Vector identities

Notation: ${\bf a}$, ${\bf b}$, ${\bf c}$, ${\bf d}$ are general vectors; $\phi$, $\psi$ are general scalar fields; ${\bf a} = a_x\,{\bf e}_x+a_y\,{\bf e}_y+a_x\,{\bf e}_z= (a_x,\,a_y,\,a_z)$, and so on, where ${\bf e}_x$, ${\bf e}_y$, and ${\bf e}_z$ are right-handed Cartesian basis vectors.

$$\vert{\bf a}\vert = \left(a_x^{\,2}+a_y^{\,2}+a_z^{\,2}\right)^{1/2}$$ (A.73)

$${\bf a}\cdot{\bf b} = a_x\,b_x+a_y\,b_y+a_z\,b_z$$ (A.74)

$${\bf a}\times {\bf b} =\left\vert\begin{array}{ccc}{\bf e}_x,& {\bf e}_y,& {\bf e}_z\\ [0.5ex] a_x,& a_y, &a_z\\ [0.5ex] b_x,& b_y,& b_z \end{array}\right\vert$$

$$= (a_y\,b_z-a_z\,b_y)\,{\bf e}_x+(a_z\,b_x-a_x\,b_z)\,{\bf e}_y+(a_x\,b_y-a_y\,b_x)\,{\bf e}_z$$ (A.75)

$${\bf a}\times ({\bf b}\times {\bf c}) = ({\bf a}\cdot{\bf c})\,{\bf b} - ({\bf a}\cdot{\bf b})\,{\bf c}$$ (A.76)

$$({\bf a}\times {\bf b})\times {\bf c} = ({\bf c}\cdot{\bf a})\,{\bf b}- ({\bf c}\cdot{\bf b})\,{\bf a}$$ (A.77)

$$({\bf a}\times {\bf b})\cdot({\bf c}\times {\bf d}) = ({\bf a}\cdot{\bf c})\,({\bf b}\cdot{\bf d}) - ({\bf a}\cdot{\bf d})\,({\bf b}\cdot{\bf c})$$ (A.78)

$$({\bf a}\times {\bf b})\times ({\bf c}\times {\bf d}) = ({\bf a}\times {\bf b}\cdot{\bf d})\,{\bf c} - ({\bf a}\times {\bf b}\cdot{\bf c})\,{\bf d}$$ (A.79)

$$\nabla\phi = \frac{\partial\phi}{\partial x}\,{\bf e}_x+\frac{\partial\phi}{\partial y}\,{\bf e}_y+\frac{\partial\phi}{\partial z}\,{\bf e}_z$$ (A.80)

$$\nabla(\phi\,\psi) =\phi\,\nabla\psi+\psi\,\nabla\phi$$ (A.81)