Identity 8: $\nabla \times \left(\mathbf{F}\times \mathbf{G}\right)\= \left(\mathbf{G}·\nabla \right)\mathbf{F}-\left(\mathbf{F}·\nabla \right)\mathbf{G}\+\mathbf{F} \nabla ·\mathbf{G}-\mathbf{G} \nabla ·\mathbf{F}$

Its left side is the curl of the cross product of F and G, so it is the vector

The right side contains the scalar operators $\mathbf{F}·\nabla \=u {\partial }_x\+v {\partial }_y\+w {\partial }_z$ and $\mathbf{G}·\nabla \=a {\partial }_x\+b {\partial }_y\+c {\partial }_z$ that act componentwise on the vectors to which they are applied. In essence, they are directional derivatives of one vector field in the direction of another. As such, they can be implemented in Maple via the DirectionalDiff command in the Student VectorCalculus package.

The first two terms on the right are

$\left(\mathbf{G}\mathbf{·}\nabla \right)\mathbf{F}\=\left[\begin{array}{c}\(a {\partial }_x\+b {\partial }_y\+c {\partial }_z\)u\\ \(a {\partial }_x\+b {\partial }_y\+c {\partial }_z\)v\\ \(a {\partial }_x\+b {\partial }_y\+c {\partial }_z\)w\end{array}\right]$ = $\left[\begin{array}{c}a u_x\+b u_y\+c u_z\\ a v_x\+b v_y\+c v_z\\ a w_x\+b w_y\+c w_z\end{array}\right]$ 

and the negative of$$

$\left(\mathbf{F}·\nabla \right)\mathbf{G}\=\left[\begin{array}{c}\(u {\partial }_x\+v {\partial }_y\+w {\partial }_z\)a\\ \(u {\partial }_x\+v {\partial }_y\+w {\partial }_z\)b\\ \(u {\partial }_x\+v {\partial }_y\+w {\partial }_z\)c\end{array}\right]$ = $\left[\begin{array}{c}u a_x\+v a_y\+w a_z\\ u b_x\+v b_y\+w b_z\\ u c_x\+v c_y\+w c_z\end{array}\right]$ 

The third term on the right, namely, $\mathbf{F} \nabla ·\mathbf{G}$, is the vector field F times the scalar "gradient of G."

The fourth term on the right, namely, $-\mathbf{G} \nabla ·\mathbf{F}$, is the negative of the vector field G times the gradient of G. Together, these two terms are

$\left[\begin{array}{c}u \(a_x\+b_y\+c_z\)\\ v \(a_x\+b_y\+c_z\)\\ w \(a_x\+b_y\+c_z\)\end{array}\right]-\left[\begin{array}{c}a \(u_x\+v_y\+w_z\)\\ b \(u_x\+v_y\+w_z\)\\ c \(u_x\+v_y\+w_z\)\end{array}\right]$ = $\left[\begin{array}{c}u a_x\+u b_y\+u c_z-a u_x-a v_y-a w_z\\ v a_x\+v b_y\+v c_z-b u_x-b v_y-b w_z\\ w a_x\+w b_y\+w c_z-c u_x-c v_y-c w_z\end{array}\right]$

The sum of the terms on the right is then

$\left[\begin{array}{c}a u_x\+b u_y\+c u_z\\ a v_x\+b v_y\+c v_z\\ a w_x\+b w_y\+c w_z\end{array}\right]-\left[\begin{array}{c}u a_x\+v a_y\+w a_z\\ u b_x\+v b_y\+w b_z\\ u c_x\+v c_y\+w c_z\end{array}\right]\+\left[\begin{array}{c}u a_x\+u b_y\+u c_z-a u_x-a v_y-a w_z\\ v a_x\+v b_y\+v c_z-b u_x-b v_y-b w_z\\ w a_x\+w b_y\+w c_z-c u_x-c v_y-c w_z\end{array}\right]$

which simplifies to

$\nabla \times \left(\mathbf{F}\times \mathbf{G}\right)\=\left[\begin{array}{c}{u}_{y}b\+u b_y-v_{y}a-v a_y-w_{z}a-w a_z\+u_{z}c\+u c_z\\ v_{z}c\+v c_z-w_{z}b-w b_z-u_{x}b-u b_x\+v_{x}a\+v a_x\\ w_{x}a\+w a_x-u_{x}c-u c_x-v_{y}c-v c_y\+w_{y}b\+w b_y\end{array}\right]$