Pre-Quantum Electrodynamics

When applied to arguments which are products of scalar/vector fields, the action of \(\nabla\), \(\nabla \cdot\) and \(\nabla \times\) can be explicited as follows:

Gradient of a product:

\[ \nabla (uv) = u \nabla v + v \nabla u \tag{grad_prod}\label{grad_prod} \]

Gradient of a scalar product:

\[ \nabla ({\bf a} \cdot {\bf b}) = ({\bf a} \cdot \nabla) {\bf b} + ({\bf b} \cdot \nabla) {\bf a} + {\bf a} \times (\nabla \times {\bf b}) + {\bf b} \times (\nabla \times {\bf a}) \tag{grad_sprod}\label{grad_sprod} \]

Divergence of a product:

\[ \nabla \cdot (\psi {\bf a}) = {\bf a} \cdot \nabla \psi + \psi \nabla \cdot {\bf a} \tag{div_prod}\label{div_prod} \]

Divergence of a cross product:

\[ \nabla \cdot ({\bf a} \times {\bf b}) = {\bf b} \cdot (\nabla \times {\bf a}) - {\bf a} \cdot (\nabla \times {\bf b}) \tag{div_xprod}\label{div_xprod} \]

Curl of a product:

\[ \nabla \times (\psi {\bf a}) = \nabla \psi \times {\bf a} + \psi \nabla \times {\bf a} \tag{curl_prod}\label{curl_prod} \]

Curl of a cross product:

\[ \nabla \times ({\bf a} \times {\bf b}) = {\bf a} (\nabla \cdot {\bf b}) - {\bf b} (\nabla \cdot {\bf a}) + ({\bf b} \cdot \nabla) {\bf a} - ({\bf a} \cdot \nabla) {\bf b} \tag{curl_xprod}\label{curl_xprod} \]