The velocity potential and the stream function are both harmonic in their two dimensional domain.

Proof

The velocity potential

\[\phi\]

and the stream function \[\psi\]

both satisfy \[\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} =f, \: \frac{\partial \phi}{\partial y} = \frac{\partial \psi}{\partial x} = -g\]

Differentiate the first with respect to

\[x\]

to give \[\frac{\partial^2 \phi}{\partial x^2} = \frac{\partial^2 \psi}{\partial x \partial y}\]

and the second with respect to

\[y\]

to give \[\frac{\partial^2 \phi}{\partial y^2} =- \frac{\partial^2 \psi}{\partial y \partial x}\]

Adding these two equation give

\[\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}=0\]

Differentiate the first with respect to

\[y\]

to give \[\frac{\partial^2 \phi}{\partial y \partial x} = \frac{\partial^2 \psi}{ \partial y^2}\]

and the second with respect to

\[s\]

to give \[\frac{\partial^2 \phi}{\partial x \partial y} =- \frac{\partial^2 \psi}{\partial x^2}\]

Adding these two equation give

\[\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}=0\]