= Surface divergence theorem for a tangent field
{title2=$\int_S\nabla_s\cdot v\,dS=\int_{\partial S}u\cdot v\,ds$}
For a smooth tangent <vector field> $v$ on a compact <orientable surface> with smooth boundary and <outward conormal of a surface boundary>, the integral of its <surface divergence> is its boundary <flux>. Extend the unit <normal vector> to a unit field $m$. The <curl of a cross product> gives $m\cdot\nabla\times(m\times v)=\nabla\cdot v-m_im_j\partial_jv_i$ on the surface: $m\cdot v=0$ there and $m\cdot\partial_jm=0$ by unit length. Apply <Stokes theorem> and use $(m\times v)\cdot d\mathbf r=v\cdot(d\mathbf r\times m)$ to obtain the formula.
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