \newcommand{\henry}{\ensuremath{\mathrm{H}}} \newcommand{\hertz}{\ensuremath{\mathrm{Hz}}} \newcommand{\hour}{\ensuremath{\mathrm{h}}} \newcommand{\joule}{\ensuremath{\mathrm{J}}} \newcommand{\kelvin}{\ensuremath{\mathrm{K}}} \newcommand{\kilogram}{\ensuremath{\mathrm{kg}}} \newcommand{\metre}{\ensuremath{\mathrm{m}}} \newcommand{\minute}{\ensuremath{\mathrm{min}}} \newcommand{\mole}{\ensuremath{\mathrm{mol}}} \newcommand{\newton}{\ensuremath{\mathrm{N}}} \newcommand{\ohm}{\ensuremath{\Omega}} \newcommand{\pascal}{\ensuremath{\mathrm{Pa}}} \newcommand{\radian}{\ensuremath{\mathrm{rad}}} \newcommand{\second}{\ensuremath{\mathrm{s}}} \newcommand{\siemens}{\ensuremath{\mathrm{S}}} \newcommand{\steradian}{\ensuremath{\mathrm{sr}}} \newcommand{\tesla}{\ensuremath{\mathrm{T}}} \newcommand{\volt}{\ensuremath{\mathrm{V}}} \newcommand{\watt}{\ensuremath{\mathrm{W}}} \newcommand{\weber}{\ensuremath{\mathrm{Wb}}} \newcommand{\C}{\coulomb} \newcommand{\F}{\farad} \newcommand{\J}{\joule} \newcommand{\N}{\newton} \newcommand{\Pa}{\pascal} \newcommand{\rad}{\radian} \newcommand{\sr}{\steradian} \newcommand{\T}{\tesla} \newcommand{\V}{\volt} \newcommand{\W}{\watt} \newcommand{\square}[1]{\ensuremath{\mathrm{#1}^{2}}} % prefix 2 \newcommand*{\cubic}[1]{\ensuremath{\mathrm{#1}^{3}}} % prefix 3 \newcommand*{\quartic}[1]{\ensuremath{\mathrm{#1}^{4}}} % prefix 4 \newcommand*{\reciprocal}[1]{\ensuremath{\mathrm{#1}^{-1}}} % prefix -1 \newcommand*{\reciprocalsquare}[1]{\ensuremath{\mathrm{#1}^{-2}}} % prefix -2 \newcommand*{\reciprocalcubic}[1]{\ensuremath{\mathrm{#1}^{-3}}} % prefix -3 \newcommand*{\reciprocalquartic}[1]{\ensuremath{\mathrm{#1}^{-4}}} % prefix -4 \newcommand*{\squared}{\ensuremath{^{\mathrm{2}}}} % postfix 2 \newcommand*{\cubed}{\ensuremath{^{\mathrm{3}}}} % postfix 3 \newcommand*{\quarted}{\ensuremath{^{\mathrm{4}}}} % postfix 4 \newcommand*{\reciprocaled}{\ensuremath{^{\mathrm{-1}}}} % postfix -1 \newcommand*{\reciprocalsquared}{\ensuremath{^{\mathrm{-2}}}} % postfix -2 \newcommand*{\reciprocalcubed}{\ensuremath{^{\mathrm{-3}}}} % postfix -3 \newcommand*{\reciprocalquarted}{\ensuremath{^{\mathrm{-4}}}} % postfix -4 \newcommand\mi@exchangeargs[2]{#2#1}% \newcommand\mi@name{}% \long\def\mi@name#1#{\romannumeral0\mi@innername{#1}}% \newcommand\mi@innername[2]{% \expandafter\mi@exchangeargs\expandafter{\csname#2\endcsname}{#1}}% \begingroup \@firstofone{% \endgroup \newcommand\mi@forkifnull[3]{% \romannumeral\iffalse{\fi\expandafter\@secondoftwo\expandafter {\expandafter{\string#1}\expandafter\@secondoftwo\string}% \expandafter\@firstoftwo\expandafter{\iffalse}\fi0 #3}{0 #2}}}% \newcommand\selectbaseunit[3]{#1} \newcommand\selectdrvdunit[3]{#2} \newcommand\selecttradunit[3]{#3} \newcommand\selectunit{} \newcommand\perpusebaseunit{\let\selectunit=\selectbaseunit} \newcommand\perpusedrvdunit{\let\selectunit=\selectdrvdunit} \newcommand\perpusetradunit{\let\selectunit=\selecttradunit} \newcommand\hereusebaseunit[1]{% \begingroup\perpusebaseunit#1\endgroup}% \newcommand\hereusedrvdunit[1]{% \begingroup\perpusedrvdunit#1\endgroup}% \newcommand\hereusetradunit[1]{% \begingroup\perpusetradunit#1\endgroup}% \newenvironment{usebaseunit}{\perpusebaseunit}{}% \newenvironment{usedrvdunit}{\perpusedrvdunit}{}% \newenvironment{usetradunit}{\perpusetradunit}{}% \newcommand*\newphysicsquantity{\definephysicsquantity{\newcommand}} \newcommand*\redefinephysicsquantity{\definephysicsquantity{\renewcommand}} \newcommandx\definephysicsquantity[5][4=,5=]{% \innerdefinewhatsoeverquantityfork{#3}{#4}{#5}{#1}{#2}{}{[1]}{##1}}% \newcommand*\newphysicsconstant{\definephysicsconstant{\newcommand}} \newcommand*\redefinephysicsconstant{\definephysicsconstant{\renewcommand}} \newcommandx\definephysicsconstant[7][6=,7=]{% \innerdefinewhatsoeverquantityfork{#5}{#6}{#7}{#1}{#2}{#3}{}{#4}}% \newcommand\innerdefinewhatsoeverquantityfork[3]{% \expandafter\innerdefinewhatsoeverquantity\romannumeral0% \mi@forkifnull{#3}{\mi@forkifnull{#2}{{#1}}{{#2}}{#1}}% {\mi@forkifnull{#2}{{#1}}{{#2}}{#3}}{#1}}% \newcommand\innerdefinewhatsoeverquantity[8]{% \mi@name#4{#5}#7{\ensuremath{\unit{#8}{\selectunit{#3}{#1}{#2}}}}% \mi@name#4{#5baseunit}#7{\ensuremath{\unit{#8}{#3}}}% \mi@name#4{#5drvdunit}#7{\ensuremath{\unit{#8}{#1}}}% \mi@name#4{#5tradunit}#7{\ensuremath{\unit{#8}{#2}}}% \mi@name#4{#5onlyunit}{\ensuremath{\selectunit{#3}{#1}{#2}}}% \mi@name#4{#5onlybaseunit}{\ensuremath{#3}}% \mi@name#4{#5onlydrvdunit}{\ensuremath{#1}}% \mi@name#4{#5onlytradunit}{\ensuremath{#2}}% \mi@name#4{#5value}#7{\ensuremath{#8}}% \mi@forkifnull{#7}{% \ifx#4\renewcommand\mi@name\let{#5mathsymbol}=\relax\fi \mi@name\newcommand{#5mathsymbol}{\ensuremath{#6}}}{}}% \ifthenelse{\boolean{@optitalicvectors}} {\typeout{mandi: You'll get italic vector kernels.}} {\typeout{mandi: You'll get Roman vector kernels.}} \ifthenelse{\boolean{@optdoubleabsbars}} {\typeout{mandi: You'll get double absolute value bars.}} {\typeout{mandi: You'll get single absolute value bars.}} \ifthenelse{\boolean{@optbaseunits}} {\perpusebaseunit % \typeout{mandi: You'll get base units.}} {\ifthenelse{\boolean{@optdrvdunits}} {\perpusedrvdunit % \typeout{mandi: You'll get derived units.}} {\perpusetradunit % \typeout{mandi: You'll get traditional units.}}} \def\resetMathstrut@{% \setbox\z@\hbox{% \mathchardef\@tempa\mathcode`\[\relax \def\@tempb##1"##2##3{\the\textfont"##3\char"}% \expandafter\@tempb\meaning\@tempa \relax }% \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@} \begingroup \catcode`(\active \xdef({\left\string(} \catcode`)\active \xdef){\right\string)} \endgroup \mathcode`(="8000 \mathcode`)="8000 \typeout{mandi: parentheses made adjustable in math mode.} \let\oldr@@t\r@@t \def\r@@t#1#2{% \setbox0=\hbox{\(\oldr@@t#1{#2\,}\)}\dimen0=\ht0 \advance\dimen0-0.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth -\dimen0}% {\box0\lower0.4pt\box2}} \LetLtxMacro{\oldsqrt}{\sqrt} \renewcommand*{\sqrt}[2][\relax]{\oldsqrt[#1]{#2}} \typeout{mandi: square root symbol fixed.} \newcommand{\m}{\metre} \newcommand{\kg}{\kilogram} \newcommand{\s}{\second} \newcommand{\A}{\ampere} \newcommand{\K}{\kelvin} \newcommand{\mol}{\mole} \newcommand{\cd}{\candela} \newphysicsquantity{displacement}{\m}[\m][\m] \newphysicsquantity{mass}{\kg}[\kg][\kg] \newphysicsquantity{duration}{\s}[\s][\s] \newphysicsquantity{current}{\A}[\A][\A] \newphysicsquantity{temperature}{\K}[\K][\K] \newphysicsquantity{amount}{\mol}[\mol][\mol] \newphysicsquantity{luminous}{\cd}[\cd][\cd] \newphysicsquantity{planeangle}{\m\usk\reciprocal\m}[\rad][\rad] \newphysicsquantity{solidangle}{\m\squared\usk\reciprocalsquare\m}[\sr][\sr] \newcommand{\indegrees}[1]{\ensuremath{\unit{#1}{\degree}}} \newcommand{\inarcminutes}[1]{\ensuremath{\unit{#1}{\arcminute}}} \newcommand{\inarcseconds}[1]{\ensuremath{\unit{#1}{\arcsecond}}} \newcommand{\ineV}[1]{\ensuremath{\unit{#1}{\electronvolt}}} \newcommand{\inAU}[1]{\ensuremath{\unit{#1}{\mathrm{AU}}}} \newcommand{\velocityc}[1]{\ensuremath{#1c}} \newphysicsquantity{velocity}{\m\usk\reciprocal\s}[\m\usk\reciprocal\s][\m\per\s] \newphysicsquantity{acceleration}{\m\usk\s\reciprocalsquared}[\N\per\kg][\m\per\s\squared] \newcommand{\gamman}[1]{\ensuremath{#1}} \newphysicsquantity{momentum}{\m\usk\kg\usk\reciprocal\s}[\N\usk\s][\kg\usk\m\per\s] \newphysicsquantity{impulse}{\m\usk\kg\usk\reciprocal\s}[\N\usk\s][\kg\usk\m\per\s] \newphysicsquantity{force}{\m\usk\kg\usk\s\reciprocalsquared}[\N][\N] \newphysicsquantity{springstiffness}{\kg\usk\s\reciprocalsquared}[\N\per\m][\N\per\m] \newphysicsquantity{springstretch}{\m} \newphysicsquantity{area}{\m\squared