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