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For smooth inviscid barotropic fluid with potential body forces, barotropic vorticity transport and $\nabla \cdot \mathbf{\omega }=0$ give $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}-\mathbf{\omega }\nabla \cdot \mathbf{u}$. Combine this with material surface element evolution to obtain $D(\mathbf{\omega }\cdot d\mathbf{S})/Dt=0$. Integrating over a material surface proves the flux law. Its boundary must move with the fluid; flux through a fixed open surface is generally not conserved. By Stokes theorem the same statement is Kelvin's circulation theorem around the material boundary.

Combine the material surface element evolution law with the product rule for $\mathbf{B}\cdot d\mathbf{S}$. The result isExpanding the curl gives the equivalent title formula. The ideal magnetohydrodynamic induction equation and Gauss's law for magnetism make the integrand zero and prove magnetic flux freezing, without requiring incompressible flow.

The surface must be a material surface $S(t)$, moving with the fluid; its boundary moves with it as well. The stated barotropic vorticity transport law, with $\mathbf{\omega }=\nabla \times \mathbf{v}$, gives $\nabla \cdot \mathbf{\omega }=0$. Expanding its curl therefore yieldsA material tangent vector $\mathbf{t}$ obeys $D\mathbf{t}/Dt=(\nabla \mathbf{v})\mathbf{t}$. Apply the product rule to the cross product of two such tangents to obtain material surface element evolution:Here $(\nabla \mathbf{v}{)}_{ij}={\partial }_j{v}_i$. The stretching and area-change terms cancel exactly in the transported flux element:Integrate over the fixed particle labels parameterizing the material surface. This proves the conservation of vorticity flux through a material surface:The conserved flux is also the circulation ${\oint }_{\partial S(t)}\mathbf{v}\cdot d\mathbf{\ell }$ by the Stokes theorem, so this agrees with Kelvin's circulation theorem. The assumptions are smooth inviscid barotropic flow and conservative body forces. A fixed open surface generally exchanges vorticity flux with its surroundings and does not satisfy this conservation statement; compressibility by itself does not invalidate the material-surface result.

This is magnetic flux freezing in ideal magnetohydrodynamics. In addition to the supplied ideal magnetohydrodynamic induction equation, a physical magnetic field obeys Gauss's law for magnetism, $\nabla \cdot \mathbf{B}=0$.

Parameterize the material surface by two particle labels $a,b$, so $d\mathbf{S}=({\partial }_a\mathbf{x}\times {\partial }_b\mathbf{x})dadb$ and $D\mathbf{x}/Dt=\mathbf{u}$. Differentiating these tangents gives the material surface element evolution lawwhere $(\nabla \mathbf{u}{)}_{ij}={\partial }_j{u}_i$. Thus the material magnetic-flux transport theorem isExpand the curl in the ideal magnetohydrodynamic induction equation:Since $D/Dt={\partial }_t+\mathbf{u}\cdot \nabla$ is the material derivative, the first three terms cancel the corresponding terms in the surface transport formula. The remaining integrand is $\mathbf{u}\nabla \cdot \mathbf{B}=0$. ThereforeNo incompressible flow assumption is needed: the changing surface area cancels the compression term. This proof assumes smooth velocity and magnetic field and a surface carried with the fluid; the ideal magnetohydrodynamic induction equation by itself, for an arbitrary non-solenoidal vector field, would leave the final divergence term.