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\flushleft\underline{Homework 11 --- MATH 1586 Spring 2018} \\
Recall that if $S$ is a surface parametrized by $\vec{r}(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle$ for $a \leq u \leq b$ and $c \leq v \leq d$, then we have two kinds of surface integral: first the scalar surface integral
$$\displaystyle\iint_S f(x,y,z) \mathrm{d}S = \displaystyle\int_a^b\int_c^d f(x(u,v),y(u,v),z(u,v)) \left\lVert \dfrac{\partial \vec{r}}{\partial u} \times \dfrac{\partial \vec{r}}{\partial v} \right\rVert \mathrm{d}v \mathrm{d}u,$$
and a vector field surface integral
$$\displaystyle\iint_S \vec{F} \cdot \mathrm{d}\vec{S} = \displaystyle\int_a^b\int_c^d \vec{F}(x(u,v),y(u,v),z(u,v)) \cdot \left( \dfrac{\partial \vec{r}}{\partial u} \times \dfrac{\partial \vec{r}}{\partial v} \right) \mathrm{d}v\mathrm{d}u.$$

\begin{enumerate}[1.]
\item Let $f(x,y,z)=x$ and let $\vec{r}(u,v)=\langle u,v,uv \rangle$ for $0 \leq u \leq 1$ and $0 \leq v \leq 1$. Set up \textbf{but do not evaluate} $\displaystyle\iint_S f(x,y,z) \mathrm{d}S$. 
\item Let $\vec{F}=\langle y,x,xyz \rangle$ and let $\vec{r}(u,v)=\langle uv, u^2, v^2 \rangle$ for $0 \leq u \leq 1$ and $0 \leq v \leq 1$. Set up \textbf{but do not evaluate} $\displaystyle\iint_S \vec{F} \cdot \mathrm{d}\vec{S}$. 
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