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\begin{center}{ \Large{Worksheets}  \\\large{10/19/2017} }\end{center}
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\begin{enumerate}
\item Compute the length of the cycloid $c(t)=(t-\sin t,1-\cos t)$, $t\in[0,2\pi]$.

[\emph{Hint: You will need the trigonometric identity $1-\cos(2\theta)=2\sin^2\theta$}.]\vspace{3.5in}
\item Let $F(x,y,z)=\langle xy^2,y+zx,zx^2\rangle$. Find $\textrm{div}~F$, $\textrm{curl}~F$ and $\textrm{div}~\nabla F$.\newpage
\item A wire is parametrized by $c(t)=(\cos t,\sin t,t)$, $0\le t\le \frac{\pi}{2}$. Let its mass density at point $(x,y,z)$ be given by $f(x,y,z)=xy$. Find the mass of the wire.

[\emph{Hint: You will need the trigonometric identity $\sin(2\theta)=2\sin\theta\cos\theta$}.]\vspace{4in}
\item (Exercise 25, p.259) Let $F:\mathbb{R}^3\to\mathbb{R}^3$ be a vector field. Which of the following expressions are meaningful, and which are nonsense? For those which are meaningful, decide whether the expression defines a scalar function or a vector field.
\begin{enumerate}
\item curl(grad F)\vspace{.5in}
\item grad(curl F)\vspace{.5in}
\item div(grad F)\vspace{.5in}
\item grad(div F)\vspace{.5in}
\item curl(div F)\vspace{.5in}
\item div(curl F)\vspace{.5in}
\end{enumerate}
\newpage
\item Find the work done by the force field $F(x,y)=\langle x,1\rangle$ on a particle that moves along the cycloid in Problem 1.\vspace{4in}
\item Fill in the blanks
\[\int_{0}^{1}{\int_{x}^{1}{\int_{0}^{y-x}{f(x,y,z)dzdydx}}}=\int_{{}}^{{}}{{}}\int_{{}}^{{}}{{}}\int_{{}}^{{}}{{}}~~~~~~~~dxdydz\]
Then sketch the region for the integral.
\newpage
\item Find a parametrization for the curve C which is the intersection of \begin{enumerate}\item the cylinder $y^2+z^2=1$ and the plane $z=x$\vspace{2in}
\item the cone $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=2$.\end{enumerate}\vspace{2in}
\item Compute the volume of the solid bounded by the planes $x=0$, $y=0$, $z=0$, $x+y=0$ and $x=z-y-1$.
\end{enumerate}
\end{document}











