= Cross-product form of Stokes theorem
{title2=$\oint_Cd\mathbf l\times\mathbf F=\int_S(d\mathbf S\times\nabla)\times\mathbf F$}
For a continuously differentiable <vector field> on an oriented surface with compatibly oriented boundary, this vector identity follows by dotting both sides with an arbitrary constant vector $a$ and applying <Stokes theorem> to $\mathbf F\times a$. The <curl> identity $\nabla\times(a\times\mathbf F)=a\,\nabla\cdot\mathbf F-(a\cdot\nabla)\mathbf F$ gives the surface integrand. In components, $[(d\mathbf S\times\nabla)\times\mathbf F]_i=dS_j\partial_iF_j-dS_i\partial_jF_j$, with the derivatives acting on $\mathbf F$. For the <position vector> field $\mathbf F=\mathbf r$, it equals $-2d\mathbf S$; hence the line integral is minus twice the oriented vector area.
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