# Vector Calculus - Use of Identities

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

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?

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
Mentor
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.

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 realise that you can put a set of bracket in like that. And the cross product of something with itself is zero. Thanks!