Find the flux of F = (3z + 1)k upward across the hemisphere x² + y² + z² = a², z ≥ 0 

(a) With the Divergence Theorem and 

(b) By evaluating the flux integral directly.

THEOREM 8-Divergence Theorem Let F be a vector field whose compo- nents have continuous first partial derivatives, and let S be a piecewise smooth oriented closed surface. The flux of F across S in the direction of the surface's outward unit normal field n equals the triple integral of the divergence V. F over the region D enclosed by the surface: [ F.ndo = Outward flux J V.F dV. Divergence integral (2)