\documentclass[12pt]{article}
\usepackage{fancyhdr}
\pagestyle{fancyplain}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ... 
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.Ïpng}
\setlength{\topmargin}{0in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{8.25in}
\setlength{\oddsidemargin}{0in}
\setlength{\evensidemargin}{0in}

% useful math stuff
\newcommand{\gv}[1]{\ensuremath{\mbox{\boldmath$ #1 $}}} 
\newcommand{\grad}[1]{\gv{\nabla} #1} % for gradient
\let\divsymb=\div % rename builtin command \div to \divsymb
\renewcommand{\div}[1]{\gv{\nabla} \cdot #1} % for divergence
\newcommand{\curl}[1]{\gv{\nabla} \times #1} % for curl


\begin{document}
\lhead{PHYS 2210   Fall 2011   Homework Set 1}
\chead{}
\rhead{ Due at 4 PM on \textbf{Aug 25, 2011}}

%\maketitle

Show and explain all of your work!   Correct answers for which we cannot follow your work are worth no credit.


\begin{enumerate}

\item (0 points)  Please install Mathematica 8.0 on your computer, or find a computer that you can regularly use that has it installed already (like the ones in the computer room).  As a registered CU student, you can obtain a (free!) student license from  \verb|http://oit.colorado.edu/software-hardware/site-licenses/mathematica|, but it can take a day or so to get the activation key.  Future problem sets will require some Mathematica use.  If you have never used Mathematica before, you might want to watch the first 2 or 3 parts to the ``Hands-on Start" Tutorial videos,  at 
\\ \verb|http://www.wolfram.com/broadcast/screencasts/handsonstart/|, which will take less than 25 minutes.


\item (0.5 point) Show that $\frac{1}{2}\sin(\frac{\pi}{2}+\alpha)+\frac{1}{2}\sin(\frac{\pi}{2}-\alpha) = \cos(\alpha).$

\item (0.5 point)  Given the two vectors $\vec{B} =\{2,2,1\}$ and $\vec{C}=\{1,2,-6\}$, what is $\vec{B}\times\vec{C}$?  

\item(0.5 point) Are $\vec{B}$ and $\vec{C}$ in the previous problem orthogonal to each other?  How can you tell?

\item (1 point) Given the function $f(x,y,z) = xy +z$, what is the gradient of $f$, $\grad f$?  Is it a vector or scalar?

\item (1 points) Given the function $\vec{F} = x\hat{i} + xy \hat{j} + \hat{k}$, what is the curl of $\vec{F}$, $\curl{\vec{F}$}?  Is it a vector or scalar?

\item (1 point) Given the function $\vec{F} = x\hat{i} + xy \hat{j} + \hat{k}$, what is the divergence of $\vec{F}$, $\div{ \vec{F}$}?  Is it a vector or scalar?

\item (2.5 points) Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle $\beta$ with its vertex at the origin.  (This intersection is sort of shaped like an ice cream cone.)  Show that in the limits where $\beta=0$  and $\beta=\pi$ that your expression gives the expected values.

\item (3 points)  A 3-kg block is moved up a 37$^\circ$ incline (relative to horizontal) under the action of a constant \emph{horizontal} force of 40 N.  The coefficient of kinetic friction is 0.1 and the block is displaced 2 m along the incline.  
	\begin{enumerate}
	  \item[a)] Calculate the work done by the 40 N force.
	  \item[b)] Calculate the work done by friction.
           \item[c)] If the block is initially at rest, how long does it take the block to travel 2 m along the incline? 
	\end{enumerate}	
\end{enumerate}


\end{document}  