Pre-Quantum Electrodynamics

\[ \int_{\cal S} ({\boldsymbol \nabla} \times {\bf v}) \cdot d{\bf a} = \oint_{\cal P} {\bf v} \cdot d{\bf l}. \tag{Stokes}\label{Stokes} \]

Corollary 1: \(\int ({\boldsymbol \nabla} \times {\bf v}) \cdot d{\bf a}\) depends only on the boundary line and not on the particular surface used.

Corollary 2: \(\oint ({\boldsymbol \nabla} \times {\bf v}) \cdot d{\bf a}= 0\) for any closed surface, since the boundary shrinks to a point.