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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?