- #1

kostoglotov

- 234

- 6

## Homework Statement

Prove the following identity

[tex]\nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \vec{F} \times (\nabla \times \vec{G}) + \vec{G}\times (\nabla \times \vec{F})[/tex]

## Homework Equations

vector triple product

[tex]\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a}\cdot \vec{c}) - \vec{c}(\vec{a}\cdot \vec{b})[/tex]

## The Attempt at a Solution

The first thing I wanted to do was investigate what expanding according to the vector triple product would do to the original statement I am trying to prove. This happens:

[tex]\nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \nabla (\vec{F}\cdot \vec{G}) - (\vec{F}\cdot \nabla)\vec{G} + \nabla (\vec{F}\cdot \vec{G}) - (\vec{G}\cdot \nabla)\vec{F} = 2\nabla (\vec{F}\cdot \vec{G}) [/tex]

What's happening here? Is it not valid to use the vector differential operator in an expansion of the vector triple product? Why not?