Identity 4: $\nabla \left(\mathbf{F}·\mathbf{G}\right)\=\left(\mathbf{F}·\nabla \right)\mathbf{G}\+\left(\mathbf{G}·\nabla \right)\mathbf{F}\+\mathbf{F}\times \left(\nabla \times \mathbf{G}\right)\+\mathbf{G}\times \left(\nabla \times \mathbf{F}\right)$

Its left side is the gradient of $u a\+v b\+w c$, so it is the vector

$\left[\begin{array}{c}{\partial }_x\(u a\+v b\+w c\)\\ {\partial }_y\(u a\+v b\+w c\)\\ {\partial }_z\(u a\+v b\+w c\)\end{array}\right]$ = $\left[\begin{array}{c}{u}_{x}a\+u a_x\+v_{x}b\+v b_x\+w_{x}c\+w c_x\\ u_{y}a\+u a_y\+v_{y}b\+v b_y\+w_{y}c\+w c_y\\ u_{z}a\+u a_z\+v_{z}b\+v b_z\+w_{z}c\+w c_z\end{array}\right]$ 

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{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]$ = $\left[\begin{array}{c}u a_x\+v a_y\+w a_z\\ u b_x\+v b_y\+w b_z\\ u c_x\+v c_y\+w c_z\end{array}\right]$ 

and

$\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]$ = $\left[\begin{array}{c}a u_x\+b u_y\+c u_z\\ a v_x\+b v_y\+c v_z\\ a w_x\+b w_y\+c w_z\end{array}\right]$ 

The third term on the right, namely, $\mathbf{F}\times \left(\nabla \times \mathbf{G}\right)$, is the cross product of F and the curl of G.

$\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ u& v& w\\ c_y-b_z& a_z-c_x& b_x-a_y\end{array}\right|$ = $\left[\begin{array}{c}v\(b_x-a_y\)-w\(a_z-c_x\)\\ w\(c_y-b_z\)-u\(b_x-a_y\)\\ u\(a_z-c_x\)-v\(c_y-b_z\)\end{array}\right]$ = $\left[\begin{array}{c}v b_x-v a_y-w a_z\+w c_x\\ w c_y-w b_z-u b_x\+u a_y\\ u a_z-u c_x-v c_y\+v b_z\end{array}\right]$

The fourth term on the right, namely, $\mathbf{G}\times \left(\nabla \times \mathbf{F}\right)$, is the cross product of G and the curl of F.

$\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a& b& c\\ w_y-v_z& u_z-w_x& v_x-u_y\end{array}\right|$ = $\left[\begin{array}{c}b\(v_x-u_y\)-c\(u_z-w_x\)\\ c\(w_y-v_z\)-a\(v_x-u_y\)\\ a\(u_z-w_x\)-b\(w_y-v_z\)\end{array}\right]$ = $\left[\begin{array}{c}b v_x-b u_y-c u_z\+c w_x\\ c w_y-c v_z-a v_x\+a u_y\\ a u_z-a w_x-b w_y\+b v_z\end{array}\right]$

The sum of the four terms on the right is then

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

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