Pre-Quantum Electrodynamics

Consider a fixed vector field \({\bf F} = {\bf F}(x, y, z) = {\bf F}({\bf r})\). Assume that the functions \({\boldsymbol \nabla} \cdot {\bf F} = D({\bf r})\) and \({\boldsymbol \nabla} \times {\bf F} = {\bf C} ({\bf r})\) are given to us everywhere within a finite volume \({\cal V}\) (for consistency, \({\bf C}\) must be divergenceless, \({\boldsymbol \nabla} \cdot {\bf C} = 0\)). Then,

\[ {\bf F} ({\bf r}) = -{\boldsymbol \nabla} \phi ({\bf r}) + {\boldsymbol \nabla} \times {\bf A} ({\bf r}) \]

where

\[ \phi({\bf r}) = \frac{1}{4\pi} \int_{\cal V} \frac{D({\bf r}') d\tau'}{|{\bf r} - {\bf r}'|}, \hspace{1cm} {\bf A} ({\bf r}) = \frac{1}{4\pi} \int_{\cal V} \frac{{\bf C}({\bf r}') d\tau'}{|{\bf r} - {\bf r}'|}. \]