Stoke's Theorem for Line Integral: Calculating F*T ds on C

[kasse]

Task:

Calculate the line integral F*T ds around C where

F=(-xz-2y,x^2-yz,z^2+1) and

C is the boundary curve between the cylinder x^2+Y^2=1 and the top half of the sphere X^2+Y^2+z^2=10.



My work:

Surely I'm supposed to use Stoke's theorem here. First I replace x^2+y^2 with 1 in the sphere equation, to find that C lies in the plane z=3.

Then I need the normal unit vector. But how do I find that?

Is it simply the partial of the surface eq. f(x,y,z)=x^2+y^2+z^2-10=0?

 

[Glass]

Isn't it just a circle on the plane z=3? Or am I missing something?

 

[HallsofIvy]

Are you really required to use Stoke's theorem? As Glass said, the boundary curve is just the circle x²+ y²= 1 with z= 3. It should be easy to integrate F around that boundary directly.

However, it is true that the gradient (which I presume is what you mean by "partial of surface") of x^2+y^2+z^2-10 is normal to that surface. It may not be a "unit" normal though.