Chapter 15.6, Problem 3QCE

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 the graph of a function $z=g(x,y)$ over a region R in the $xy\text{-Plane}$ and that n has a positive k-component for every point on $\sigma$ . Then a double integral over R whose value is $\Phi \text{is}\text{\_\_\_\_\_}\text{.}$

(b) Suppose that $\sigma$ is the triangular region with vertices $(1,0,0),(0,1,0),\text{and}(0,0,1)$ with upward orientation. T hen the flux of

$\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k}$

across $\sigma \text{is}\Phi =\_\_\_\_\_.$