## Proof That the Velocity Potential and Stream Functions Are Harmonic

Theorem
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
$\frac{\partial^2 \phi}{\partial y^2} =- \frac{\partial^2 \psi}{\partial y \partial x}$

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

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
$\frac{\partial^2 \phi}{\partial x \partial y} =- \frac{\partial^2 \psi}{\partial x^2}$

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