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