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)

\[ ∫_{\cal V} {\boldsymbol ∇} ⋅ (f{\bf A}) dτ = ∫_{\cal V} f({\boldsymbol ∇} ⋅ {\bf A}) dτ

• ∫_{\cal V} {\bf A} ⋅ ({\boldsymbol ∇} f) dτ = \oint f{\bf A} ⋅ d{\bf a},

\]

or in other words

\[ ∫_{\cal V} f({\boldsymbol ∇} ⋅ {\bf A}) dτ = -∫_{\cal V} {\bf A} ⋅ ({\boldsymbol ∇} f) dτ

• \oint_{\cal S} f{\bf A} ⋅ d{\bf a}.

\label{Gr(1.59)} \]