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{\bf Quiz \#12; Tuesday, date: 04/17/2018}
{\bf MATH 53 Multivariable Calculus with Stankova}
{\bf Section \#117; time: 5 -- 6:30 pm}
{\bf GSI name: Kenneth Hung}
{\bf Student name:}
\vspace*{0.25in}
\begin{enumerate}
\item Show that the line integral is independent of path and evaluate the integral.
\[
\int_C -\cos y \,dx + (x \sin y - \cos y) \,dy,
\]
where $C$ is any path from $(3, 0)$ to $(1, \pi)$.
\item {\em True / False?} Fix two points $A$ and $B$ in a domain $D$. If $\int_C \mathbf{F} \cdot d\rr$ is the same for all paths $C$ from $A$ to $B$, then $\mathbf{F}$ must be conservative on $D$.
\item {\em True / False?} Here is another proof of Green's Theorem with holes in it: Suppose the region with hole is $D'$, the hole itself is $D_2$ and the region $D'$ with the hole filled is $D_1$. The outer and inner boundaries are $C_1$ and $C_2$. We can then apply Green's Theorem to $D_1$ and $D_2$, and subtract one integral from the other.
\end{enumerate}
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