Vector identity involving grad and a function

[ck99]

Homework Statement

The question is to use index notation to show that the following is true, where a is a three-vector and f is some function.

Homework Equations

The Attempt at a Solution

Hmmmm . . . I haven't really got anything to put here!

I am starting to get to grips with the basics of index notation, and using the Levi-Civitas identities for other identity proofs. I haven't ever worked with grad before though, and I can't find any help online because I can't find any other identities that look like this one. Does it have a name or something to help me search?

The closest I have found is "The product of a vector and a scalar" on wikipedia

but my question states that f is a function, not a scalar, which must make some difference I guess! I just don't know where to start here, do any of the operations within my question have an index-notation version using the LC tensor or similar?

Any help much appreciated :)


PS: You will need to click the thumbnails in the post to see the full pictures of the equations. I am not much of a computer guy. Or a maths guy, it seems :(

 

[vela]

f is a scalar function, as opposed to, say, a vector function.

The i-th component of $\nabla f$ is $\partial_i$f. Use that and the Levi-Civita symbol to prove the identity.

 

[ck99]

f is a scalar function, as opposed to, say, a vector function.

The i-th component of $\nabla f$ is $\partial_i$f. Use that and the Levi-Civita symbol to prove the identity.

I think the root of my problem here is that I have no idea how to write out the starting equation in ordinary longhand notation (as components of vectors), let alone in index notation.

Is $\nabla f$ equal to $f \nabla$?

And what do I get from multiplying $\nabla$ with fa?

 

[vela]

If you don't understand the notation or what the gradient is, a good place to start is looking up what it means.

http://en.wikipedia.org/wiki/Gradient

Your textbooks will probably have a more accessible discussion.

 

[paranormal]

Hello, it seems like you are trying to prove the following vector identity:

∇(fA) = (∇f)A + f∇A

This is known as the product rule for gradients. To prove this, we can use index notation as follows:

∇(fA) = (∂/∂x^i)(fA^j) = (∂f/∂x^i)A^j + f(∂/∂x^i)A^j = (∇f)A + f∇A

Where we have used the product rule for derivatives in the second step, and the definition of the gradient in the third step.

I hope this helps, let me know if you have any further questions. Good luck with your studies!