(949, #15) Evaluate

where F(x,y,z) = < e^y, y e^x , x² y> and S is the part of the paraboloid z = x² + y² that lies abve the square x = 0..1, y = 0..1 and has upward orientation.

Solution. The parametrization of the surface is

In addition r_x x r_z = < -2 x , -2 y, 1>, so the orientation is upward. Consequently