} \newphysicsquantity{volume}{\cubic\m} \newphysicsquantity{linearmassdensity}{\reciprocal\m\usk\kg}[\kg\per\m][\kg\per\m] \newphysicsquantity{areamassdensity}{\m\reciprocalsquared\usk\kg}[\kg\per\m\squared] [\kg\per\m\squared] \newphysicsquantity{volumemassdensity}{\m\reciprocalcubed\usk\kg}[\kg\per\m\cubed] [\kg\per\m\cubed] \newphysicsquantity{youngsmodulus}{\reciprocal\m\usk\kg\usk\s\reciprocalsquared} [\N\per\m\squared][\Pa] \newphysicsquantity{work}{\m\squared\usk\kg\usk\s\reciprocalsquared}[\J] [\N\usk\m] \newphysicsquantity{energy}{\m\squared\usk\kg\usk\s\reciprocalsquared}[\J] [\N\usk\m] \newphysicsquantity{power}{\m\squared\usk\kg\usk\s\reciprocalcubed}[\W][\J\per\s] \newphysicsquantity{angularvelocity}{\rad\usk\reciprocal\s}[\rad\per\s][\rad\per\s] \newphysicsquantity{angularacceleration}{\rad\usk\s\reciprocalsquared}[\rad\per\s\squared] [\rad\per\s\squared] \newphysicsquantity{angularmomentum}{\m\squared\usk\kg\usk\reciprocal\s}[\J\usk\s] [\kg\usk\m\squared\per\s] \newphysicsquantity{momentofinertia}{\m\squared\usk\kg}[\J\usk\s\squared] [\kg\usk\m\squared] \newphysicsquantity{torque}{\m\squared\usk\kg\usk\s\reciprocalsquared}[\J\per\rad] [\N\usk\m] \newphysicsquantity{entropy}{\m\squared\usk\kg\usk\s\reciprocalsquared\usk\reciprocal\K} [\J\per\K][\J\per\K] \newphysicsquantity{wavelength}{\m}[\m][\m] \newphysicsquantity{wavenumber}{\reciprocal\m}[\per\m][\per\m] \newphysicsquantity{frequency}{\reciprocal\s}[\hertz][\hertz] \newphysicsquantity{angularfrequency}{\rad\usk\reciprocal\s}[\rad\per\s][\rad\per\s] \newphysicsquantity{charge}{\A\usk\s}[\C][\C] \newphysicsquantity{permittivity}{\m\reciprocalcubed\usk\reciprocal\kg\usk\s \reciprocalquarted\usk\A\squared}[\F\per\m][\C\squared\per\N\usk\m\squared] \newphysicsquantity{permeability}{\m\usk\kg\usk\s\reciprocalsquared\usk\A \reciprocalsquared}[\henry\per\m][\T\usk\m\per\A] \newphysicsquantity{electricfield}{\m\usk\kg\usk\s\reciprocalcubed\usk\reciprocal\A} [\V\per\m][\N\per\C] \newphysicsquantity{electricdipolemoment}{\m\usk\s\usk\A}[\C\usk\m][\C\usk\m] \newphysicsquantity{magneticfield}{\kg\usk\s\reciprocalsquared\usk\reciprocal\A}[\T] [\N\per\C\usk(\m\per\s)] \newphysicsquantity{cmagneticfield}{\m\usk\kg\usk\s\reciprocalcubed\usk\reciprocal\A} [\V\per\m][\N\per\C] \newphysicsquantity{linearchargedensity}{\reciprocal\m\usk\s\usk\A}[\C\per\m][\C\per\m] \newphysicsquantity{areachargedensity}{\reciprocalsquare\m\usk\s\usk\A} [\C\per\square\m][\C\per\square\m] \newphysicsquantity{volumechargedensity}{\reciprocalcubic\m\usk\s\usk\A} [\C\per\cubic\m][\C\per\cubic\m] \newphysicsquantity{mobility} {\m\squared\usk\kg\usk\s\reciprocalquarted\usk\reciprocal\A}[\m\squared\per\volt\usk\s] [(\m\per\s)\per(\N\per\C)] \newphysicsquantity{numberdensity}{\reciprocalcubic\m}[\per\cubic\m][\per\cubic\m] \newphysicsquantity{polarizability}{\reciprocal\kg\usk\s\quarted\usk\square\A} [\C\usk\square\m\per\V][\C\usk\m\per(\N\per\C)] \newphysicsquantity{electricpotential} {\square\m\usk\kg\usk\reciprocalcubic\s\usk\reciprocal\A}[\J\per\C][\V] \newphysicsquantity{emf}{\square\m\usk\kg\usk\reciprocalcubic\s\usk\reciprocal\A} [\J\per\C][\V] \newphysicsquantity{dielectricconstant}{}[][] \newphysicsquantity{indexofrefraction}{}[][] \newphysicsquantity{relativepermittivity}{}[][] \newphysicsquantity{relativepermeability}{}[][] \newphysicsquantity{energydensity}{\m\reciprocaled\usk\kg\usk\reciprocalsquare\s} [\J\per\cubic\m][\J\per\cubic\m] \newphysicsquantity{electroncurrent}{\reciprocal\s} [\ensuremath{\mathrm{e}}\per\s][\ensuremath{\mathrm{e}}\per\s] \newphysicsquantity{conventionalcurrent}{\A}[\C\per\s][\A] \newphysicsquantity{magneticdipolemoment}{\square\m\usk\A}[\J\per\T][\A\usk\square\m] \newphysicsquantity{currentdensity}{\reciprocalsquare\m\usk\A}[\C\usk\s\per\square\m] [\A\per\square\m] \newphysicsquantity{capacitance} {\reciprocalsquare\m\usk\reciprocal\kg\usk\quartic\s\usk\square\A}[\C\per\V][\F] \newphysicsquantity{conductivity} {\reciprocalcubic\m\usk\reciprocal\kg\usk\cubic\s\usk\square\A}[\siemens\per\m] [\A\per\V\usk\m] \newphysicsquantity{resistivity} {\cubic\m\usk\kg\usk\reciprocalcubic\s\usk\reciprocalsquare\A}[\m\per\siemens] [\ohm\usk\m] \newphysicsquantity{resistance} {\square\m\usk\kg\usk\reciprocalcubic\s\usk\reciprocalsquare\A}[\V\per\A][\ohm] \newphysicsquantity{conductance} {\reciprocalsquare\m\usk\reciprocal\kg\usk\cubic\s\usk\square\A}[\A\per\V][\siemens] \newcommand{\lv}{\ensuremath{\left\langle}} \newcommand{\rv}{\ensuremath{\right\rangle}} \newcommand{\symvect}{\mivector} \newcommand{\ncompsvect}{\mivector} \ExplSyntaxOn \NewDocumentCommand{\magvectncomps}{ m O{} } {% \sum_of_squares:nn { #1 }{ #2 } }% \cs_new:Npn \sum_of_squares:nn #1 #2 {% \tl_if_empty:nTF { #2 } {% \clist_set:Nn \l_tmpa_clist { #1 } \ensuremath{% \sqrt{(\clist_use:Nnnn \l_tmpa_clist { )^2+( } { )^2+( } { )^2+( } )^2 } }% }% {% \clist_set:Nn \l_tmpa_clist { #1 } \ensuremath{% \sqrt{(\clist_use:Nnnn \l_tmpa_clist {\;{ #2 })^2+(} {\;{ #2 })^2+(} {\;{ #2 })^2+(} \;{ #2 })^2} }% }% }% \ExplSyntaxOff \newcommand{\zerovect}{\vect{0}} \newcommand{\ncompszerovect}{\mivector{0,0,0}} \ifthenelse{\boolean{@optitalicvectors}} {\newcommand{\vect}[1]{\ensuremath{\vec{#1}}}} {\newcommand{\vect}[1]{\ensuremath{\vec{\mathrm{#1}}}}} \ifthenelse{\boolean{@optdoubleabsbars}} {\newcommand{\magvect}[1]{\ensuremath{\magof{\vect{#1}}}}} {\newcommand{\magvect}[1]{\ensuremath{\abs{\vect{#1}}}}} \newcommand{\dmagvect}[1]{\ensuremath{\dx{\magvect{#1}}}} \newcommand{\Dmagvect}[1]{\ensuremath{\Delta\!\magvect{#1}}} \ifthenelse{\boolean{@optitalicvectors}} {\newcommand{\dirvect}[1]{\ensuremath{\widehat{{#1}}}}} {\newcommand{\dirvect}[1]{\ensuremath{\widehat{\mathrm{#1}}}}} \ifthenelse{\boolean{@optitalicvectors}} {\newcommand{\compvect}[2]{\ensuremath{\ssub{#1}{\(#2\)}}}} {\newcommand{\compvect}[2]{\ensuremath{\ssub{\mathrm{#1}}{\(#2\)}}}} \newcommand{\scompsvect}[1]{\ensuremath{\lv \compvect{#1}{x}, \compvect{#1}{y}, \compvect{#1}{z}\rv}} \newcommand{\magvectscomps}[1]{\ensuremath{\sqrt{ \msup{\compvect{#1}{x}}{2}+ \msup{\compvect{#1}{y}}{2}+ \msup{\compvect{#1}{z}}{2}}}} \newcommand{\dvect}[1]{\ensuremath{\mathrm{d}\vect{#1}}} \newcommand{\Dvect}[1]{\ensuremath{\Delta\vect{#1}}} \newcommand{\dirdvect}[1]{\ensuremath{\widehat{\dvect{#1}}}} \newcommand{\dirDvect}[1]{\ensuremath{\widehat{\Dvect{#1}}}} \newcommand{\ddirvect}[1]{\ensuremath{\mathrm{d}\dirvect{E}}} \newcommand{\Ddirvect}[1]{\ensuremath{\Delta\dirvect{E}}} \ifthenelse{\boolean{@optdoubleabsbars}} {\newcommand{\magdvect}[1]{\ensuremath{\magof{\dvect{#1}}}} \newcommand{\magDvect}[1]{\ensuremath{\magof{\Dvect{#1}}}}} {\newcommand{\magdvect}[1]{\ensuremath{\abs{\dvect{#1}}}} \newcommand{\magDvect}[1]{\ensuremath{\abs{\Dvect{#1}}}}} \newcommand{\compdvect}[2]{\ensuremath{\mathrm{d}\compvect{#1}{#2}}} \newcommand{\compDvect}[2]{\ensuremath{\Delta\compvect{#1}{#2}}} \newcommand{\scompsdvect}[1]{\ensuremath{\lv \compdvect{#1}{x}, \compdvect{#1}{y}, \compdvect{#1}{z}\rv}} \newcommand{\scompsDvect}[1]{\ensuremath{\lv \compDvect{#1}{x}, \compDvect{#1}{y}, \compDvect{#1}{z}\rv}} \newcommand{\dervect}[2]{\ensuremath{\frac{\dvect{#1}}{\mathrm{d}{#2}}}} \newcommand{\Dervect}[2]{\ensuremath{\frac{\Dvect{#1}}{\Delta{#2}}}} \newcommand{\compdervect}[3]{\ensuremath{\dbyd{\compvect{#1}{#3}}{#2}}} \newcommand{\compDervect}[3]{\ensuremath{\DbyD{\compvect{#1}{#3}}{#2}}} \newcommand{\scompsdervect}[2]{\ensuremath{\lv \compdervect{#1}{#2}{x}, \compdervect{#1}{#2}{y}, \compdervect{#1}{#2}{z}\rv}} \newcommand{\scompsDervect}[2]{\ensuremath{\lv \compDervect{#1}{#2}{x}, \compDervect{#1}{#2}{y}, \compDervect{#1}{#2}{z}\rv}} \ifthenelse{\boolean{@optdoubleabsbars}} {\newcommand{\magdervect}[2]{\ensuremath{\magof{\dervect{#1}{#2}}}} \newcommand{\magDervect}[2]{\ensuremath{\magof{\Dervect{#1}{#2}}}}} {\newcommand{\magdervect}[2]{\ensuremath{\abs{\dervect{#1}{#2}}}} \newcommand{\magDervect}[2]{\ensuremath{\abs{\Dervect{#1}{#2}}}}} \newcommand{\dermagvect}[2]{\ensuremath{\dbyd{\magvect{#1}}{#2}}} \newcommand{\Dermagvect}[2]{\ensuremath{\DbyD{\magvect{#1}}{#2}}} \newcommand{\scompspos}{\mivector{x,y,z}} \newcommand{\comppos}[1]{\ensuremath{{#1}}} \newcommand{\scompsdpos}{\mivector{\mathrm{d}x,\mathrm{d}y,\mathrm{d}z}} \newcommand{\scompsDpos}{\mivector{\Delta x,\Delta y,\Delta z}} \newcommand{\compdpos}[1]{\ensuremath{\mathrm{d}{#1}}} \newcommand{\compDpos}[1]{\ensuremath{\Delta{#1}}} \newcommand{\scompsderpos}[1]{\ensuremath{\lv \frac{\mathrm{d}x}{\mathrm{d}{#1}},\frac{\mathrm{d}y}{\mathrm{d}{#1}}, \frac{\mathrm{d}z}{\mathrm{d}{#1}}\rv}} \newcommand{\scompsDerpos}[1]{\ensuremath{\lv \frac{\Delta x}{\Delta{#1}},\frac{\Delta y}{\Delta{#1}}, \frac{\Delta z}{\Delta{#1}}\rv}} \newcommand{\compderpos}[2]{\ensuremath{\frac{\mathrm{d}{#1}}{\mathrm{d}{#2}}}} \newcommand{\compDerpos}[2]{\ensuremath{\frac{\Delta{#1}}{\Delta{#2}}}} \newcommand{\vectsub}[2]{\ensuremath{\ssub{\vect{#1}}{#2}}} \ifthenelse{\boolean{@optitalicvectors}} {\newcommand{\compvectsub}[3]{\ensuremath{\ssub{#1}{#2,\(#3\)}}}} {\newcommand{\compvectsub}[3]{\ensuremath{\ssub{\mathrm{#1}}{#2,\(#3\)}}}} \newcommand{\scompsvectsub}[2]{\ensuremath{\lv \compvectsub{#1}{#2}{x}, \compvectsub{#1}{#2}{y}, \compvectsub{#1}{#2}{z}\rv}} \ifthenelse{\boolean{@optdoubleabsbars}} {\newcommand{\magvectsub}[2]{\ensuremath{\magof{\vectsub{#1}{#2}}}}} {\newcommand{\magvectsub}[2]{\ensuremath{\abs{\vectsub{#1}{#2}}}}} \newcommand{\magvectsubscomps}[2]{\ensuremath{\sqrt{ \msup{\compvectsub{#1}{#2}{x}}{2}+ \msup{\compvectsub{#1}{#2}{y}}{2}+ \msup{\compvectsub{#1}{#2}{z}}{2}}}} \ifthenelse{\boolean{@optitalicvectors}} {\newcommand{\dirvectsub}[2]{\ensuremath{\ssub{\widehat{#1}}{#2}}}} {\newcommand{\dirvectsub}[2]{\ensuremath{\ssub{\widehat{\mathrm{#1}}}{#2}}}} \newcommand{\dvectsub}[2]{\ensuremath{\mathrm{d}\vectsub{#1}{#2}}} \newcommand{\Dvectsub}[2]{\ensuremath{\Delta\vectsub{#1}{#2}}} \newcommand{\compdvectsub}[3]{\ensuremath{\mathrm{d}\compvectsub{#1}{#2}{#3}}} \newcommand{\compDvectsub}[3]{\ensuremath{\Delta\compvectsub{#1}{#2}{#3}}} \newcommand{\scompsdvectsub}[2]{\ensuremath{\lv \compdvectsub{#1}{#2}{x}, \compdvectsub{#1}{#2}{y}, \compdvectsub{#1}{#2}{z}\rv}} \newcommand{\scompsDvectsub}[2]{\ensuremath{\lv \compDvectsub{#1}{#2}{x}, \compDvectsub{#1}{#2}{y}, \compDvectsub{#1}{#2}{z},\rv}} \newcommand{\dermagvectsub}[3]{\ensuremath{\dbyd{\magvectsub{#1}{#2}}{#3}}} \newcommand{\Dermagvectsub}[3]{\ensuremath{\DbyD{\magvectsub{#1}{#2}}{#3}}} \newcommand{\dervectsub}[3]{\ensuremath{\dbyd{\vectsub{#1}{#2}}{#3}}} \newcommand{\Dervectsub}[3]{\ensuremath{\DbyD{\vectsub{#1}{#2}}{#3}}} \ifthenelse{\boolean{@optdoubleabsbars}} {\newcommand{\magdervectsub}[3]{\ensuremath{\magof{\dervectsub{#1}{#2}{#3}}}} \newcommand{\magDervectsub}[3]{\ensuremath{\magof{\Dervectsub{#1}{#2}{#3}}}}} {\newcommand{\magdervectsub}[3]{\ensuremath{\abs{\dervectsub{#1}{#2}{#3}}}} \newcommand{\magDervectsub}[3]{\ensuremath{\abs{\Dervectsub{#1}{#2}{#3}}}}} \newcommand{\compdervectsub}[4]{\ensuremath{\dbyd{\compvectsub{#1}{#2}{#4}}{#3}}} \newcommand{\compDervectsub}[4]{\ensuremath{\DbyD{\compvectsub{#1}{#2}{#4}}{#3}}} \newcommand{\scompsdervectsub}[3]{\ensuremath{\lv \compdervectsub{#1}{#2}{#3}{x}, \compdervectsub{#1}{#2}{#3}{y}, \compdervectsub{#1}{#2}{#3}{z}\rv}} \newcommand{\scompsDervectsub}[3]{\ensuremath{\lv \compDervectsub{#1}{#2}{#3}{x}, \compDervectsub{#1}{#2}{#3}{y}, \compDervectsub{#1}{#2}{#3}{z}\rv}} \newcommand{\comppossub}[2]{\ensuremath{\ssub{#2}{#1}}} \newcommand{\scompspossub}[1]{\ensuremath{\lv \comppossub{#1}{x}, \comppossub{#1}{y}, \comppossub{#1}{z}\rv}} \newcommand{\compdpossub}[2]{\ensuremath{\mathrm{d}\comppossub{#1}{#2}}} \newcommand{\compDpossub}[2]{\ensuremath{\Delta\comppossub{#1}{#2}}} \newcommand{\scompsdpossub}[1]{\ensuremath{\lv \compdpossub{#1}{x}, \compdpossub{#1}{y}, \compdpossub{#1}{z}\rv}} \newcommand{\scompsDpossub}[1]{\ensuremath{\lv \compDpossub{#1}{x}, \compDpossub{#1}{y}, \compDpossub{#1}{z}\rv}} \newcommand{\compderpossub}[3]{\ensuremath{\dbyd{\comppossub{#1}{#3}}{#2}}} \newcommand{\compDerpossub}[3]{\ensuremath{\DbyD{\comppossub{#1}{#3}}{#2}}} \newcommand{\scompsderpossub}[2]{\ensuremath{\lv \compderpossub{#1}{#2}{x}, \compderpossub{#1}{#2}{y}, \compderpossub{#1}{#2}{z}\rv}} \newcommand{\scompsDerpossub}[2]{\ensuremath{\lv \compDerpossub{#1}{#2}{x}, \compDerpossub{#1}{#2}{y}, \compDerpossub{#1}{#2}{z}\rv}} \newcommand{\relpos}[1]{\ensuremath{\vectsub{r}{#1}}} \newcommand{\relvel}[1]{\ensuremath{\vectsub{v}{#1}}} \newcommand{\relmom}[1]{\ensuremath{\vectsub{p}{#1}}} \newcommand{\relfor}[1]{\ensuremath{\vectsub{F}{#1}}} \newcommand{\vectdotvect}[2]{\ensuremath{{#1}\bullet{#2}}} \newcommand{\vectdotsvect}[2]{\ensuremath{\scompsvect{#1}\bullet\scompsvect{#2}}} \newcommand{\vectdotevect}[2]{\ensuremath{ \compvect{#1}{x}\compvect{#2}{x}+ \compvect{#1}{y}\compvect{#2}{y}+ \compvect{#1}{z}\compvect{#2}{z}}} \newcommand{\vectdotspos}[1]{\ensuremath{\scompsvect{#1}\bullet\scompspos}} \newcommand{\vectdotepos}[1]{\ensuremath{ \compvect{#1}{x}\comppos{x}+ \compvect{#1}{y}\comppos{y}+ \compvect{#1}{z}\comppos{z}}} \newcommand{\vectdotsdvect}[2]{\ensuremath{\scompsvect{#1}\bullet\scompsdvect{#2}}} \newcommand{\vectdotsDvect}[2]{\ensuremath{\scompsvect{#1}\bullet\scompsDvect{#2}}} \newcommand{\vectdotedvect}[2]{\ensuremath{ \compvect{#1}{x}\compdvect{#2}{x}+ \compvect{#1}{y}\compdvect{#2}{y}+ \compvect{#1}{z}\compdvect{#2}{z}}} \newcommand{\vectdoteDvect}[2]{\ensuremath{ \compvect{#1}{x}\compDvect{#2}{x}+ \compvect{#1}{y}\compDvect{#2}{y}+ \compvect{#1}{z}\compDvect{#2}{z}}} \newcommand{\vectdotsdpos}[1]{\ensuremath{\scompsvect{#1}\bullet\scompsdpos}} \newcommand{\vectdotsDpos}[1]{\ensuremath{\scompsvect{#1}\bullet\scompsDpos}} \newcommand{\vectdotedpos}[1]{\ensuremath{ \compvect{#1}{x}\compdpos{x}+ \compvect{#1}{y}\compdpos{y}+ \compvect{#1}{z}\compdpos{z}}} \newcommand{\vectdoteDpos}[1]{\ensuremath{ \compvect{#1}{x}\compDpos{x}+ \compvect{#1}{y}\compDpos{y}+ \compvect{#1}{z}\compDpos{z}}} \newcommand{\vectsubdotsvectsub}[4]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsvectsub{#3}{#4}}} \newcommand{\vectsubdotevectsub}[4]{\ensuremath{ \compvectsub{#1}{#2}{x}\compvectsub{#3}{#4}{x}+ \compvectsub{#1}{#2}{y}\compvectsub{#3}{#4}{y}+ \compvectsub{#1}{#2}{z}\compvectsub{#3}{#4}{z}}} \newcommand{\vectsubdotsdvectsub}[4]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsdvectsub{#3}{#4}}} \newcommand{\vectsubdotsDvectsub}[4]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsDvectsub{#3}{#4}}} \newcommand{\vectsubdotedvectsub}[4]{\ensuremath{ \compvectsub{#1}{#2}{x}\compdvectsub{#3}{#4}{x}+ \compvectsub{#1}{#2}{y}\compdvectsub{#3}{#4}{y}+ \compvectsub{#1}{#2}{z}\compdvectsub{#3}{#4}{z}}} \newcommand{\vectsubdoteDvectsub}[4]{\ensuremath{ \compvectsub{#1}{#2}{x}\compDvectsub{#3}{#4}{x}+ \compvectsub{#1}{#2}{y}\compDvectsub{#3}{#4}{y}+ \compvectsub{#1}{#2}{z}\compDvectsub{#3}{#4}{z}}} \newcommand{\vectsubdotsdvect}[3]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsdvect{#3}}} \newcommand{\vectsubdotsDvect}[3]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsDvect{#3}}} \newcommand{\vectsubdotedvect}[3]{\ensuremath{ \compvectsub{#1}{#2}{x}\compdvect{#3}{x}+ \compvectsub{#1}{#2}{y}\compdvect{#3}{y}+ \compvectsub{#1}{#2}{z}\compdvect{#3}{z}}} \newcommand{\vectsubdoteDvect}[3]{\ensuremath{ \compvectsub{#1}{#2}{x}\compDvect{#3}{x}+ \compvectsub{#1}{#2}{y}\compDvect{#3}{y}+ \compvectsub{#1}{#2}{z}\compDvect{#3}{z}}} \newcommand{\vectsubdotsdpos}[2]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsdpos}} \newcommand{\vectsubdotsDpos}[2]{\ensuremath{ \scompsvectsub{#1}{#2}\bullet\scompsDpos}} \newcommand{\vectsubdotedpos}[2]{\ensuremath{ \compvectsub{#1}{#2}{x}\compdpos{x}+ \compvectsub{#1}{#2}{y}\compdpos{y}+ \compvectsub{#1}{#2}{z}\compdpos{z}}} \newcommand{\vectsubdoteDpos}[2]{\ensuremath{ \compvectsub{#1}{#2}{x}\compDpos{x}+ \compvectsub{#1}{#2}{y}\compDpos{y}+ \compvectsub{#1}{#2}{z}\compDpos{z}}} \newcommand{\dervectdotsvect}[3]{\ensuremath{ \scompsdervect{#1}{#2}\bullet\scompsvect{#3}}} \newcommand{\Dervectdotsvect}[3]{\ensuremath{ \scompsDervect{#1}{#2}\bullet\scompsvect{#3}}} \newcommand{\dervectdotevect}[3]{\ensuremath{ \compdervect{#1}{#2}{x}\compvect{#3}{x}+ \compdervect{#1}{#2}{y}\compvect{#3}{y}+ \compdervect{#1}{#2}{z}\compvect{#3}{z}}} \newcommand{\Dervectdotevect}[3]{\ensuremath{ \compDervect{#1}{#2}{x}\compvect{#3}{x}+ \compDervect{#1}{#2}{y}\compvect{#3}{y}+ \compDervect{#1}{#2}{z}\compvect{#3}{z}}} \newcommand{\vectdotsdervect}[3]{\ensuremath{ \scompsvect{#1}\bullet\scompsdervect{#2}{#3}}} \newcommand{\vectdotsDervect}[3]{\ensuremath{ \scompsvect{#1}\bullet\scompsDervect{#2}{#3}}} \newcommand{\vectdotedervect}[3]{\ensuremath{ \compvect{#1}{x}\compdervect{#2}{#3}{x}+ \compvect{#1}{y}\compdervect{#2}{#3}{y}+ \compvect{#1}{z}\compdervect{#2}{#3}{z}}} \newcommand{\vectdoteDervect}[3]{\ensuremath{ \compvect{#1}{x}\compDervect{#2}{#3}{x}+ \compvect{#1}{y}\compDervect{#2}{#3}{y}+ \compvect{#1}{z}\compDervect{#2}{#3}{z}}} \newcommand{\dervectdotspos}[2]{\ensuremath{ \scompsdervect{#1}{#2}\bullet\scompspos}} \newcommand{\Dervectdotspos}[2]{\ensuremath{ \scompsDervect{#1}{#2}\bullet\scompspos}} \newcommand{\dervectdotepos}[2]{\ensuremath{ \compdervect{#1}{#2}{x}\comppos{x}+ \compdervect{#1}{#2}{y}\comppos{y}+ \compdervect{#1}{#2}{z}\comppos{z}}} \newcommand{\Dervectdotepos}[2]{\ensuremath{ \compDervect{#1}{#2}{x}\comppos{x}+ \compDervect{#1}{#2}{y}\comppos{y}+ \compDervect{#1}{#2}{z}\comppos{z}}} \newcommand{\dervectdotsdvect}[3]{\ensuremath{ \scompsdervect{#1}{#2}\bullet\scompsdvect{#3}}} \newcommand{\DervectdotsDvect}[3]{\ensuremath{ \scompsDervect{#1}{#2}\bullet\scompsDvect{#3}}} \newcommand{\dervectdotedvect}[3]{\ensuremath{ \compdervect{#1}{#2}{x}\compdvect{#3}{x}+ \compdervect{#1}{#2}{y}\compdvect{#3}{y}+ \compdervect{#1}{#2}{z}\compdvect{#3}{z}}} \newcommand{\DervectdoteDvect}[3]{\ensuremath{ \compDervect{#1}{#2}{x}\compDvect{#3}{x}+ \compDervect{#1}{#2}{y}\compDvect{#3}{y}+ \compDervect{#1}{#2}{z}\compDvect{#3}{z}}} \newcommand{\dervectdotsdpos}[2]{\ensuremath{ \scompsdervect{#1}{#2}\bullet\scompsdpos}} \newcommand{\DervectdotsDpos}[2]{\ensuremath{ \scompsDervect{#1}{#2}\bullet\scompsDpos}} \newcommand{\dervectdotedpos}[2]{\ensuremath{ \compdervect{#1}{#2}{x}\compdpos{x}+ \compdervect{#1}{#2}{y}\compdpos{y}+ \compdervect{#1}{#2}{z}\compdpos{z}}} \newcommand{\DervectdoteDpos}[2]{\ensuremath{ \compDervect{#1}{#2}{x}\compDpos{x}+ \compDervect{#1}{#2}{y}\compDpos{y}+ \compDervect{#1}{#2}{z}\compDpos{z}}} \newcommand{\ezero}{\ensuremath{\msub{\mathbf{e}}{0}}} \newcommand{\eone}{\ensuremath{\msub{\mathbf{e}}{1}}} \newcommand{\etwo}{\ensuremath{\msub{\mathbf{e}}{2}}} \newcommand{\ethree}{\ensuremath{\msub{\mathbf{e}}{3}}} \newcommand{\efour}{\ensuremath{\msub{\mathbf{e}}{4}}} \newcommand{\ek}[1]{\ensuremath{\msub{\mathbf{e}}{#1}}} \newcommand{\e}{\ek} \newcommand{\uezero}{\ensuremath{\msub{\widehat{\mathbf{e}}}{0}}} \newcommand{\ueone}{\ensuremath{\msub{\widehat{\mathbf{e}}}{1}}} \newcommand{\uetwo}{\ensuremath{\msub{\widehat{\mathbf{e}}}{2}}} \newcommand{\uethree}{\ensuremath{\msub{\widehat{\mathbf{e}}}{3}}} \newcommand{\uefour}{\ensuremath{\msub{\widehat{\mathbf{e}}}{4}}} \newcommand{\uek}[1]{\ensuremath{\msub{\widehat{\mathbf{e}}}{#1}}} \newcommand{\ue}{\uek} \newcommand{\ezerozero}{\ek{00}} \newcommand{\eoneone}{\ek{11}} \newcommand{\eonetwo}{\ek{12}} \newcommand{\eonethree}{\ek{13}} \newcommand{\eonefour}{\ek{14}} \newcommand{\etwoone}{\ek{21}} \newcommand{\etwotwo}{\ek{22}} \newcommand{\etwothree}{\ek{23}} \newcommand{\etwofour}{\ek{24}} \newcommand{\ethreeone}{\ek{31}} \newcommand{\ethreetwo}{\ek{32}} \newcommand{\ethreethree}{\ek{33}} \newcommand{\ethreefour}{\ek{34}} \newcommand{\efourone}{\ek{41}} \newcommand{\efourtwo}{\ek{42}} \newcommand{\efourthree}{\ek{43}} \newcommand{\efourfour}{\ek{44}} \newcommand{\euzero}{\ensuremath{\msup{\mathbf{e}}{0}}} \newcommand{\euone}{\ensuremath{\msup{\mathbf{e}}{1}}} \newcommand{\eutwo}{\ensuremath{\msup{\mathbf{e}}{2}}} \newcommand{\euthree}{\ensuremath{\msup{\mathbf{e}}{3}}} \newcommand{\eufour}{\ensuremath{\msup{\mathbf{e}}{4}}} \newcommand{\euk}[1]{\ensuremath{\msup{\mathbf{e}}{#1}}} \newcommand{\eu}{\euk} \newcommand{\euzerozero}{\euk{00}} \newcommand{\euoneone}{\euk{11}} \newcommand{\euonetwo}{\euk{12}} \newcommand{\euonethree}{\euk{13}} \newcommand{\euonefour}{\euk{14}} \newcommand{\eutwoone}{\euk{21}} \newcommand{\eutwotwo}{\euk{22}} \newcommand{\eutwothree}{\euk{23}} \newcommand{\eutwofour}{\euk{24}} \newcommand{\euthreeone}{\euk{31}} \newcommand{\euthreetwo}{\euk{32}} \newcommand{\euthreethree}{\euk{33}} \newcommand{\euthreefour}{\euk{34}} \newcommand{\eufourone}{\euk{41}} \newcommand{\eufourtwo}{\euk{42}} \newcommand{\eufourthree}{\euk{43}} \newcommand{\eufourfour}{\euk{44}} \newcommand{\gzero}{\ensuremath{\msub{\mathbf{\gamma}}{0}}} \newcommand{\gone}{\ensuremath{\msub{\mathbf{\gamma}}{1}}} \newcommand{\gtwo}{\ensuremath{\msub{\mathbf{\gamma}}{2}}} \newcommand{\gthree}{\ensuremath{\msub{\mathbf{\gamma}}{3}}} \newcommand{\gfour}{\ensuremath{\msub{\mathbf{\gamma}}{4}}} \newcommand{\gk}[1]{\ensuremath{\msub{\mathbf{\gamma}}{#1}}} \newcommand{\g}{\gk} \newcommand{\gzerozero}{\gk{00}} \newcommand{\goneone}{\gk{11}} \newcommand{\gonetwo}{\gk{12}} \newcommand{\gonethree}{\gk{13}} \newcommand{\gonefour}{\gk{14}} \newcommand{\gtwoone}{\gk{21}} \newcommand{\gtwotwo}{\gk{22}} \newcommand{\gtwothree}{\gk{23}} \newcommand{\gtwofour}{\gk{24}} \newcommand{\gthreeone}{\gk{31}} \newcommand{\gthreetwo}{\gk{32}} \newcommand{\gthreethree}{\gk{33}} \newcommand{\gthreefour}{\gk{34}} \newcommand{\gfourone}{\gk{41}} \newcommand{\gfourtwo}{\gk{42}} \newcommand{\gfourthree}{\gk{43}} \newcommand{\gfourfour}{\gk{44}} \newcommand{\guzero}{\ensuremath{\msup{\mathbf{\gamma}}{0}}} \newcommand{\guone}{\ensuremath{\msup{\mathbf{\gamma}}{1}}} \newcommand{\gutwo}{\ensuremath{\msup{\mathbf{\gamma}}{2}}} \newcommand{\guthree}{\ensuremath{\msup{\mathbf{\gamma}}{3}}} \newcommand{\gufour}{\ensuremath{\msup{\mathbf{\gamma}}{4}}} \newcommand{\guk}[1]{\ensuremath{\msup{\mathbf{\gamma}}{#1}}} \newcommand{\gu}{\guk} \newcommand{\guzerozero}{\guk{00}} \newcommand{\guoneone}{\guk{11}} \newcommand{\guonetwo}{\guk{12}} \newcommand{\guonethree}{\guk{13}} \newcommand{\guonefour}{\guk{14}} \newcommand{\gutwoone}{\guk{21}} \newcommand{\gutwotwo}{\guk{22}} \newcommand{\gutwothree}{\guk{23}} \newcommand{\gutwofour}{\guk{24}} \newcommand{\guthreeone}{\guk{31}} \newcommand{\guthreetwo}{\guk{32}} \newcommand{\guthreethree}{\guk{33}} \newcommand{\guthreefour}{\guk{34}} \newcommand{\gufourone}{\guk{41}} \newcommand{\gufourtwo}{\guk{42}} \newcommand{\gufourthree}{\guk{43}} \newcommand{\gufourfour}{\guk{44}} \ExplSyntaxOn \NewDocumentCommand{\mivector}{ O{,} m o }% {% \mi_vector:nn { #1 } { #2 } \IfValueT{#3}{\;{#3}} }% \seq_new:N \l__mi_list_seq \cs_new_protected:Npn \mi_vector:nn #1 #2 {% \ensuremath{% \seq_set_split:Nnn \l__mi_list_seq { , } { #2 } \int_compare:nF { \seq_count:N \l__mi_list_seq = 1 } { \left\langle } \seq_use:Nnnn \l__mi_list_seq { #1 } { #1 } { #1 } \int_compare:nF { \seq_count:N \l__mi_list_seq = 1 } { \right\rangle } }% }% \ExplSyntaxOff \ExplSyntaxOn \seq_new:N \l__vector_arg_seq \cs_new_protected:Npn \vector_main:nnnn #1 #2 #3 #4 {% \seq_set_split:Nnn \l__vector_arg_seq { #3 } { #4 } \begin{#1matrix} \seq_use:Nnnn \l__vector_arg_seq { #2 } { #2 } { #2 } \end{#1matrix} }% \NewDocumentCommand{\rowvector}{ O{,} m } {% \ensuremath{ \vector_main:nnnn { p } { \,\, } { #1 } { #2 } }% }% \NewDocumentCommand{\colvector}{ O{,} m } {% \ensuremath{ \vector_main:nnnn { p } { \\ } { #1 } { #2 } }% }% \ExplSyntaxOff \newcommandx{\scompscvect}[2][1,usedefault]{% \ifthenelse{\equal{#1}{}}% {% \colvector{\msub{#2}{1},\msub{#2}{2},\msub{#2}{3}}% }% {% \colvector{\msub{#2}{0},\msub{#2}{1},\msub{#2}{2},\msub{#2}{3}}% }% }% \newcommandx{\scompsrvect}[2][1,usedefault]{% \ifthenelse{\equal{#1}{}}% {% \rowvector[,]{\msub{#2}{1},\msub{#2}{2},\msub{#2}{3}}% }% {% \rowvector[,]{\msub{#2}{0},\msub{#2}{1},\msub{#2}{2},\msub{#2}{3}}% }% }% \newphysicsconstant{oofpez}{\ensuremath{\frac{1}{\phantom{_o}4\pi\ssub{\epsilon}{o}}}} {\scin[9]{9}}{\ensuremath{\m\cubed\usk\kg\usk\reciprocalquartic\s\usk\\reciprocalsquared}} [\m\per\farad][\newton\usk\m\squared\per\coulomb\squared] \newphysicsconstant{oofpezcs}{\ensuremath{\frac{1}{\phantom{_o}4\pi\ssub{\epsilon}{o} c^2\phantom{_o}}}}{\scin{-7}}{\m\usk\kg\usk\s\reciprocalsquared\usk\A\reciprocalsquared} [\T\usk\m\squared][\N\usk\s\squared\per\C\squared] \newphysicsconstant{epsz}{\ensuremath{\ssub{\epsilon}{o}}}{\scin[9]{-12}} {\m\reciprocalcubed\usk\reciprocal\kg\usk\s\quarted\usk\A\squared}[\F\per\m] [\C\squared\per\N\usk\m\squared] \newphysicsconstant{mzofp}{\ensuremath{\frac{\phantom{_oo}\ssub{\mu}{o}\phantom{_o}} {4\pi}}}{\scin{-7}}{\m\usk\kg\usk\s\reciprocalsquared\usk\A\reciprocalsquared} [\henry\per\m] [\tesla\usk\m\per\A] \newphysicsconstant{muz}{\ensuremath{\ssub{\mu}{o}}}{\scin[4\pi]{-7}} {\m\usk\kg\usk\s\reciprocalsquared\usk\A\reciprocalsquared}[\henry\per\m] [\T\usk\m\per\A] \newphysicsconstant{kboltz}{\ensuremath{\ssub{k}{B}}}{\scin[1.38]{-23}} {\kg\usk\m\squared\usk\reciprocalsquare\s\usk\reciprocal\K}[\joule\per\K][\J\per\K] \newcommand{\kboltznev}{\ensuremath{\scin[8.62]{-5}{\eV\per\K}}} \newphysicsconstant{stefan}{\ensuremath{\sigma}}{\scin[5.67]{-8}} {\kg\usk\s\reciprocalcubed\usk\K\reciprocalquarted}[\W\per\m\squared\usk\K^4] [\W\per\m\squared\usk\K\quarted] \newphysicsconstant{planck}{\ensuremath{h}}{\scin[6.62]{-34}} {\m\squared\usk\kg\usk\reciprocal\s}[\J\usk\s][\kg\usk\m\squared\per\s] \newcommand{\plancknev}{\ensuremath{\scin[4.136]{-15}{\eV\usk\s}}} \newphysicsconstant{planckbar}{\ensuremath{\hbar}}{\scin[1.05]{-34}} {\m\squared\usk\kg\usk\reciprocal\s}[\joule\usk\s][\kg\usk\m\squared\per\s] \newcommand{\planckbarnev}{\ensuremath{\scin[4.136]{-15}{\eV\usk\s}}} \newphysicsconstant{Navogadro}{\ensuremath{\ssub{N}{A}}}{\scin[6.022]{23}} {\reciprocal\mol}[\reciprocal\mol][\reciprocal\mol] \newphysicsconstant{bigG}{\ensuremath{G}}{\scin[6.67]{-11}} {\m\cubed\usk\reciprocal\kg\usk\s\reciprocalsquared}[\J\usk\m\per\kg\squared] [\N\usk\m\squared\per\kg\squared] \newphysicsconstant{littleg}{\ensuremath{g}}{9.80}{\m\usk\s\reciprocalsquared} [\N\per\kg][\m\per\s\squared] \newphysicsconstant{clight}{\ensuremath{c}}{\scin[3.00]{8}}{\m\usk\reciprocal\s} [\m\usk\reciprocal\s][\m\per\s] \newcommand{\clightnfn}{\ensuremath{\unit{1}{\mathrm{ft}\per\mathrm{n}\s}}} \newphysicsconstant{Ratom}{\ensuremath{\ssub{r}{atom}}}{\scin{-10}}{\m}[\m][\m] \newphysicsconstant{Mproton}{\ensuremath{\ssub{m}{proton}}}{\scin[1.673]{-27}} {\kg}[\kg][\kg] \newphysicsconstant{Mneutron}{\ensuremath{\ssub{m}{neutron}}}{\scin[1.675]{-27}} {\kg}[\kg][\kg] \newphysicsconstant{Mhydrogen}{\ensuremath{\ssub{m}{hydrogen}}}{\scin[1.673]{-27}} {\kg}[\kg][\kg] \newphysicsconstant{Melectron}{\ensuremath{\ssub{m}{electron}}}{\scin[9.109]{-31}} {\kg}[\kg][\kg] \newphysicsconstant{echarge}{\ensuremath{e}}{\scin[1.602]{-19}}{\A\usk\s}[\C][\C] \newphysicsconstant{Qelectron}{\ensuremath{\ssub{Q}{electron}}}{-\echargevalue} {\A\usk\s}[\C][\C] \newphysicsconstant{qelectron}{\ensuremath{\ssub{q}{electron}}}{-\echargevalue} {\A\usk\s}[\C][\C] \newphysicsconstant{Qproton}{\ensuremath{\ssub{Q}{proton}}}{+\echargevalue} {\A\usk\s}[\C][\C] \newphysicsconstant{qproton}{\ensuremath{\ssub{q}{proton}}}{+\echargevalue} {\A\usk\s}[\C][\C] \newphysicsconstant{MEarth}{\ensuremath{\ssub{M}{Earth}}}{\scin[6]{24}}{\kg}[\kg][\kg] \newphysicsconstant{MMoon}{\ensuremath{\ssub{M}{Moon}}}{\scin[7]{22}}{\kg}[\kg][\kg] \newphysicsconstant{MSun}{\ensuremath{\ssub{M}{Sun}}}{\scin[2]{30}}{\kg}[\kg][\kg] \newphysicsconstant{REarth}{\ensuremath{\ssub{R}{Earth}}}{\scin[6.4]{6}}{\m}[\m][\m] \newphysicsconstant{RMoon}{\ensuremath{\ssub{R}{Moon}}}{\scin[1.75]{6}}{\m}[\m][\m] \newphysicsconstant{RSun}{\ensuremath{\ssub{R}{Sun}}}{\scin[7]{8}}{\m}[\m][\m] \newphysicsconstant{ESdist}{\magvectsub{r}{ES}}{\scin[1.5]{11}}{\m}[\m][\m] \newphysicsconstant{SEdist}{\magvectsub{r}{SE}}{\scin[1.5]{11}}{\m}[\m][\m] \newphysicsconstant{EMdist}{\magvectsub{r}{EM}}{\scin[4]{8}}{\m}[\m][\m] \newphysicsconstant{MEdist}{\magvectsub{r}{ME}}{\scin[4]{8}}{\m}[\m][\m] \newcommand{\lightyear}{\ensuremath{\mathrm{ly}}} \newcommand{\Lightyear}{\ensuremath{\mathrm{LY}}} \newcommand{\cyear}{\ensuremath{c\usk\mathrm{year}}} \newcommand{\cyr}{\ensuremath{c\usk\mathrm{yr}}} \newcommand{\yyear}{\ensuremath{\mathrm{year}}} \newcommand{\yr}{\ensuremath{\mathrm{yr}}} \newcommand{\parsec}{\ensuremath{\mathrm{pc}}} \newphysicsconstant{LSun}{\ensuremath{\ssub{L}{Sun}}}{\scin[4]{26}} {\m\squared\usk\kg\usk\s\reciprocalcubed}[\W][\J\per\s] \newphysicsconstant{TSun}{\ensuremath{\ssub{T}{Sun}}}{5800}{\K}[\K][\K] \newphysicsconstant{MagSun}{\ensuremath{\ssub{M}{Sun}}}{+4.83}{}[][] \newphysicsconstant{magSun}{\ensuremath{\ssub{m}{Sun}}}{-26.74}{}[][] \newcommand{\Lstar}[1][\(\star\)]{\ensuremath{\ssub{L}{#1}}} \newcommand{\Lsolar}{\ensuremath{\Lstar[\(\odot\)]}} \newcommand{\Tstar}[1][\(\star\)]{\ensuremath{\ssub{T}{#1}}} \newcommand{\Tsolar}{\ensuremath{\Tstar[\(\odot\)]}} \newcommand{\Rstar}[1][\(\star\)]{\ensuremath{\ssub{R}{#1}}} \newcommand{\Rsolar}{\ensuremath{\Rstar[\(\odot\)]}} \newcommand{\Mstar}[1][\(\star\)]{\ensuremath{\ssub{M}{#1}}} \newcommand{\Msolar}{\ensuremath{\Mstar[\(\odot\)]}} \newcommand{\Fstar}[1][\(\star\)]{\ensuremath{\ssub{F}{#1}}} \newcommand{\fstar}[1][\(\star\)]{\ensuremath{\ssub{f}{#1}}} \newcommand{\FSun}{\ensuremath{\Fstar[Sun]}} \newcommand{\fSun}{\ensuremath{\fstar[Sun]}} \newcommand{\Fsolar}{\ensuremath{\Fstar[\(\odot\)]}} \newcommand{\fsolar}{\ensuremath{\fstar[\(\odot\)]}} \newcommand{\Magstar}[1][\(\star\)]{\ensuremath{\ssub{M}{#1}}} \newcommand{\magstar}[1][\(\star\)]{\ensuremath{\ssub{m}{#1}}} \newcommand{\Magsolar}{\ensuremath{\Magstar[\(\odot\)]}} \newcommand{\magsolar}{\ensuremath{\magstar[\(\odot\)]}} \newcommand{\Dstar}[1][\(\star\)]{\ensuremath{\ssub{D}{#1}}} \newcommand{\dstar}[1][\(\star\)]{\ensuremath{\ssub{d}{#1}}} \newcommand{\Dsolar}{\ensuremath{\Dstar[\(\odot\)]}} \newcommand{\dsolar}{\ensuremath{\dstar[\(\odot\)]}} \newcommand{\onehalf}{\ensuremath{\frac{1}{2}}\xspace} \newcommand{\onethird}{\ensuremath{\frac{1}{3}}\xspace} \newcommand{\onefourth}{\ensuremath{\frac{1}{4}}\xspace} \newcommand{\onefifth}{\ensuremath{\frac{1}{5}}\xspace} \newcommand{\onesixth}{\ensuremath{\frac{1}{6}}\xspace} \newcommand{\oneseventh}{\ensuremath{\frac{1}{7}}\xspace} \newcommand{\oneeighth}{\ensuremath{\frac{1}{8}}\xspace} \newcommand{\oneninth}{\ensuremath{\frac{1}{9}}\xspace} \newcommand{\onetenth}{\ensuremath{\frac{1}{10}}\xspace} \newcommand{\twothirds}{\ensuremath{\frac{2}{3}}\xspace} \newcommand{\twofifths}{\ensuremath{\frac{2}{5}}\xspace} \newcommand{\twosevenths}{\ensuremath{\frac{2}{7}}\xspace} \newcommand{\twoninths}{\ensuremath{\frac{2}{9}}\xspace} \newcommand{\threehalves}{\ensuremath{\frac{3}{2}}\xspace} \newcommand{\threefourths}{\ensuremath{\frac{3}{4}}\xspace} \newcommand{\threefifths}{\ensuremath{\frac{3}{5}}\xspace} \newcommand{\threesevenths}{\ensuremath{\frac{3}{7}}\xspace} \newcommand{\threeeighths}{\ensuremath{\frac{3}{8}}\xspace} \newcommand{\threetenths}{\ensuremath{\frac{3}{10}}\xspace} \newcommand{\fourthirds}{\ensuremath{\frac{4}{3}}\xspace} \newcommand{\dx}[1]{\ensuremath{\,\mathrm{d}{#1}}} \newcommand{\evalfromto}[3]{\ensuremath{\Bigg.