## Proof That Integral of Conjugal Veclocity Around a Contour is Zero

Theorem
Let
$\mathbf{v} = f \mathbf{i} = g \mathbf{j}$
be the velocity fluid of an incompressible vector field.
Consider the irrotational vector field
$\mathbf{u} = g \mathbf{i} + f \mathbf{j}$
where
$f, \: g$
are differentiable.
$\oint \mathbf{u} \cdot \mathbf{r} =0$
.
Proof
Since
$\mathbf{u}$
is irrotational
$\mathbf{\nabla} \times \mathbf{u} ==(\frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}) \mathbf{k} =0$

Since
$\mathbf{v} = f \mathbf{i} - g \mathbf{j}$
is incompressible
$\mathbf{\nabla} \cdot \mathbf{v} = (\frac{\partial }{\partial x} \mathbf{i} + \frac{\partial }{\partial x} \mathbf{i}) \cdot ( f \mathbf{i} - g \mathbf{j}) = \frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}$

Bu Green's Theorem,
$\oint_C g dx + f dy = \int \int_A (\frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}) dx dy$

Hence
$\oint \mathbf{u} \cdot \mathbf{r} = \oint_C g dx + f dy = \int \int_A (\frac{\partial f}{\partial x} - \frac{\partial g}{\partial y}) dx dy = 0$
.