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Surface integral differential

  • #1

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. :(
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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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 [itex]\vec{r}= cos(u+v)\vec{i}+ sin(u+v)\vec{j}+ uv\vec{k}[/itex].
Find the derivative of that vector with respect to each of the parameters:
[tex]\vec{r}_u= -sin(u+v)\vec{i}+ cos(u+v)\vec{j}+ v\vec{k}[/tex]
[tex]\vec{r}_v= -sin(u+v)\vec{i}+ cos(u+v)\vec{j}+ u\vec{k}[/tex]

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.
 

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