Chapter 14.5, Problem 40E

Another derivative combination Let F = (f, g, h) and let u be a differentiable scalar-valued function.

a.    Take the dot product of F and the del operator; then apply the result to u to show that

$\begin{array}{c}\left(F\cdot \nabla \right)u=\left(f\frac{\partial }{\partial x}+g\frac{\partial }{\partial y}+h\frac{\partial }{\partial z}\right)u\\ =f\frac{\partial u}{\partial x}+g\frac{\partial u}{\partial y}+h\frac{\partial u}{\partial z}.\end{array}$

b.    Evaluate (F·▿)(xy²z³) at (1, 1, 1), where F = (1, 1, 1).