{#1}\Bigg\rvert_{#2}^{#3}}} \@ifpackageloaded{physymb}{% \typeout{mandi: Package physymb detected. Its commands will be used.} }{% \newcommand{\evalat}[2]{\ensuremath{\Bigg.{#1}\Bigg\rvert_{#2}}} }% \newcommand{\evaluatedat}[1]{\ensuremath{\Bigg.\Bigg\rvert_{#1}}} \newcommandx{\integral}[4][1,2,usedefault]{\ensuremath{ \int_{\ifthenelse{\equal{#1}{}}{}{#4=#1}}^{\ifthenelse{\equal{#2}{}}{}{#4=#2}}} {#3}\dx{#4}} \newcommandx{\Integral}[4][1,2,usedefault]{\ensuremath{ \bigint_{\ifthenelse{\equal{#1}{}}{}{#4=#1}}^{\ifthenelse{\equal{#2}{}}{} {#4=#2}}}{#3}\dx{#4}} \newcommand{\opensurfintegral}[2]{\ensuremath{ \int\nolimits_{#1}\vectdotvect{\vect{#2}}{\dirvect{n}}\dx{A}}} \newcommand{\opensurfIntegral}[2]{\ensuremath{ \bigint\nolimits_{\mskip -25.00mu\displaystyle\mathbf{#1}} \vectdotvect{\vect{#2}}{\dirvect{n}} \dx{A}}} \newcommand{\closedsurfintegral}[2]{\ensuremath{ \oint\nolimits_{#1}\vectdotvect{\vect{#2}}{\dirvect{n}}\dx{A}}} \newcommand{\closedsurfIntegral}[2]{\ensuremath{ \bigoint\nolimits_{\mskip -25.00mu\displaystyle\mathbf{#1}}\;\; \vectdotvect{\vect{#2}}{\dirvect{n}}\dx{A}}} \newcommand{\openlineintegral}[2]{\ensuremath{ \int\nolimits_{#1}\vectdotvect{\vect{#2}}{\dirvect{t}} \dx{\ell}}} \newcommand{\openlineIntegral}[2]{\ensuremath{ \bigint\nolimits_{\mskip -25.00mu\displaystyle\mathbf{#1}} \vectdotvect{\vect{#2}}{\dirvect{t}}\dx{\ell}}} \newcommand{\closedlineintegral}[2]{\ensuremath{ \oint\nolimits_{#1}\vectdotvect{\vect{#2}}{\dirvect{t}}\dx{\ell}}} \newcommand{\closedlineIntegral}[2]{\ensuremath{ \bigoint\nolimits_{\mskip -25.00mu\displaystyle\mathbf {#1}}\;\; \vectdotvect{\vect{#2}}{\dirvect{t}}\dx{\ell}}} \newcommandx{\dbydt}[1][1]{\ensuremath{\frac{\mathrm{d}{#1}}{\mathrm{d}t}}} \newcommandx{\DbyDt}[1][1]{\ensuremath{\frac{\Delta{#1}}{\Delta t}}} \newcommandx{\ddbydt}[1][1]{\ensuremath{\frac{\mathrm{d}^{2}{#1}}{\mathrm{d}t^{2}}}} \newcommandx{\DDbyDt}[1][1]{\ensuremath{\frac{\Delta^{2}{#1}}{\Delta t^{2}}}} \newcommandx{\pbypt}[1][1]{\ensuremath{\frac{\partial{#1}}{\partial t}}} \newcommandx{\ppbypt}[1][1]{\ensuremath{\frac{\partial^{2}{#1}}{\partial t^{2}}}} \newcommand{\dbyd}[2]{\ensuremath{\frac{\mathrm{d}{#1}}{\mathrm{d}{#2}}}} \newcommand{\DbyD}[2]{\ensuremath{\frac{\Delta{#1}}{\Delta{#2}}}} \newcommand{\ddbyd}[2]{\ensuremath{\frac{\mathrm{d}^{2}{#1}}{\mathrm{d}{#2}^{2}}}} \newcommand{\DDbyD}[2]{\ensuremath{\frac{\Delta^{2}{#1}}{\Delta{#2}^{2}}}} \newcommand{\pbyp}[2]{\ensuremath{\frac{\partial{#1}}{\partial{#2}}}} \newcommand{\ppbyp}[2]{\ensuremath{\frac{\partial^{2}{#1}}{\partial{#2}^{2}}}} \newcommand{\seriesfofx}{\ensuremath{% f(x) \approx f(a) + \frac{f^\prime (a)}{1!}(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots}\xspace} \newcommand{\seriesexpx}{\ensuremath{% e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots}\xspace} \newcommand{\seriessinx}{\ensuremath{% \sin x \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots}\xspace} \newcommand{\seriescosx}{\ensuremath{% \cos x \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots}\xspace} \newcommand{\seriestanx}{\ensuremath{% \tan x \approx x + \frac{x^3}{3} + \frac{2x^5}{15} + \ldots}\xspace} \newcommand{\seriesatox}{\ensuremath{% a^x \approx 1 + x \ln{a} + \frac{(x \ln a)^2}{2!} + \frac{(x \ln a)^3}{3!} + \ldots} \xspace} \newcommand{\serieslnoneplusx}{\ensuremath{% \ln(1 \pm x) \approx \pm\; x - \frac{x^2}{2} \pm \frac{x^3}{3} - \frac{x^4}{4} \pm \ldots} \xspace} \newcommand{\binomialseries}{\ensuremath{% (1 + x)^n \approx 1 + nx + \frac{n(n-1)}{2!}x^2 + \ldots}\xspace} \@ifpackageloaded{physymb}{% \typeout{mandi: Package physymb detected. Its commands will be used.} }{% \newcommand{\gradient}{\ensuremath{\nabla}} \newcommand{\divergence}{\ensuremath{\nabla\bullet}} \newcommand{\curl}{\ensuremath{\nabla\times}} \newcommand{\laplacian}{\ensuremath{\msup{\nabla}{2}}} \newcommand{\dalembertian}{\ensuremath{\Box}} }% \newcommand{\diracdelta}[1]{\ensuremath{\boldsymbol{\delta}\quant{#1}}} \@ifpackageloaded{physymb}{% \typeout{mandi: Package physymb detected. Its commands will be used.} }{% \DeclareMathOperator{\asin}{\sin^{-1}} \DeclareMathOperator{\acos}{\cos^{-1}} \DeclareMathOperator{\atan}{\tan^{-1}} \DeclareMathOperator{\asec}{\sec^{-1}} \DeclareMathOperator{\acsc}{\csc^{-1}} \DeclareMathOperator{\acot}{\cot^{-1}} \DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\csch}{csch} \DeclareMathOperator{\asinh}{\sinh^{-1}} \DeclareMathOperator{\acosh}{\cosh^{-1}} \DeclareMathOperator{\atanh}{\tanh^{-1}} \DeclareMathOperator{\asech}{\sech^{-1}} \DeclareMathOperator{\acsch}{\csch^{-1}} \DeclareMathOperator{\acoth}{\coth^{-1}} \DeclareMathOperator{\sgn}{sgn} }% \DeclareMathOperator{\dex}{dex} \newcommand{\eV}{\electronvolt} \newcommand{\ev}{\electronvolt} \newcommand{\logb}[1][\relax]{\ensuremath{\log_{_{#1}}}} \ifthenelse{\boolean{@optitalicvectors}} {\newcommand{\cB}{\ensuremath{c\mskip -5.00mu B}}} {\newcommand{\cB}{\ensuremath{\textsf{c}\mskip -3.00mu\mathrm{B}}}} \newcommand{\newpi}{\ensuremath{\pi\mskip -7.8mu\pi}} \newcommand{\scripty}[1]{\ensuremath{\mathcalligra{#1}}} \newcommandx{\flux}[1][1]{\ensuremath{\ssub{\Phi}{#1}}} \@ifpackageloaded{physymb}{% \typeout{mandi: Package physymb detected. Its commands will be used.} }{% \newcommand{\abs}[1]{\ensuremath{\left\lvert{#1}\right\rvert}} }% \newcommand{\magof}[1]{\ensuremath{\left\lVert{#1}\right\rVert}} \newcommand{\dimsof}[1]{\ensuremath{\left[{#1}\right]}} \newcommand{\unitsof}[1]{\ensuremath{\left[{#1}\right]_{_{u}}}} \newcommand{\quant}[1]{\ensuremath{\left({#1}\right)}} \newcommand{\bquant}[1]{\ensuremath{\left[{#1}\right]}} \newcommand{\changein}[1]{\ensuremath{\delta{#1}}} \newcommand{\Changein}[1]{\ensuremath{\Delta{#1}}} \newcommandx{\scin}[3][1,3=\!