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