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\begin{center}{ \Large{Worksheets}  \\\large{12/7/2017} }\end{center}
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\begin{enumerate}
\item  Let $\vec{F}:\mathbb{R}^3\to\mathbb{R}^3$ be a vector field, and $c=c(t)$ be a curve. You have learned three methods to compute $\int_c \vec{F}\cdot d\vec{s}$.\begin{enumerate}
\item What are they?\vspace{2in}
\item In what situations is one method more favorable than the others?\vspace{2in}
\item Use your judgment to choose a method to compute $\int_C \vec{F}\cdot d\vec{s}$ where 
\begin{enumerate}
\item $\vec{F}(x,y,z)=(x,~y,~z)$, and $c(t)=(t^2,3t,2t^3)$, $0\le t\le 1$.\vspace{3in}
\item $\vec{F}(x,y,z)=(yz,~zx,~xy)$, and $c$ consists of straight lines joining (1,0,0), (0,1,0), (0,0,1) (in this order).\vspace{3in}
\end{enumerate}
\end{enumerate}
\item  Let $\vec{F}:\mathbb{R}^3\to\mathbb{R}^3$ be a vector field, and $S=\Phi(u,v)$ be a surface. You have learned two methods to compute $\int_S \vec{F}\cdot d\vec{S}$.\begin{enumerate}
\item What are they?\vspace{2in}
\item In what situations is one method more favorable than the other?\vspace{2in}\newpage
\item Use your judgment to choose a method to compute $\int_S \vec{F}\cdot d\vec{S}$ where 
\begin{enumerate}
\item $\vec{F}(x,y,z)=(2x,~-2y,~z^2)$, and $S$ is the cylinder $x^2+y^2=4$ with $0\le z\le 1$.\vspace{3in}
\item $\vec{F}(x,y,z)=(2x,~y^2,~z^2)$, and $S$ is the unit sphere $x^2+y^2+z^2=1$.\vspace{3in}
\end{enumerate}
\end{enumerate}
\item Give a parametrization to the following shape (curve/ surface/ solid):\begin{enumerate}
\item The intersection of the cylinder $x^2+(y-1)^2$=1 and the graph $z=x^2-y^2$.\vspace{2in}\newpage
\item The triangle with vertices (1,0,0), (0,2,0), (0,1,1).\vspace{1.5in}
\item The upper half ellipsoid \[E=\left\{(x,y,z):\frac{x^2}{4}+(y-1)^2+4z^2=1,~~z\ge 0\right\}\]\vspace{1.5in}
\end{enumerate}
\item Let $f:\mathbb{R}^2\to\mathbb{R}$. Denote $\vec{X}=\begin{bmatrix}x\\y\end{bmatrix}$ and $\vec{a}=\begin{bmatrix}a_1\\a_2\end{bmatrix}$. Recall that the \emph{second-order} Taylor approximation of $f$ near $\vec{a}$ is
\[f(\vec{X})\approx f(\vec{a})+\nabla f(\vec{a})(\vec{X}-\vec{a})+\frac{1}{2}(\vec{X}-\vec{a})^T~Hf(\vec{a})~(\vec{X}-\vec{a})\]
Here $Hf(\vec{a})$ is the Hessian matrix.\\
Now compute the second-order Taylor approximation for the function $f(x,y)=e^x\cos y$ near the point $\vec{a}=(0,0)$. Then find an approximate value of $f$ at (.1, -.01).
\end{enumerate}
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