# Gradient of a dot product identity proof?

1. Jan 28, 2013

### Libohove90

Gradient of a dot product identity proof???

1. The problem statement, all variables and given/known data
I have been given a E&M homework assignment to prove all the vector identities in the front cover of Griffith's E&M textbook. I have trouble proving:

(1) ∇(A$\bullet$B) = A×(∇×B)+B×(∇×A)+(A$\bullet$∇)B+(B$\bullet$∇)A

2. Relevant equations
(2) A×(B×C) = B(A$\bullet$C)-C(A$\bullet$B)

3. The attempt at a solution
I applied the identity in equation (2) to the first two terms on the right hand side of equation (1), and that allowed me to cancel out 4 terms. Yet, I end up with:

∇(A$\bullet$B) = ∇(A$\bullet$B)+∇(B$\bullet$A)

....which I know cannot be correct. How can I prove this identity in a relatively straightforward way? I have seen other pages asking this yet they all involved the use of Levi-Cevita symbols and the Kronecker Delta, which I am trying not to use because we haven't learned them. I would gladly appreciate anyone's effort to help me out.

2. Jan 28, 2013

### dextercioby

Re: Gradient of a dot product identity proof???

The only sensible way I can see is to do it by hand for let's say the <x> component in both sides and show they are the same.

3. Jan 28, 2013

### Libohove90

Re: Gradient of a dot product identity proof???

I was hoping I can get around the long calculations lol

4. Jan 28, 2013

### Staff: Mentor

Re: Gradient of a dot product identity proof???

You have to be careful with the ∇ operator in vector identities, as it is not commutative. I think this caused the problem here.
The long calculation will work, and it is sufficient to consider one component.

5. Jan 28, 2013

### vela

Staff Emeritus
Re: Gradient of a dot product identity proof???

I know you wanted avoid them, but it's definitely worth learning about the Kronecker delta and Levi-Civita symbol. Using them makes verifying the identity much easier.