\!,usedefault]{\ensuremath{ \ifthenelse{\equal{#1}{}} {\unit{\msup{10}{#2}}{#3}} {\unit{\msup{{#1}\times 10}{#2}}{#3}}}} \newcommand{\ee}[2]{\texttt{{#1}e{#2}}} \newcommand{\EE}[2]{\texttt{{#1}E{#2}}} \newcommand{\dms}[3]{\ensuremath{\indegrees{#1}\inarcminutes{#2}\inarcseconds{#3}}} \newcommand{\hms}[3]{\ensuremath{{#1}^{\hour}{#2}^{\mathrm{m}}{#3}^{\s}}} \newcommand{\clockreading}{\hms} \newcommand{\latitude}[1]{\ensuremath{\unit{#1}{\degree}}} \newcommand{\latitudeN}[1]{\ensuremath{\unit{#1}{\degree\; N}}} \newcommand{\latitudeS}[1]{\ensuremath{\unit{#1}{\degree\; S}}} \newcommand{\longitude}[1]{\ensuremath{\unit{#1}{\degree}}} \newcommand{\longitudeE}[1]{\ensuremath{\unit{#1}{\degree\; E}}} \newcommand{\longitudeW}[1]{\ensuremath{\unit{#1}{\degree\; W}}} \newcommand{\ssub}[2]{\ensuremath{{#1}_{_{_{\mbox{\tiny{#2}}}}}}} \newcommand{\ssup}[2]{\ensuremath{{#1}^{^{^{\mbox{\tiny{#2}}}}}}} \newcommand{\ssud}[3]{\ensuremath{{#1}^{^{^{\mbox{\tiny{#2}}}}}_{_{_{\mbox{\tiny{#3}}}}}}} \newcommand{\msub}[2]{\ensuremath{#1^{^{\scriptstyle{{}}}}_{_{_{\scriptstyle{#2}}}}}} \newcommand{\msup}[2]{\ensuremath{#1^{^{\scriptstyle{#2}}}}} \newcommand{\msud}[3]{\ensuremath{#1^{^{\scriptstyle{#2}}}_{_{_{\scriptstyle{#3}}}}}} \newcommand{\levicivita}[1]{\ensuremath{\msub{\varepsilon}{#1}}} \newcommand{\xaxis}{\ensuremath{x\mbox{-axis }}} \newcommand{\yaxis}{\ensuremath{y\mbox{-axis }}} \newcommand{\zaxis}{\ensuremath{z\mbox{-axis }}} \newcommand{\naxis}[1]{\ensuremath{{#1}\mbox{-axis}}} \newcommand{\xyplane}{\ensuremath{xy\mbox{-plane }}} \newcommand{\yzplane}{\ensuremath{yz\mbox{-plane }}} \newcommand{\zxplane}{\ensuremath{zx\mbox{-plane }}} \newcommand{\yxplane}{\ensuremath{yx\mbox{-plane }}} \newcommand{\zyplane}{\ensuremath{zy\mbox{-plane }}} \newcommand{\xzplane}{\ensuremath{xz\mbox{-plane }}} \newcommand{\cuberoot}[1]{\ensuremath{\sqrt[3]{#1}}} \newcommand{\fourthroot}[1]{\ensuremath{\sqrt[4]{#1}}} \newcommand{\fifthroot}[1]{\ensuremath{\sqrt[5]{#1}}} \newcommand{\fsqrt}[1]{\ensuremath{\msup{#1}{\onehalf}}} \newcommand{\fcuberoot}[1]{\ensuremath{\msup{#1}{\onethird}}} \newcommand{\ffourthroot}[1]{\ensuremath{\msup{#1}{\onefourth}}} \newcommand{\ffifthroot}[1]{\ensuremath{\msup{#1}{\onefifth}}} \newcommand{\relgamma}[1]{\ensuremath{ \frac{1}{\sqrt{1-\msup{\quant{\frac{#1}{c}}}{2}}}}} \newcommand{\frelgamma}[1]{\ensuremath{ \msup{\quant{1-\frac{\msup{{#1}}{2}}{\msup{c}{2}}}}{-\onehalf}}} \newcommand{\oosqrtomxs}[1]{\ensuremath{\frac{1}{\sqrt{1-\msup{#1}{2}}}}} \newcommand{\oosqrtomx}[1]{\ensuremath{\frac{1}{\sqrt{1-{#1}}}}} \newcommand{\ooomx}[1]{\ensuremath{\frac{1}{1-{#1}}}} \newcommand{\ooopx}[1]{\ensuremath{\frac{1}{1+{#1}}}} \newcommand{\isequals}{\wordoperator{?}{=}\xspace} \newcommand{\wordoperator}[2]{\ensuremath{% \mathrel{\vcenter{\offinterlineskip \halign{\hfil\tiny\upshape##\hfil\cr\noalign{\vskip-.5ex} {#1}\cr\noalign{\vskip.5ex}{#2}\cr}}}}} \newcommand{\definedas}{\wordoperator{defined}{as}\xspace} \newcommand{\associated}{\wordoperator{associated}{with}\xspace} \newcommand{\adjustedby}{\wordoperator{adjusted}{by}\xspace} \newcommand{\earlierthan}{\wordoperator{earlier}{than}\xspace} \newcommand{\laterthan}{\wordoperator{later}{than}\xspace} \newcommand{\forevery}{\wordoperator{for}{every}\xspace} \newcommand{\pwordoperator}[2]{\ensuremath{\left(% \mathrel{\vcenter{\offinterlineskip \halign{\hfil\tiny\upshape##\hfil\cr\noalign{\vskip-.5ex} {#1}\cr\noalign{\vskip.5ex}{#2}\cr}}}\right)}}% \newcommand{\pdefinedas}{\pwordoperator{defined}{as}\xspace} \newcommand{\passociated}{\pwordoperator{associated}{with}\xspace} \newcommand{\padjustedby}{\pwordoperator{adjusted}{by}\xspace} \newcommand{\pearlierthan}{\pwordoperator{earlier}{than}\xspace} \newcommand{\platerthan}{\pwordoperator{later}{than}\xspace} \newcommand{\pforevery}{\pwordoperator{for}{every}\xspace} \newcommand{\defines}{\ensuremath{\stackrel{\text{\tiny{def}}}{=}}\xspace} \newcommand{\inframe}[1][\relax]{\ensuremath{\xrightarrow[\tiny{\mathcal #1}]{}}\xspace} \newcommand{\associates}{\ensuremath{\xrightarrow{\text{\tiny{assoc}}}}\xspace} \newcommand{\becomes}{\ensuremath{\xrightarrow{\text{\tiny{becomes}}}}\xspace} \newcommand{\rrelatedto}[1]{\ensuremath{\xLongrightarrow{\text{\tiny{#1}}}}} \newcommand{\lrelatedto}[1]{\ensuremath{\xLongleftarrow[\text{\tiny{#1}}]{}}} \newcommand{\brelatedto}[2]{\ensuremath{ \xLongleftrightarrow[\text{\tiny{#1}}]{\text{\tiny{#2}}}}} \newcommand{\momprinciple}{\ensuremath{ \vectsub{p}{sys,f}=\vectsub{p}{sys,i}+\Fnetsys\Delta t}} \newcommand{\LHSmomprinciple}{\ensuremath{ \vectsub{p}{sys,f}}} \newcommand{\RHSmomprinciple}{\ensuremath{ \vectsub{p}{sys,i}+\Fnetsys\Delta t}} \newcommand{\energyprinciple}{\ensuremath{\ssub{E}{sys,f}=\ssub{E}{sys,i}+ \ssub{W}{ext}+Q}} \newcommand{\LHSenergyprinciple}{\ensuremath{\ssub{E}{sys,f}}} \newcommand{\RHSenergyprinciple}{\ensuremath{\ssub{E}{sys,i}+\ssub{W}{ext}+Q}} \newcommand{\angularmomprinciple}{\ensuremath{\vectsub{L}{sys,A,f}=\vectsub{L}{sys,A,i}+ \Tnetsys\Delta t}} \newcommand{\LHSangularmomprinciple}{\ensuremath{\vectsub{L}{sys,A,f}}} \newcommand{\RHSangularmomprinciple}{\ensuremath{\vectsub{L}{sys,A,i}+\Tnetsys\Delta t}} \newcommand{\gravinteraction}{\ensuremath{ \bigGmathsymbol\frac{\msub{M}{1}\msub{M}{2}}{\msup{\magvectsub{r}{12}}{2}} \quant{-\dirvectsub{r}{12}}}} \newcommand{\elecinteraction}{\ensuremath{ \oofpezmathsymbol\frac{\msub{Q}{1}\msub{Q}{2}}{\msup{\magvectsub{r}{12}}{2}} \dirvectsub{r}{12}}} \newcommand{\Bfieldofparticle}{\ensuremath{ \mzofpmathsymbol\frac{Q\magvect{v}}{\msup{\magvect{r}}{2}}\dirvect{v}\times\dirvect{r}}} \newcommand{\Efieldofparticle}{\ensuremath{ \oofpezmathsymbol\frac{Q}{\msup{\magvect{r}}{2}}\dirvect{r}}} \newcommand{\Esys}{\ssub{E}{sys}} \newcommandx{\Us}[1][1]{\ssub{\ssub{U}{s}}{#1}} \newcommandx{\Ug}[1][1]{\ssub{\ssub{U}{g}}{#1}} \newcommandx{\Ue}[1][1]{\ssub{\ssub{U}{e}}{#1}} \newcommandx{\Ktrans}[1][1]{\ssub{\ssub{K}{trans}}{#1}} \newcommandx{\Krot}[1][1]{\ssub{\ssub{K}{rot}}{#1}} \newcommandx{\Eparticle}[1][1]{\ssub{\ssub{E}{particle}}{#1}} \newcommandx{\Einternal}[1][1]{\ssub{\ssub{E}{internal}}{#1}} \newcommandx{\Erest}[1][1]{\ssub{\ssub{E}{rest}}{#1}} \newcommandx{\Echem}[1][1]{\ssub{\ssub{E}{chem}}{#1}} \newcommandx{\Etherm}[1][1]{\ssub{\ssub{E}{therm}}{#1}} \newcommandx{\Evib}[1][1]{\ssub{\ssub{E}{vib}}{#1}} \newcommandx{\Ephoton}[1][1]{\ssub{\ssub{E}{photon}}{#1}} \newcommand{\DEsys}{\Changein\Esys} \newcommand{\DUs}{\Changein\Us} \newcommand{\DUg}{\Changein\Ug} \newcommand{\DUe}{\Changein\Ue} \newcommand{\DKtrans}{\Changein\Ktrans} \newcommand{\DKrot}{\Changein\Krot} \newcommand{\DEparticle}{\Changein\Eparticle} \newcommand{\DEinternal}{\Changein\Einternal} \newcommand{\DErest}{\Changein\Erest} \newcommand{\DEchem}{\Changein\Echem} \newcommand{\DEtherm}{\Changein\Etherm} \newcommand{\DEvib}{\Changein\Evib} \newcommand{\DEphoton}{\Changein\Ephoton} \newcommand{\Usfinal}{\ssub{\left(\onehalf\ks \msup{s}{2}\right)}{f}} \newcommand{\Usinitial}{\ssub{\left(\onehalf\ks \msup{s}{2}\right)}{i}} \newcommand{\Ugfinal}{\ssub{\left(-G\frac{\msub{M}{1}\msub{M}{2}} {\magvectsub{r}{12}}\right)}{f}} \newcommand{\Uginitial}{\ssub{\left(-G\frac{\msub{M}{1}\msub{M}{2}} {\magvectsub{r}{12}}\right)}{i}} \newcommand{\Uefinal}{\ssub{\left(\oofpezmathsymbol\frac{\ssub{Q}{1}\ssub{Q}{2}} {\magvectsub{r}{12}}\right)}{f}} \newcommand{\Ueinitial}{\ssub{\left(\oofpezmathsymbol\frac{\ssub{Q}{1}\ssub{Q}{2}} {\magvectsub{r}{12}}\right)}{i}} \newcommand{\ks}{\ssub{k}{s}} \newcommand{\Fnet}{\ensuremath{\vectsub{F}{net}}} \newcommand{\Fnetext}{\ensuremath{\vectsub{F}{net,ext}}} \newcommand{\Fnetsys}{\ensuremath{\vectsub{F}{net,sys}}} \newcommand{\Fsub}[1]{\ensuremath{\vectsub{F}{#1}}} \newcommand{\Tnet}{\ensuremath{\vectsub{T}{net}}} \newcommand{\Tnetext}{\ensuremath{\vectsub{T}{net,ext}}} \newcommand{\Tnetsys}{\ensuremath{\vectsub{T}{net,sys}}} \newcommand{\Tsub}[1]{\ensuremath{\vectsub{T}{#1}}} \newcommand{\vpythonline}{\lstinline[language=Python,numbers=left,numberstyle=\tiny, upquote=true,breaklines]} \lstnewenvironment{vpythonblock}{\lstvpython}{} \newcommand{\vpythonfile}{\lstinputlisting[language=Python,numbers=left, numberstyle=\tiny,upquote=true,breaklines]} \newcommandx{\emptyanswer}[2][1=0.80,2=0.1,usedefault] {\begin{minipage}{#1\textwidth}\hfill\vspace{#2\textheight}\end{minipage}} \newenvironmentx{activityanswer}[5][1=white,2=black,3=black,4=0.90,5=0.10,usedefault]{% \def\skipper{#5}% \def\response@fbox{\fcolorbox{#2}{#1}}% \begin{center}% \begin{lrbox}{\@tempboxa}% \begin{minipage}[c][