Compute the flux of curl(F) through the part of the paraboloid z = x 2 + y 2 that lies below the plane z = 4 with upward-pointing unit normal vector and F = h3z,5x,−2yi.

Parameterize this surface (call it S) by

with and .

The normal vector to S is

Compute the curl of F :

So the flux of curl(F) is

Alternatively, you can apply Stokes' theorem, which reduces the surface integral of the curl of F to the line integral of F along the intersection of the paraboloid with the plane z = 4. Parameterize this curve (call it C) by

with . Then

the answer is simply just 6n add the two together as they have an n