How do I find the circulation of the field F= x^2i + 2xj + z^2k around the ellipse 4x^2 + y^2 =4 in the xy plane, counterclockwise when viewed from above? I need to use Stoke's theorem, and I learned that there are two ways to use Stoke's theorem, one being the (integral over C) of (F (dot) dr), and the other is the (double integral over the surface) of (del (cross) F (dot) n * dsigma). Since I am given the curve, can I only use the first form of the theorem, or can I treat the ellipse like a surface in the xy plane and use the second form?

Question

anonymous · Answer

curlF=(0,0,2) , n=k=(0,0,1) then curlF.n=2  so intint(2dsig)=2*A=2*pi*ab=4pi