\[ \frac{D\Gamma}{Dt} = 0 \]

$$ \oint_C \frac{Du_i}{Dt} \, dx_i = \oint_C \left( -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + g_i + \frac{1}{\rho} \frac{\partial \tau_{ij}}{\partial x_j} \right) dx_i = -\frac{1}{\rho} \oint_C dp - \oint_C d\Phi + \oint_C \left( \frac{1}{\rho} \frac{\partial \tau_{ij}}{\partial x_j} \right) dx_i $$

For a barotropic fluid VorticityGenerationBarotropic , the first term on the right side of the equation above is zero because $C$ is a closed contour, and $\rho$ and $p$ are single valued at each point in space. Similarly, the second integral on the right side is zero since $\Phi$ is also single valued at each point in space.

\[\frac{D\Gamma}{Dt} = \frac{D}{Dt} \oint_C u_i \, dx_i = \oint_C \frac{Du_i}{Dt} \, dx_i + \oint_C u_i \frac{D}{Dt} (dx_i)\]

Now consider the second term on the right side resulting from time differentiating the definition of the circulation. The velocity at point $\mathbf{x} + d\mathbf{x}$ on $C$ is: $\boxed{ \mathbf{u} + d\mathbf{u} = \frac{D}{Dt} (\mathbf{x} + d\mathbf{x}) = \frac{D\mathbf{x}}{Dt} + \frac{D}{Dt} (d\mathbf{x}), \quad \text{so} \quad d\mathbf{u}_i = \frac{D}{Dt} (dx_i)} $
Thus, the last term then becomes: $\boxed{ \oint_C u_i \frac{D}{Dt} (dx_i) = \oint_C u_i \, du_i = \oint_C d \left( \frac{1}{2} u_i^2 \right) = 0} $
where the final equality again follows because $C$ is a closed contour and $\mathbf{u}$ is a single-valued vector function, the expression obtained by time differentiating the definition of the circulation simplifies to: $$ \frac{D\Gamma}{Dt} = \oint_C \left( \frac{1}{\rho} \frac{\partial \tau_{ij}}{\partial x_j} \right) dx_i $$ and Kelvin’s theorem $\frac{D\Gamma}{Dt} = 0$ is proved when the fluid is inviscid ($\mu = \mu_v = 0$) or when the integrated viscous force $(\partial \tau_{ij} / \partial x_j)$ is zero around the contour $C$. This latter condition can occur when $C$ lies entirely in irrotational fluid.