How Do You Calculate the Surface Integral of a Parametric Surface?

[chancellorpho]

Homework Statement

For the parametrically defined surface S given by r(u,v) = <cos(u+v), sin(u+v), uv>, find the following differential:

In double integral over S of f(x, y, z)dS, dS =

Homework Equations

Above

The Attempt at a Solution

I thought I needed to put x, y, and z all in terms of two variables, (all three in terms of x and y, or y and z, or x and z), so that I can find dz/dx and dz/dy, but I don't know how to do this. :(

 

[HallsofIvy]

You are given x, y and z "in terms of two variables": x= cos(u+v), y= sin(u+v), z= uv. Do everything in terms of u and v, not x, y, and z.


The simplest way to find dS is this:
You are given $\vec{r}= cos(u+v)\vec{i}+ sin(u+v)\vec{j}+ uv\vec{k}$.
Find the derivative of that vector with respect to each of the parameters:
$$\vec{r}_u= -sin(u+v)\vec{i}+ cos(u+v)\vec{j}+ v\vec{k}$$
$$\vec{r}_v= -sin(u+v)\vec{i}+ cos(u+v)\vec{j}+ u\vec{k}$$

The "fundamental vector product" is the cross product of those two derivative vectors. It is perpendicular to the surface at each point and dS is its length times dudv.