Question

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above.
F(x, y, z) = yzi + 9xzj + exyk, C is the circle x2 + y2 = 1, z = 3.

Answer 1

S has a unit normal k , so we need the k component of curl F only 
curl F ( k direction) = (3z-z) =2z at z=3 , = 6 

So INT_C F.dr =INT_S curlF .<n> dS = INT:A curl F. (k) dA , 
= INT6 dA = 6A 
The circle ( S) hasa radius R=4 , so A= piR^2 = 16pi 

Work = 6*16pi = 96pi ( stokes gives Counterclockwise , so we use +k , clockwise , -k , or simply factor by -1 )