Chapter 15.6, Problem 2QCE

In these exercises, $\text{F(}x,y,z)$ denotes a vector field defined on a surface $\sigma$ oriented by a unit normal vector field $\text{n}(x,y,z),\text{and}\Phi$ denotes the flux of $\text{F}$ across $\sigma .$

(a) Assume that $\sigma$ is parametrized by a vector-value function $\text{r}(u,v)$ whose domain is a region R in the $uv\text{-plane}$ and that n is a positive multiple of

$\frac{\partial r}{\partial u}\times \frac{\partial r}{\partial v}$

Then the double integral over R whose value is $\Phi$ is $\_\_\_\_\_.$

(b) Suppose that $\sigma$ is the parametric surface

$\text{r}(u,v)=u\text{i}+v\text{j}+\text{(}u+v)\text{k}(0\le u^2+v^2\le 1)$

and that n is a positive multiple of

$\frac{\partial r}{\partial u}\times \frac{\partial r}{\partial v}$

Then the flux of $\text{F(}x,y,z)=x\text{i}+y\text{j}+z\text{k across }\sigma \text{ is }\Phi =\_\_\_\_\_.$