Stokes Theorem

[bluestribute]

So I'm having trouble with Stoke's Theorem and finding my bounds. Any help (using this as an example)?:

Use Stokes' Theorem to compute the flux ofcurl(F)
through the portion of the plane

x

7

+

y

3

+ z = 1
where x, y, z ≥ 0
for F =

yz, 0, x

.
(Assume counterclockwise orientation.)

I have my curl no problem (check my math if you want, but I got 0, y-1, -z), and DS I'm just solving for Z, right (algebraically)? Then are my bounds, after solving for z . . . I don't know. Do I switch to polar, or do I switch to spherical, and rewrite everything then?

 

[HallsofIvy]

First, you mean the plane (x/7)+ (y/3)+ z= 1, right?

Since the problem says to use Stoke's theorem, you shouldn't be calculating the curl at all! Stokes theorem says that \(\displaystyle \int\int \nabla\times\vec{F}\cdot dS= \int \vec{F}\cdot ds\).

To find the integral of curl F over the surface, integrate F itself over the boundary.

Here, the plane intersect the plane z= 0 in the line (x/7)+ (y/3)= 1 (or 3x+ 7y= 21), the plane y= 0 in the line (x/7)+ z= 1 (or x+ 7z= 7), and the plane x= 0 in the line (y/3)+ z= 1 (or y+ 3z= 3). To find the integral of curl F over the surface, integrate F over those three line segments. For, example, for the first one, you could use the parameterization x= 7(t+ 1), y= -3t, z= 0.