= Solenoidal flux depends on the oriented boundary
{title2=$\int_{S_1}\mathbf F\cdot d\mathbf S=\int_{S_2}\mathbf F\cdot d\mathbf S$}
Suppose two oriented surfaces have the same oriented boundary and $S_1\cup(-S_2)$ bounds a volume inside a domain where the <vector field> $\mathbf F$ is smooth and <solenoidal>. The <divergence theorem> gives zero flux through their closed union, proving the displayed equality. If a global <vector potential> $\mathbf F=\nabla\times\mathbf G$ exists on the relevant surfaces, the <Stokes theorem> proves the equality directly from their common boundary. The domain condition matters: a singularity inside the joining volume can obstruct the divergence argument.
Back to article page
