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