(955, #1) Use Stokes Theorem to evaluate to

where F = < x y z, x, e^xy cos(z) >, and S is the hemisphere x² + y² + z² = 1, z >= 0,oriented upwards.

Solution:

The boundary C is the circle x2 + y2 = 1, with the parametrization x = cos(t), y = sin(t), t = 0..2 Pi. It is positively oriented with respect to the upwardly oriented sphere.

Thus we have