### Kelvin’s Circulation Theorem

Prove Kelvin’s Circulation Theorem, assuming that the fluid in question is barotropic and there is no rotation. So

$$\frac{D\Gamma}{Dt} = \frac{D}{Dt} \oint \vec{v} \cdot d\vec{l} = 0$$

where $$\vec{v}$$ is the velocity vector and $$\Gamma$$ is the circulation.

From Wikipedia.

From Wikipedia.