Curl On Bottom of Sphere: Determine F(x,y,z)

[bfr]

Homework Statement

Determine the curl on the surface of the bounded region consisting of the bottom part of the sphere with equation 625=z^2+x^2+y^2 where z<=20, in the force field F(x,y,z)=<x^2 * y,x*y^2 * z,2x>

Homework Equations

http://img187.imageshack.us/img187/291/1fdf437d8e18a23191b63dfnj8.png

The Attempt at a Solution

I used Stokes' theorem to change the double integral for curl into a single circulation circle around the top of the bottom section of the sphere:

625=z^2+x^2+y^2; z=20; 225=r^2

Let x=15cos t; y=15sin t; z=20 (I'm still writing x,y, and z instead of their substituted values in the following integral though): integral (<x^2*y,xy^2z,2z> dot <-15sin(t),15cos(t)>) dt from t=0 to t=2pi ("dot" represents a dot product). Is this right?

 

[gabbagabbahey]

Homework Statement

Determine the curl on the surface of the bounded ...

That doesn't make much sense to me. The curl of a vector function is simply the curl of the vector function. It has different values at different point on the surface, so to determine "the curl of a vector on a surface" doesn't make a whole lot of sense.

Surely the question is supposed to be to "find the flux of curl(F) through the surface"?

The Attempt at a Solution

I used Stokes' theorem to change the double integral for curl into a single circulation circle around the top of the bottom section of the sphere:

625=z^2+x^2+y^2; z=20; 225=r^2

Let x=15cos t; y=15sin t; z=20 (I'm still writing x,y, and z instead of their substituted values in the following integral though): integral (<x^2*y,xy^2z,2z> dot <-15sin(t),15cos(t)>) dt from t=0 to t=2pi ("dot" represents a dot product). Is this right?

Assuming that you are indeed trying to find the flux of curl(F) through the surface (as opposed to trying to find the value of curl(F) at every point on the surface); then this approach looks correct.

However, since x and y depend on the parameter t, you need to make sure you substitute x=15cos t, y=15sin t and z=20 into the integral before computing it.