Proof That Integral Around a Closed Plane Curve is Zero for an Incompressible Fluid With Irrotational Velocity

If an in compressible fluid has velocity given by the irrational vector field
$\mathbf{v} = f \mathbf{i} - g \mathbf{j}$
in a three dimensional region then
$\oint f dx - g dy =0$

Proof
Green's Theorem states
$\oint_C P dx + Q dy = \int \int_A \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dx dy$
where
$C$
is the boundary of
$A$
and
$P,Q$
are differentiable.
Since
$\mathbf{v}$
is irrational,
$\mathbf{v} = f \mathbf{i} - g \mathbf{j} = \frac{\partial \phi}{\partial x} \mathbf{i} + \frac{\partial \phi}{\partial y} \mathbf{j} \rightarrow f = \frac{\partial \phi}{\partial x}, \: g = - \frac{\partial \phi}{\partial y}$

Then
\begin{aligned} \oint f dx - g dy &= - \int \int_A \frac{\partial g}{\partial x} + \frac{\partial f}{\partial y} dx dy \\ &= \int \int_A \frac{\partial^2 f}{\partial x \partial y} - \frac{\partial^2 f}{\partial y \partial x} dx dy \\ &= 0 \end{aligned}