##### Vector identities
c.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*}

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42