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\flushleft\underline{Lab 8 --- MATH 1586 Spring 2018} \\
Recall: this site plots surfaces nicely: \href{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/}{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/} \\
Also, it is helpful to use \href{http://www.wolframalpha.com/}{WolframAlpha} to compute integrals. You may also of course use Mathcad to do it. \\
Recall that if $S$ is a surface parametrized by
$$\left\{ \begin{array}{ll}
\vec{r}(u,v) \\
a \leq u \leq b \\
c \leq v \leq d,
\end{array} \right.$$
then the vector field surface integral (``flux integral") is defined by
$$\displaystyle\iint_S \vec{F} \cdot \mathrm{d}\vec{r} = \displaystyle\int_a^b \displaystyle\int_c^d \vec{F}\left(\vec{r}(u,v) \right) \cdot \left( \dfrac{\partial \vec{r}}{\partial u} \times \dfrac{\partial \vec{r}}{\partial v} \right) \mathrm{d}t.$$
\begin{enumerate}
\item In the theory of electrostatics, Gauss' law says that the net charge enclosed by a closed surface $S$ is 
$$Q = \epsilon_0 \displaystyle\iint_S \vec{E} \cdot \mathrm{d}\vec{S},$$
where $\epsilon_0$ is a constant called the ``permittivity of free space".

In this problem, you will find the flux (i.e. vector field surface integral) of the vector field $\vec{F}=\langle z, y , x \rangle$ across the unit sphere parametrized by $\vec{r}(u,v)=\langle \sin(v)\cos(u), \sin(v)\sin(u), \cos(v)\rangle$ for $0 \leq u \leq 2\pi$ and $0 \leq \phi \leq \pi$. 

\begin{enumerate}
\item Draw the parametrized surface in this problem and attach an image of it to your lab.
\item Use WolframAlpha (or MathCad) to compute $\dfrac{\partial \vec{r}}{\partial u} \times \dfrac{\partial \vec{r}}{\partial v}$.
\item Write the vector field surface integral for this problem.
\item Use the computer to find the electric flux across the surface.
\end{enumerate}

\item If the temperature at a point in space is $T(x,y,z)$ then the ``heat flow" is defined as the vector field $\vec{F}=-K \nabla T$, where $K$ is an experimentally derived constant called the ``conductivity" of the substance. In this situation, the ``heat flow across the surface $S$" is given by
$$\displaystyle\iint_S \vec{F} \cdot d\vec{S} = -K \displaystyle\iint_S \nabla u \cdot d\vec{S}.$$

Consider the metal cylinder parametrized as 
$$\left\{ \begin{array}{ll}
\vec{r}(u,v)=\langle u, \sqrt{6}\cos(v), \sqrt{6}\sin(v)\rangle \\
0 \leq u \leq 4 \\
0 \leq v \leq 2\pi
\end{array} \right.$$
This cylinder has conductivity constant $K=6.5$. The temperature $T$ is $T(x,y,z)=2y^2+2z^2$. 

\begin{enumerate}
\item Plot the surface and attach a picture of it.
\item Find the heat flow vector field.
\item Use WolframAlpha (or MathCad) to compute $\dfrac{\partial \vec{r}}{\partial u} \times \dfrac{\partial \vec{r}}{\partial v}$.
\item Write the vector field surface integral for this problem.
\item Use the computer to find the rate of heat flow across the cylindrical surface.
\end{enumerate}

\end{enumerate}

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