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{\bf Math 222 in class problems} \hspace{.125in} \textbf{Week: March 22, 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 
 Let $C$ be the circle formed by intersecting the plane $x+z=1$ with the sphere $x^2+y^2+z^2=1$. Find a parametrization of $C$.
 
 \item Find the arclength of $\mathbf{r}(t) = (\ln t, 2t, t^2), \  t\in[1, e]$.  (Folland: $e^2$)
 
 \item Compute $\int_C \sqrt{z} ds$, where $C$ is parametrized by $\mathbf{r}(t) = (2\cos t, 2\sin t, t^2)$, $0\leq t\leq 2\pi$.  (Folland: $\frac{2}{3}[(1+4\pi^2)^{3/2}-1]$)

\item Find the work done by the vector field $F(x,y,z) = \langle y^2, 2xy+e^{3z}, 3ye^{3z} \rangle$
along the line segment starting at $(0,0,1)$ and ending at $(2,1,0)$


\item On HW 7: Jones 12.E, 12-7 \\ 
Define a 1-form $\alpha$ on the punctured plane $\R^2 \setminus \{ 0 \}$ by
\[
\alpha = \left( \frac{-y}{x^2 + y^2} \right) dx +  \left( \frac{x}{x^2 + y^2} \right) dy. 
\]

\begin{enumerate}
\item
 Calculate $\int_C \alpha$ for any circle $C$ of radius $R$ around the origin. 

\item Prove that in the half plane $\{ x >0 \}, \ \alpha$ is the differential of a function.  
\end{enumerate} 
\item Match the equations of vector fields with their graphs.  Determine which vector fields in 11-18 are conservative, and for the ones which are, find their potential functions.
\end{enumerate}


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