456 CHAPTER 21. FUNCTIONS OF MANY VARIABLES

21.7 Partial Differential EquationsPartial differential equations are equations which involve the partial derivatives of somefunction. The most famous partial differential equations involve the Laplacian, namedafter Laplace3.

Definition 21.7.1 Let u be a function of n variables. Then

∆u ≡n

∑k=1

uxkxk

This is also written as ∇2u. The symbol ∆ or ∇

2 is called the Laplacian. When ∆u = 0the function u is called harmonic. Laplace’s equation is ∆u = 0. The heat equation isut −∆u = 0 and the wave equation is utt −∆u = 0.

Example 21.7.2 Find the Laplacian of u(x,y) = x2 − y2.

uxx = 2 while uyy = −2. Therefore, ∆u = uxx + uyy = 2− 2 = 0. Thus this function isharmonic, ∆u = 0.

Example 21.7.3 Find ut −∆u where u(t,x,y) = e−t cosx.

In this case, ut = −e−t cosx while uyy = 0 and uxx = −e−t cosx therefore, ut −∆u = 0and so u solves the heat equation ut −∆u = 0.

Example 21.7.4 Let u(t,x) = sin t cosx. Find utt −∆u.

In this case, utt =−sin t cosx while ∆u =−sin t cosx. Therefore, u is a solution of thewave equation utt −∆u = 0.

21.8 Exercises1. Find fx, fy, fz, fxy, fyx, fxz, fzx, fzy, fyz for the following. Verify the mixed partial der-

ivatives are equal.

(a) x2y3z4 + sin(xyz)

(b) sin(xyz)+ x2yz

(c) z ln∣∣x2 + y2 +1

∣∣(d) ex2+y2+z2

(e) tan(xyz)

2. Suppose f is a continuous function and f : U → R where U is an open set andsuppose that x ∈U has the property that for all y near x, f (x) ≤ f (y). Prove thatif f has all of its partial derivatives at x, then fxi (x) = 0 for each xi. Hint: This isjust a repeat of the usual one variable theorem seen in beginning calculus. You justdo this one variable argument for each variable to get the conclusion.

3Laplace was a great physicist and mathematician of the 1700’s. He made fundamental contributions to me-chanics and astronomy.