- #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?