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