Chapter 15.5, Problem 1QCE

Consider the surface integral ${\displaystyle {\iint }_{\sigma }f\left(x,y,z\right)dS.}$

(a) If $\sigma$ is a parametric surface whose vector equation is

$\text{r}=x\left(u,v\right)\text{i}+y\left(u,v\right)\text{j}+z\left(u,v\right)\text{k}$

to evaluate the integral replace $dS$ by $\text{\_\_\_\_\_\_}\text{.}$

(b) If $\sigma$ is the graph of a function $z=g\left(x,y\right)$ with continuous first partial derivatives, to evaluate the integral replace $dS$ by $\text{\_\_\_\_\_\_}\text{.}$