## Curl and Divergence for Velocity and Stream Functions

Suppose
$\phi$
is the velocity function and
$\psi$
is the velocity function of a two dimensional flow in the
$xy$
plan.
Let
$\omega = \phi \mathbf{i} - \psi \mathbf{j}$
then
1.
$\mathbf{\nabla} \times \mathbf{\omega} =0$

21.
$\mathbf{\nabla} \cdot \mathbf{\omega} =0$

There exist scalar functions
$\phi , \: \psi$
satisfying
$\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}=1$
and
$\frac{\partial \phi}{\partial y} = - \frac{\partial \psi}{\partial x}=-g$

Then
\begin{aligned} \mathbf{\nabla} \cdot \mathbf{\omega} &= ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j}) \cdot (\phi \mathbf{i} - \psi \mathbf{j}) \\ &= \frac{\partial \phi}{\partial x} - \frac{\partial \psi}{\partial y} \\ &=0 \end{aligned}

and
\begin{aligned} \mathbf{\nabla} \times \omega &= (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{h} ) \times (\phi \mathbf{i} - \psi \mathbf{j}) \\ &=(- \frac{\partial \psi}{\partial x} - \frac{\partial \phi}{\partial y}) \mathbf{k} \\ &=0 \end{aligned}