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{\bf Math 222 in class problems} \hspace{.125in} \textbf{Week: April 12, 2021} \hspace{.5in}  \textbf{Name:} %outlier malcom
\begin{enumerate}

%DO CHANGE OF COORDINATES IN WEEK 7

%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.


\item (Similar to HW \#9) Use Stokes' theorem to evaluate $\displaystyle \int_C F \cdot \mathbf{dr}$ where $F(x,y,z)= \langle -y^2, x, z^2\rangle$ and $C$ is the curve of intersection of the plane $y+z =2$ and the cylinder $x^2+y^2=1$.  Orient $C$ to be counterclockwise when viewed from above.  (Answer: $\pi$)

\item Use Stokes' theorem to compute $\displaystyle \iint_S \mbox{curl }F \cdot dS$ where $F (x,y,z)= \langle xz, yz, xy \rangle$ and $S$ is the portion of the sphere $x^2+y^2+z^2=4$ that lies inside the cylinder $x^2+y^2=1$ and above the $xy$-plane.  (The answer is 0.)


\item Jones Problem 13-6 \\
Let $m$ be a fixed real number and let $\gamma$ be the curve of intersection of the paraboloid $z=x^2+y^2$ and the plane $z=mx$.  Assume the curve has counterclockwise orientation as viewed from above, e.g. viewed from $(0,0,r)$ for large positive $r$.  Compute directly the line integral $\displaystyle \int_\gamma y \ dz$. Also compute the same line integral using Stokes' theorem.  (Answer: $-\pi m^3/4$.)

\item Let $C_r$ denote the circle of radius $r$ about the origin in the $xz$-plane, oriented counterclockwise as viewed from the positive $y$-axis.  Suppose $F$ is a $C^1$ vector field on the complement of the $y$-axis in $\R^3$ such that $\int_{C_1} F \cdot \mathbf{dx} = 5$ and curl $F(x,yz) = 3 \mathbf{j} +\dfrac{z\mathbf{i}-x\mathbf{k}}{(x^2+z^2)^2}$.  Compute $\int_{C_r} F \cdot \mathbf{dx}$ for every $r$.  (Answer: $5+3\pi(r^2-1).$)


 \item Is there a vector field $F$ on $\R^3$ such that curl $F = \langle x\sin y, \cos y, z-xy \rangle$? 


\item Show that 
\[
\mbox{div }( F \times B) = B \cdot \mbox{curl }F - F \cdot \mbox{curl }G.
\]



\end{enumerate}


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