Hmm...was hoping someone might be able to answer this by now! ;}
Well, here's a simpler question that is related to the more general case given above:
Compute the line integral of the tangential component of the vector field F: R3 -> R3 = yi-xj+3yk along the curve of intersection between the sphere x^2 + y^2 + z^2 = 9 and the plane ax + by + z = 0. Evaluate the line integral by direct evaluation and by the "Curl" Theorem.
The former method requires you to parametrize the circle of intersection (a "great circle" or rotated circle lying on the sphere/plane) and the ladder method requires you to parametrize the hemisphere bisected by the plane through the center of the sphere (a rotated half-sphere).
If someone could help me obtain the correct parametrizations, I would appreciate it! I would then probably be able to determine the more general case in the OP myself.
EDIT: For anyone who is attempting the problem, the answer is [9*pi (3a - 2)] / [sqrt(1 + a^2 + b^2)]