Pre-Quantum Electrodynamics
Vector identitiesc.m.uf.vi
\begin{align*}
(1)&{\bf A} \cdot ({\bf B} \times {\bf C}) = {\bf B} \cdot ({\bf C} \times {\bf A}) = {\bf C} \cdot ({\bf A} \times {\bf B})
\nonumber \\
(2)&{\bf A} \times ({\bf B} \times {\bf C}) = {\bf B} ({\bf A} \cdot {\bf C}) - {\bf C} ({\bf A} \cdot {\bf B})
\nonumber \\
(3)&{\boldsymbol \nabla} (fg) = f ({\boldsymbol \nabla} g) + g ({\boldsymbol \nabla} f)
\nonumber \\
(4)&{\boldsymbol \nabla}({\bf A} \cdot {\bf B}) = {\bf A} \times ({\boldsymbol \nabla} \times {\bf B}) + {\bf B} \times ({\boldsymbol \nabla} \times {\bf A}) + ({\bf A} \cdot {\boldsymbol \nabla}) {\bf B} + ({\bf B} \cdot {\boldsymbol \nabla}) {\bf A}
\nonumber \\
(5)&{\boldsymbol \nabla} \cdot (f{\bf A}) = f ({\boldsymbol \nabla} \cdot {\bf A}) + {\bf A} \cdot ({\boldsymbol \nabla} f)
\nonumber \\
(6)&{\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf A}) - {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B})
\nonumber \\
(7)&{\boldsymbol \nabla} \times (f {\bf A}) = f ( {\boldsymbol \nabla} \times {\bf A}) - {\bf A} \times ({\boldsymbol \nabla} f)
\nonumber \\
(8)&{\boldsymbol \nabla} \times ({\bf A} \times {\bf B}) = ({\bf B} \cdot {\boldsymbol \nabla}) {\bf A} - ({\bf A} \cdot {\boldsymbol \nabla}) {\bf B} + {\bf A} ({\boldsymbol \nabla} \cdot {\bf B}) - {\bf B} ({\boldsymbol \nabla} \cdot {\bf A})
\nonumber \\
(9)&{\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} \times {\bf A}) = 0
\nonumber \\
(10)&{\boldsymbol \nabla} \times ({\boldsymbol \nabla} f) = 0
\nonumber \\
(11)&{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf A}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf A}) - {\boldsymbol \nabla}^2 {\bf A}
\end{align*}

Created: 2024-02-27 Tue 10:31