\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts}
\usepackage[pdftex]{graphics}
\usepackage{geometry,bm}
\geometry{paperheight=11in,paperwidth=8.5in,tmargin=1.0in,bmargin=1.0in,inner=1.00in,outer=1.00in}
\renewcommand{\today}{Fall 2017}
\newcommand{\field}[1]{\mathbb{#1}}
\newcommand{\nat}{\field{N}}
\newcommand{\zet}{\field{Z}}
\newcommand{\real}{\field{R}}
\newcommand{\rat}{\field{Q}}
\newcommand{\cpl}{\field{C}}
% \ilf is inline formula in displaystyle
\newcommand{\ilf}[1]{$\displaystyle #1 $}
% \nlf is  a non-centered version of \[..\]
\newcommand{\nlf}[1]{\\[.7\baselineskip]$\displaystyle #1
$\\[.7\baselineskip]}
% \nlfind is  a non-centered version of \[..\], indented 1 cm
\newcommand{\nlfind}[1]%
                 {\\[.7\baselineskip]$\hspace*{1cm}\displaystyle #1%
                 $\\[.7\baselineskip]}
\renewcommand{\theequation}{\arabic{equation}}
\newtheorem{prob}{Problem}
\newenvironment{pro}[1]{\begin{prob}[#1 points]\em }{\end{prob}}
\newtheorem{ext}[prob]{Extra Credit Problem}
\newenvironment{ex}[1]{\begin{ext}[#1 points]\em }{\end{ext}}
\newcommand{\vc}[1]{\bm{\vec{#1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\


\begin{document}
\thispagestyle{empty}
\noindent{\bf Math 3335\hfill Formula Sheet for Test 2\hfill\today}\bigskip\\
\begin{description}
\item{\textbf{Integrals.}} Let $\vc{r}(t)=(x(t),y(t),z(t)):[a,b]\to\real^3$ parametrize a curve $C$.
\begin{description}
\item{\textbf{Arclength. }} $\int_C ds=\int_a^b \|\vc{r'}(t)\|\,dt$.
\item{\textbf{Line integral with respect to arclength. }} For a scalar-valued function $f:\real^3\to\real$, \[\int_C f\, ds=\int_a^b f(\vc{r}(t))  \|\vc{r'}(t)\|\,dt.\]
\item{\textbf{Line integral. }} For a vector field $\vc{F}=(F_1,F_2,F_3):\real^3\to\real^3$, \[\int_C \vc{F}\cdot d\vc{r}=\int_C F_1\,dx+F_2\, dy+F_3\, dz=\int_a^b \vc{F}(\vc{r}(t))\cdot \vc{r'}(t)\,dt.\]
\end{description}
\item{\textbf{Integral Theorems in the Plane.}} Let $\vc{F}=(F_1,F_2):\real^2\to\real^2$ be a vector field, that is smooth on a bounded region $R$ in $\real^2$. Let $C$ denote the boundary curve of $R$, oriented in the standard way.
\begin{description}
\item{\textbf{Green's Theorem. }} \[\iint_R \left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\,dx\,dy=\oint_{C} F_1\, dx+F_2\, dy=\oint_C \vc{F}\cdot d\vc{r}.\]
\item{\textbf{Area Theorem. }} In particular, 
\[\mathop{Area}(R)=\iint_R dx\,dy=\oint_{C} x\,dy=-\oint_{C}y\,dx=\frac{1}{2}\oint_{C} x\,dy-y\,dx.\]
\end{description}
\end{description}
\begin{description}
\item{\textbf{More Integrals.}}  Let $R$ be a subset of $\real^2$, and let $\vc{r}(u,v)=(x(u,v),y(u,v),z(u,v)):R\to S$ parametrize a surface $S\subseteq\real^3$.
\begin{description}
\item{\textbf{Surface area. }} $\iint_S dA=\iint_R \|\vc{r_u}\times\vc{r_v}\| du\,dv$.
\item{\textbf{Surface integral. }} For a scalar-valued function $f:S\to\real$,
 \[\iint_S f\, dA=\iint_R f(\vc{r}(u,v))\|\vc{r_u}\times\vc{r_v}\|du\,dv\]
\item{\textbf{Flux integral. }} Let $\vc{F}=(F_1,F_2,F_3):S\to\real^3$ be a vector field, and let $\vc{n}$ be a choice of unit normal for the orientable surface $S$. Then 
\begin{eqnarray*}\iint_S \vc{F}\cdot\vc{n}\,dA&=&\iint_R \vc{F}(\vc{r}(u,v))\cdot(\vc{r_u}\times\vc{r_v})\, du\,dv\\
\end{eqnarray*}
\end{description}
\newpage
\item{\textbf{Integral Theorems in $\real^3$.}} 
\begin{description}
\item{\textbf{Gauss' Divergence Theorem.} } Let $T$ be a closed bounded solid in $\real^3$ with boundary surface $\partial T$, oriented such that its unit normal points outward. Let $\vc{F}=(F_1,F_2,F_3):T\to\real^3$ be a vector field.
\[\iiint_T \mathop{div} \vc{F}\,dV=\iiint_T (\nabla\cdot \vc{F})\,dV=\iint_{\partial T} \vc{F}\cdot\vc{n}\ dA.\]
\end{description}
\begin{description}
\item{\textbf{Stokes' Theorem.}} Let $S\subseteq\real^3$ be an orientable surface with the simple closed curve $\partial S$ as its boundary. Let $\vc{F}=S\to\real^3$ be a vector field. Then
\[\iint_S (\mathop{curl}\vc{F})\cdot\vc{n} \,dA=\iint_S (\nabla\times\vc{F})\cdot\vc{n} \,dA=\oint_{\partial S} \vc{F}\cdot d\vc{r}.\]
Here $\vc{n}$ is a choice of unit normal for $S$, and $\partial S$ is being traversed such that $\vc{n}$ is to the left of the curve $\partial S$.
\end{description}
\end{description}
\end{document}

