Introduction to Electrodynamics · Problem 1.22

Vector Analysis — David J. Griffiths

Physics · vector-calculus-directional-derivative

Problem

(a) If

\mathbf{A}

and

\mathbf{B}

are two vector functions, what does the expression

(\mathbf{A} \cdot \nabla)\mathbf{B}

mean? (That is, what are its

x

y

, and

z

components, in terms of the Cartesian components of

\mathbf{A}

\mathbf{B}

, and

\nabla

?) (b) Compute

(\hat{\mathbf{r}} \cdot \nabla)\hat{\mathbf{r}}

, where

\hat{\mathbf{r}}

is the unit vector defined in Eq. 1.21. (c) For the functions in Prob. 1.15, evaluate

(\mathbf{v}_a \cdot \nabla)\mathbf{v}_b