Calculate the circulation, f · dr, c in two ways, directly and using stokes' theorem. f = y i + z j + xk and c is the boundary of s, the paraboloid z = 4 − (x2 + y2), z ≥ 0 oriented upward. (hint: use polar coordinates.) f · dr c =

, the boundary of , is a circle in the plane centered at the origin and with radius 2, hence we can parameterize it by

with . Then the line integral is

By Stokes' theorem, the line integral of along is equal to the surface integral of the curl of across :

Parameterize by

with and . Take the normal vector to to be

The curl is

Then the surface integral is

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step-by-step explanation: