Suppose you are told that curl(F) = 2y, -2z, 3
where S is the upper hemisphere x^2 + y^2 + z^2 = 4 and z is
greater than 0 with upward pointing normal dS.
Note that you don't need t out a candidate vector F to
answer this problem. You can just
compute the surface integral above directly to obtain an answer,
or you can follow the steps below.
1. Write down (but do not evaluate) a path integral which has
the same value as the surface integral
above. What result did you use here?
2. Write down (but do not evaluate) a path integral which has
the same value as the following surface
integral. ZZ
S2
curl(F) dS
where S2 is the disk x^2 + y^2 is less than 4 and z =
0 in the xy-plane with upward pointing normal dS. What
result did you use here?
3. Evaluate the surface integral in part 2 above. What is the
answer to the original question? Why?