Use the Divergence Theorem to calculate the surface integral ∫∫_S F · ds; that is, calculate the flux of F across S.

F(x, y, z) = x³yi – x²y² j – x²yz k, S is the surface of the solid bounded by the hyperboloid x² + y² – z² = 1 and the planes z = –2 and z = 2

Data from the Divergence Theorem

Let E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation. Let F be a vector field whose component functions have continuous partial derivatives on an open region that contains E. Then

S F. dS SSS E div F dV