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

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

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