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**1. The problem statement, all variables and given/known data**

Prove the following vector identity:

**[tex]\nabla[/tex]**x(

**A**x

**B**) = (

**B.[tex]\nabla[/tex]**)

**A**- (

**A.[tex]\nabla[/tex]**)

**B**+

**A**(

**[tex]\nabla[/tex].B**) -

**B**(

**[tex]\nabla[/tex].A**)

Where

**A**and

**B**are vector fields.

**2. Relevant equations**

Curl, divergence, gradient

**3. The attempt at a solution**

I think I know how to do this: I have to expand out the LHS and the RHS and show that they equal one another. To do this I need to use the product rule when taking the gradient of components with more than one term multiplied together.

What I don't understand is what's going on on the RHS: doesn't (

**B.[tex]\nabla[/tex]**)

**A**=

**B**(

**[tex]\nabla[/tex].A**) ? (Obviously this can't be the case since then all the components would cancel to zero.) So how does this really work?

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