\documentclass[fleqn]{article}

\setlength{\topmargin}{-.3in}
\setlength{\oddsidemargin}{-.2in}
\setlength{\textheight}{7.5in}
\setlength{\textwidth}{6.5in}



\usepackage{amssymb}
\usepackage{amsmath}

\newcommand{\dotprod}{\bullet}

\begin{document}

\textbf{Reveiw for Final}

\newcommand{\R}{\mathbb{R}}

\begin{enumerate}

\item  Final: Friday, August 9.  12:30-2:30PM
\item  You can have a sheet of notes for the final.
\item  Look at the old finals on the web.
	Our final could differ significantly, but this should help you study
	(we have emphasized different things this semester then previous semesters).
\item  Curves in space:
	\begin{enumerate}
	\item  Graphing
	\item  Curvature (what does it tell you?)
	\item  Unit tangent, unit normal
	\item  Many curvature formulas (for example all the formulas in section 14.5)
	\end{enumerate}

\item  Derivatives of functions of more then one variable.
	\begin{enumerate}
	\item  Partial derivatives
	\item  Gradiant (what does it tell you?)
	\item  Directional deriviative (what does it tell you?)
	\item  Chain rule
	\item  Using derivatives to find maxima and minima of functions.
	\end{enumerate}

\item  Integration 
	\begin{enumerate}
	\item  Double integrals
	\item  Triple integrals
	\item  (Higher dimensional integrals?)
	\item  Non-rectangular regions (finding the correct limits for the iterated integral)
	\item  Change of variables (Jacobians).  Mainly polar coordinates, cylindrical coordinates, spherical coordinates.
	\item  Line integrals (how to compute a line integral in various different forms)
	\item  Surface integrals (how to compute one!)
	\item  Flux through a curve (a line integral)
	\item  Flux through a surface (a surface integral)
	\end{enumerate}



\item  Fundamental theorem (what do all these things mean?)
	\begin{enumerate}
	\item  \[ \int_C \nabla f(r) \dotprod d\mathbf{r} = f(\mathbf{b}) - f(\mathbf{a}) \]
	\item  When is a vector field conservative?
	\item  \[ \iint_S \left(N_x - M_y \right) \; dA = \oint_C M \; dx + N\; dy \]
	\item  \[ \oint_C \mathbf{F} \dotprod \mathbf{n} \; ds = \iint_S \mathrm{div } \mathbf{F} \; dA \]
	\item  \[ \oint_C \mathbf{F} \dotprod \mathbf{T} \; ds = \iint_S (\mathrm{curl } \mathbf{F}) \dotprod \mathbf{k} \; dA \]
	\item  \[ \iint_{\partial S} \mathbf{F} \dotprod \mathbf{n} \; dS = \iiint_S \mathrm{div } \mathbf{F} \; dV \]
	\end{enumerate}


\end{enumerate}







\end{document}