Suppose that f :Rn → R is a function of class C2. The Laplacian of f , denoted ∇2 f , is...

Suppose that f :Rⁿ → R is a function of class C². The Laplacian of f , denoted ∇² f , is defined to be

When n = 2 or 3, this construction is important when studying certain differential equations that model physical phenomena, such as the heat or wave equations. (See Exercises 28 and 29 of §2.4.) Now suppose that f depends only on the distance x = (x₁, . . . , x_n) is from the origin in Rⁿ; that is, suppose that f (x) = g(r ) for some function g, where r = ||x||. Show that for all x ≠ 0, the Laplacian is given by