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

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

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