Vector Calculus - Use of Identities

[12x4]

Homework Statement

By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that

1. ∇ × (φ∇φ) = 0,
   where φ(r) is any scalar field.

Homework Equations

∇×(φA) = (∇φ)×A+φ(∇×A)?

The Attempt at a Solution

I honestly have no idea how to even start this one. Any help would be hugely appreciated.

 

[mfb]

You can apply the relevant equation to the expression you are asked to evaluate.

 

[12x4]

You can apply the relevant equation to the expression you are asked to evaluate.

Hi mfb, thank you for your reply. That is what I tried to do before but thought I must have been doing it wrong as i ended up with something that didn't cancel.

Okay so can I assume A = ∇φ
and the identity is ∇×(φA) = (∇φ)×A+φ(∇×A)

and follow that through like this:?

∇ x (φ∇φ) = ∇φ x ∇φ + φ∇ x ∇φ

I feel as if there is another identity that I must need to complete the question but have searched through my notes and online and can't find anything. Does ∇φ x ∇φ = -(φ∇ x ∇φ) in some way?

 

[12x4]

I've just had a thought. Are you able to say

A = ∇φ and therefore -A = φ∇

Allowing us to complete the question like this.

∇ x (φ∇φ) = (A x A) + (-A x A)
∇ x (φ∇φ) = (A x A) - (A x A)
= 0

 

[mfb]

Hi mfb, thank you for your reply. That is what I tried to do before but thought I must have been doing it wrong as i ended up with something that didn't cancel.

So you had an idea where to start, and even a start. Why didn't you write that down? Then it would have been possible to see what went wrong.

∇ x (φ∇φ) = ∇φ x ∇φ + φ∇ x ∇φ

I would probably add brackets to make it more clear: (∇φ) x (∇φ) + φ(∇ x ∇φ)
What is the cross-product of something with itself?

A = ∇φ and therefore -A = φ∇

No. The right equation is not even well-defined.

 

[12x4]

So you had an idea where to start, and even a start. Why didn't you write that down? Then it would have been possible to see what went wrong.

Sorry mfb, I thought my working was complete rubbish and irrelevant so didn't want to embarrass myself on here.

I would probably add brackets to make it more clear: (∇φ) x (∇φ) + φ(∇ x ∇φ)
What is the cross-product of something with itself?

No. The right equation is not even well-defined.

I think I understand now, I didn't realize that you can put a set of bracket in like that. And the cross product of something with itself is zero. Thanks!