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