The scalar function div(grad 1) = ∇ ∙ ∇f (also written ∇2f) is called the Laplacian, and a function f satisfying ∇2f = 0 is said to be harmonic, concepts important in physics. Show that ∇2f = fxx + fyy + fn. Then find ∇2f for each of the following functions and decide which are harmonic.
(a) f(x, y, z) = 2x2 - y2 - z2
(b) f(x, y, z) = xyz
(c) f(x, y, z) = x3 - 3xy2 + 3z
(d) f(x, y, = (x2 + y2 + z2)-1/2