Proposition

Show that all partial derivatives of a harmonic function are harmonic.

Solution

We are given that \(u_{xx} + u_{yy} = 0\). Thus \((u_{xx} + u_{yy})_x = 0\), so \(u_{xxx} + u_{yyx} = 0\). By changing the order of partial derivatives, \((u_x)_{xx} + (u_x)_{yy} = 0\). Thus $u_x$ is harmonic.

Similarly, $u_y$ is harmonic.