Pre-Quantum Electrodynamics

Example: \({\boldsymbol \nabla} \cdot (f{\bf A}) = f({\boldsymbol \nabla} \cdot {\bf A}) + {\bf A} \cdot ({\boldsymbol \nabla} f)\) implies (via Gauss' theorem)

\begin{equation*} \int_{\cal V} {\boldsymbol \nabla} \cdot (f{\bf A}) d\tau = \int_{\cal V} f({\boldsymbol \nabla} \cdot {\bf A}) d\tau + \int_{\cal V} {\bf A} \cdot ({\boldsymbol \nabla} f) d\tau = \oint f{\bf A} \cdot d{\bf a}, \end{equation*}

or in other words

\begin{equation} \int_{\cal V} f({\boldsymbol \nabla} \cdot {\bf A}) d\tau = -\int_{\cal V} {\bf A} \cdot ({\boldsymbol \nabla} f) d\tau + \oint_{\cal S} f{\bf A} \cdot d{\bf a}. \label{Gr(1.59)} \end{equation}