For vector fields and with continuous first partial derivatives

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Conditions of Validity, Ch Gradient, divergence and curl without coordinates, and with any coordinates

Problem

For vector fields

\mathbf A

and

\mathbf B

with continuous first partial derivatives, write

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

for the vector with Cartesian components

B_j\partial_jA_i

. ((a)) Prove, in Cartesian index notation, that

\begin{aligned}\nabla\times(\mathbf A\times\mathbf B)&=\mathbf A(\nabla\cdot\mathbf B)-\mathbf B(\nabla\cdot\mathbf A)\\&\quad+(\mathbf B\cdot\nabla)\mathbf A-(\mathbf A\cdot\nabla)\mathbf B.\end{aligned}

((b)) Check the identity on

\mathbf A=(0,x,0)

and

\mathbf B=(0,0,z)

by computing both sides directly.

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