#5\textheight][c]{#4\textwidth}\color{#3}% \vspace{#5\textheight}}{% \vspace{\skipper\textheight}% \end{minipage}% \end{lrbox}% \response@fbox{\usebox{\@tempboxa}}% \end{center}% }% \newenvironmentx{adjactivityanswer}[5][1=white,2=black,3=black,4=0.90,5=0.00, usedefault]{% \def\skipper{#5}% \def\response@fbox{\fcolorbox{#2}{#1}}% \begin{center}% \begin{lrbox}{\@tempboxa}% \begin{minipage}[c]{#4\textwidth}\color{#3}% \vspace{#5\textheight}}{% \vspace{\skipper\textheight}% \end{minipage}% \end{lrbox}% \response@fbox{\usebox{\@tempboxa}}% \end{center}% }% \newcommandx{\emptybox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.10,usedefault] {\begin{center} \fcolorbox{#3}{#2}{% \begin{minipage}[c][#6\textheight][c]{#5\textwidth}\color{#4}% {#1}% \end{minipage}}% \vspace{\baselineskip}% \end{center}% }% \newcommandx{\adjemptybox}[7][1=\hfill,2=white,3=black,4=black,5=0.90,6=,7=0.0,usedefault] {\begin{center} \fcolorbox{#3}{#2}{% \begin{minipage}[c]{#5\textwidth}\color{#4}% \vspace{#7\textheight}% {#1}% \vspace{#7\textheight}% \end{minipage}}% \vspace{\baselineskip}% \end{center}% }% \newcommandx{\answerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.1,usedefault] {\ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]}% \vspace{\baselineskip}% \end{center}}% {\emptybox[#1][#2][#3][#4][#5][#6]}% }% \newcommandx{\adjanswerbox}[7][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.1,7=0.0, usedefault] {\ifthenelse{\equal{#1}{}}% {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]}% \vspace{\baselineskip}% \end{center}}% {\adjemptybox[#1][#2][#3][#4][#5][#6][#7]}% }% \newcommandx{\smallanswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.10, usedefault] {\ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]}% \vspace{\baselineskip}% \end{center}}% {\emptybox[#1][#2][#3][#4][#5][#6]}% }% \newcommandx{\mediumanswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.20, usedefault]{% \ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]% }% \vspace{\baselineskip}% \end{center}% }% {\emptybox[#1][#2][#3][#4][#5][#6] }% }% \newcommandx{\largeanswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.25, usedefault]{% \ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]% }% \vspace{\baselineskip}% \end{center}% }% {\emptybox[#1][#2][#3][#4][#5][#6] }% }% \newcommandx{\largeranswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.33, usedefault]{% \ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]% }% \vspace{\baselineskip}% \end{center}% }% {\emptybox[#1][#2][#3][#4][#5][#6] }% }% \newcommandx{\hugeanswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.50, usedefault]{% \ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]% }% \vspace{\baselineskip}% \end{center}% }% {\emptybox[#1][#2][#3][#4][#5][#6] }% }% \newcommandx{\hugeranswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=0.75, usedefault]{% \ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]% }% \vspace{\baselineskip}% \end{center}% }% {\emptybox[#1][#2][#3][#4][#5][#6] }% }% \newcommandx{\fullpageanswerbox}[6][1=\hfill,2=white,3=black,4=black,5=0.90,6=1.00, usedefault]{% \ifthenelse{\equal{#1}{}} {\begin{center}% \fcolorbox{#3}{#2}{% \emptyanswer[#5][#6]}% \vspace{\baselineskip}% \end{center}}% {\emptybox[#1][#2][#3][#4][#5][#6]}% }% \mdfdefinestyle{miinstructornotestyle}{% hidealllines=false, skipbelow=\baselineskip, skipabove=\baselineskip, leftmargin=40pt, rightmargin=40pt, linewidth=1, roundcorner=10, frametitle={INSTRUCTOR NOTE}, frametitlebackgroundcolor=cyan!60, frametitlerule=true, frametitlerulewidth=1, backgroundcolor=cyan!25, linecolor=black, fontcolor=black, shadow=true} \NewEnviron{miinstructornote}{% \begin{mdframed}[style=miinstructornotestyle] \begin{adjactivityanswer}[cyan!25][cyan!25][black] \BODY \end{adjactivityanswer} \end{mdframed} }% \mdfdefinestyle{mistudentnotestyle}{% hidealllines=false, skipbelow=\baselineskip, skipabove=\baselineskip, leftmargin=40pt, rightmargin=40pt, linewidth=1, roundcorner=10, frametitle={STUDENT NOTE}, frametitlebackgroundcolor=cyan!60, frametitlerule=true, frametitlerulewidth=1, backgroundcolor=cyan!25, linecolor=black, fontcolor=black, shadow=true} \NewEnviron{mistudentnote}{% \begin{mdframed}[style=mistudentnotestyle] \begin{adjactivityanswer}[cyan!25][cyan!25][black] \BODY \end{adjactivityanswer} \end{mdframed} }% \mdfdefinestyle{miderivationstyle}{% hidealllines=false, skipbelow=\baselineskip, skipabove=\baselineskip, leftmargin=0pt, rightmargin=0pt, linewidth=1, roundcorner=10, frametitle={DERIVATION}, frametitlebackgroundcolor=orange!60, frametitlerule=true, frametitlerulewidth=1, backgroundcolor=orange!25, linecolor=black, fontcolor=black, shadow=true} \NewEnviron{miderivation}{% \begin{mdframed}[style=miderivationstyle] \setcounter{equation}{0} \begin{align} \BODY \end{align} \end{mdframed} }% \mdfdefinestyle{bwinstructornotestyle}{% hidealllines=false, skipbelow=\baselineskip, skipabove=\baselineskip, leftmargin=40pt, rightmargin=40pt, linewidth=1, roundcorner=10, frametitle={INSTRUCTOR NOTE}, frametitlebackgroundcolor=gray!50, frametitlerule=true, frametitlerulewidth=1, backgroundcolor=gray!20, linecolor=black, fontcolor=black, shadow=true} \NewEnviron{bwinstructornote}{% \begin{mdframed}[style=bwinstructornotestyle] \begin{adjactivityanswer}[gray!20][gray!20][black] \BODY \end{adjactivityanswer} \end{mdframed} }% \mdfdefinestyle{bwstudentnotestyle}{% hidealllines=false, skipbelow=\baselineskip, skipabove=\baselineskip, leftmargin=40pt, rightmargin=40pt, linewidth=1, roundcorner=10, frametitle={STUDENT NOTE}, frametitlebackgroundcolor=gray!50, frametitlerule=true, frametitlerulewidth=1, backgroundcolor=gray!20, linecolor=black, fontcolor=black, shadow=true} \NewEnviron{bwstudentnote}{% \begin{mdframed}[style=bwstudentnotestyle] \begin{adjactivityanswer}[gray!20][gray!20][black] \BODY \end{adjactivityanswer} \end{mdframed} }% \mdfdefinestyle{bwderivationstyle}{% hidealllines=false, skipbelow=\baselineskip, skipabove=\baselineskip, leftmargin=0pt, rightmargin=0pt, linewidth=1, roundcorner=10, frametitle={DERIVATION}, frametitlebackgroundcolor=gray!50, frametitlerule=true, frametitlerulewidth=1, backgroundcolor=gray!20, linecolor=black, fontcolor=black, shadow=true} \NewEnviron{bwderivation}{% \begin{mdframed}[style=bwderivationstyle] \setcounter{equation}{0} \begin{align} \BODY \end{align} \end{mdframed} }% \newcommand{\checkpoint}{% \vspace{1cm}\begin{center}|--------- CHECKPOINT ---------|\end{center}}% \newcommand{\image}[2]{% \begin{figure}[h!] \begin{center}% \includegraphics[scale=1]{#1}% \caption{#2}% \label{#1}% \end{center}% \end{figure}} \newcommand{\sneakyone}[1]{\ensuremath{\cancelto{1}{\frac{#1}{#1}}}} \newcommand{\chkphysicsquantity}[1]{% \cs{#1} }% \newcommandx{\vecto}[2][2,usedefault]{\ensuremath{% \ifthenelse{\equal{#2}{}}% {\vec{\mathrm #1}}% {\ssub{\vec{\mathrm #1}}{#2}}}}% \newcommandx{\compvecto}[3][2,usedefault]{\ensuremath{% \ifthenelse{\equal{#2}{}}% {\ssub{\mathrm #1}{\(#3\)}}% {\ssub{\mathrm #1}{#2,\(#3\)}}}}% \newcommandx{\scompsvecto}[2][2,usedefault]{\ensuremath{% \ifthenelse{\equal{#2}{}}% {\lv\compvecto{#1}{x},\compvecto{#1}{y},\compvecto{#1}{z}\rv}% {\lv\compvecto{#1}[#2]{x},\compvecto{#1}[#2]{y},\compvecto{#1}[#2]{z}\rv}}}% \newcommandx{\compposo}[2][1,usedefault]{\ensuremath{% \ifthenelse{\equal{#1}{}}% {#2}% {\ssub{#2}{#1}}}}% \newcommandx{\scompsposo}[1][1,usedefault]{\ensuremath{% \ifthenelse{\equal{#1}{}}% {\lv\compposo{x},\compposo{y},\compposo{z}\rv}% {\lv\compposo[#1]{x},\compposo[#1]{y},\compposo[#1]{z}\rv}}}% \endinput %% %% End of file `mandi.sty'.