# Surface integral differential

• chancellorpho
In summary, to find dS for the given parametrically defined surface, first find the derivatives of the vector r(u,v) with respect to each parameter, then take the cross product of those two derivative vectors to find the fundamental vector product, which is perpendicular to the surface at each point. The length of this vector multiplied by the differential of the parameters, dudv, will give you the differential dS for the double integral over S.

## 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 =

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. :(

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.