\[\phi\]
is the velocity function and \[\psi\]
is the velocity function of a two dimensional flow in the \[xy\]
plan.Let
\[\omega = \phi \mathbf{i} - \psi \mathbf{j}\]
then1.
\[\mathbf{\nabla} \times \mathbf{\omega} =0 \]
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
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \cdot \mathbf{\omega} &= ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j}) \cdot (\phi \mathbf{i} - \psi \mathbf{j}) \\ &= \frac{\partial \phi}{\partial x} - \frac{\partial \psi}{\partial y} \\ &=0 \end{aligned} \end{equation}\]
and
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \times \omega &= (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{h} ) \times (\phi \mathbf{i} - \psi \mathbf{j}) \\ &=(- \frac{\partial \psi}{\partial x} - \frac{\partial \phi}{\partial y}) \mathbf{k} \\ &=0 \end{aligned} \end{equation}\]