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

Chapter 2, Problem 32

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Suppose that f : Rn R is a function of class C2. The Laplacian of f , denoted 2 f , is defined to be 2 f = 2 f x 2 1 + 2 f x 2 2 ++ 2 f x 2 n . 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,..., xn) 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 2 f = n 1 r g (r) + g(r).