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Existence of Scalar Functions for a Two Dimensional Irrotational, Velocity of an Incompressible Fluid

Theorem
For an incompressible irrotational fluid with velocity vector field  
\[\mathbf{v} = f\mathbf{i} - g \mathbf{j} \]
  with corresponding irrotational vector field  
\[\mathbf{u} = g\mathbf{i} + \mathbf{j} \]
, continuous, differentiable functions  
\[\phi , \: \psi\]
  exists satisfying
\[\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}=f, \: \frac{\partial \phi}{\partial y} =- \frac{\partial \psi}{\partial x}= - g\]

Proof
Since  
\[\mathbf{v} \]
  is irrotational,  
\[\mathbf{\nabla} \times \mathbf{v} =0\]
  hence a scalar function  
\[\phi\]
  exists such that  
\[\mathbf{v} = \mathbf{\nabla} \phi = \frac{\partial \phi}{\partial x} \mathbf{i} + \frac{\partial \phi}{\partial y} \mathbf{j} = f \mathbf{i} - g \mathbf{j} \]

We can take  
\[f = \frac{\partial \phi}{\partial x} , \: g = - \frac{\partial \phi}{\partial y}[\]

The vector field  
\[\mathbf{u} = g\mathbf{i} + \mathbf{j} \]
  is also irrotational so a scalar function  
\[\psi\]
  exists such that  
\[\mathbf{u} = \mathbf{\nabla} \psi = \frac{\partial \psi}{\partial x} \mathbf{i} + \frac{\partial \psi}{\partial y} \mathbf{j} = g \mathbf{i} +f \mathbf{j} \]

We can take  
\[g = \frac{\partial \psi}{\partial x} , \: f= \frac{\partial \phi}{\partial y}[\]