(a) A flux density field is given as F₁ = 5a_z. Evaluate the outward flux of F₁ through the hemispherical surface, r = a, 0 < θ < π/2, 0 < ϕ < 2π.

(b) What simple observation would have saved a lot of work in part a?

(c) Now suppose the field is given by F₂ = 5za_z. Using the appropriate surface integrals, evaluate the net outward flux of F₂ through the closed surface consisting of the hemisphere of part a and its circular base in the x_y plane.

(d) Repeat part c by using the divergence theorem and an appropriate volume integral.