Question

Which of the following functions is not harmonic?
(a)u=x² - y²
(b) u= x - y
(c)u = x² + y²
(d) u = x + y​

Answer 1

d) u=x+y because it is not harmonic

Answer 2

The functions which are not harmonic are;

(b) u = x-y and

(d) u = x+y

• For a function to be harmonic, we should be able to differentiate it two times continuously. (b) and (d) cannot be differentiated twice continuously.
• The function has to satisfy Laplace's equation.
• ∇² = $u_{xx}$ + $u_{yy}$ = 0 ⇒ eq. 1, where ∇² is the Laplacian operator.

        eq. 1 is the Laplace equation. For a function to be harmonic it has to obey this equation. (b) and (d) does not obey the Laplace test.