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\begin{center}{ \Large{Worksheets}  \\\large{11/2/2017} }\end{center}
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\begin{enumerate}
\item Evaluate the line integral $\int_{c}{x^2ydx+ydy}$, where $c$ is the boundary of the region between the curves $y=x$ and $y=x^3$, $0\le x\le 1$, oriented counter-clockwise, by
\begin{enumerate}
\item Using the definition of line integral (of a vector field)\vspace{3.5in}
\item Using Green's theorem\vspace{3.5in}
\end{enumerate}
\item Evaluate the circulation of $F(x,y)=\langle x+y,~xy\rangle$ around the rectangle $0\le x\le2$, $0\le y\le 3$, by
\begin{enumerate}
\item Using the definition of line integral (calculate the line integral on each edge of the rectangle, then sum all up).\vspace{4in}
\item Using Green's theorem\vspace{2in}
\end{enumerate}
\item Recall that a vector field is called conservative if it is the gradient of a scalar function. If a vector field is defined in the whole plane or space, it is conservative if and only if its curl is equal to zero.\\

Decide if each of following vector field is conservative. If it is, find a function $f$ such that $F=\nabla f$
\begin{enumerate}
\item $F(x,y)=\langle y^2e^{xy^2},~2ye^{xy^2}\rangle$\vspace{2in}
\item $F(x,y,z)=\langle 3x^2y,~x,~5\rangle$\vspace{3in}
\item $F(x,y,z)=\langle x+z,~-y-z,~x-y\rangle$\vspace{4in}
\end{enumerate}
\item \begin{enumerate}
\item Compute the Jacobian determinant $\frac{\partial(x,y)}{\partial(r,\theta)}$ of the change of variables to polar coordinates $x=r\cos\theta$, $y=r\sin\theta$\vspace{2in}
\item Let $D$ be the unit disk $x^2+y^2\le 1$. Evaluate\[\iint_{D}{\exp ({{x}^{2}}+{{y}^{2}})dxdy}\] by making the change of variables to polar coordinates.\vspace{3in}
\end{enumerate}
\item Integrate $x^2+y^2+z^2$ over the cylinder $x^2+y^2\le 2$, $-2\le z\le 3$.

\end{enumerate}

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