Exercise 1084Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

The Laplacian

#Laplacian#harmonic functions#second derivatives

Statement

Define the Laplacian $\Delta f$ in $\mathbb{R}^2$ and show that $f(x,y)=x^2-y^2$ is harmonic.

Hint

$\Delta f=f_{xx}+f_{yy}$ .

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Definition. $\Delta f=f_{xx}+f_{yy}$ .

Computation. $f_{xx}=2$ , $f_{yy}=-2$ , so $\Delta f=2-2=0$ .

Conclusion. $f$ is harmonic: $\Delta f=0$ .