Stoke's Theorem

  • #1
3,003
6

Homework Statement


Use Stoke's Theorem to calculate [itex]\int_C\vec{F}\cdot\, dr[/itex], where
[itex]\vec{F}=<x^2z\, ,xy^2\,,z^2>[/itex] and C is the curve of the intersection of the plane [itex]x+y+z=1[/itex] and the cylinder
[itex]x^2+^2=9[/itex].
(C is oriented clockwise when viewed from above.)
Answer: [itex]\frac{81}{2}\pi[/itex].



Homework Equations

Stoke's Theorem



The Attempt at a Solution



Okay, let me try to explain where I am getting lost. Firstly, I know that the premise of Stoke's Theorem is that is relates a line integral to a Surface integral.

When I graph this, I get a cylinder that is symmetrical about the z-axis and it is intersected by a plane which results in an ellipse. (see terrible drawing below)

Now, I need to parametrize (how do you spell that anyway?) S. Now S is the surface that is bounded by C right? If not, please stop me here.

Photo1.jpg
 

Answers and Replies

  • #2
Defennder
Homework Helper
2,591
5
Now, I need to parametrize (how do you spell that anyway?) S. Now S is the surface that is bounded by C right? If not, please stop me here.
Yes. S is any surface that is bounded by C. Usually the easiest surface to work with is the plane surface bounded by C if C can be visualised as lying on a plane.
 
  • #3
3,003
6
Yes. S is any surface that is bounded by C. Usually the easiest surface to work with is the plane surface bounded by C if C can be visualised as lying on a plane.

I know form doing this in class that we parametrized S as r=<x, y, 1-x-y>

but I am not so sure why. Oh wait... is that the surface bounded by the ellipse?

So now all I need to do is find dr/dx x dr/dy and compute the dot product of the resultant with F(r(x,y) and integrate?
 
  • #4
623
0
You are told that x + y + z = 1 and now you parametrize that by <x,y,f(x,y)> and z = f(x,y) = 1 - x - y by algebra (is that part of your question?)
 
  • #5
3,003
6
I guess my question is what is S? Is it the the surface of the ellipse? I believe it is, so I guess my question has been answered.
:smile:
 

Related Threads on Stoke's Theorem

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
734
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
861
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
650
  • Last Post
Replies
1
Views
1K
Top