# Identity by using gradient

1. Mar 15, 2015

### Flotensia

1. The problem statement, all variables and given/known data
Let us consider three scalar fields u(x), v(x), and w(x). Show that they have a relationship such that f(u, v, w) = 0 if and only if

(∇u) × (∇v) · (∇w) = 0.

2. Relevant equations

3. The attempt at a solution
I could think nothing but just writing the components of (∇u) × (∇v) · (∇w) = 0. How could i prove this?

Last edited: Mar 15, 2015
2. Mar 15, 2015

### Flotensia

1. The problem statement, all variables and given/known data
Let us consider three scalar fields u(x), v(x), and w(x).
Show that they have a relationship such that f(u, v, w) = 0 if and only if
(∇u) × (∇v) · (∇w) = 0.

2. Relevant equations

3. The attempt at a solution
I can do nothing but just writing components of (∇u) × (∇v) · (∇w). How can I prove this identity?

3. Mar 15, 2015

### Ray Vickson

If $\vec{A} = \nabla u$, $\vec{B} = \nabla v$ and $\vec{C} = \nabla w$, what does the given condition say about the directions of the vectors $\vec{A}, \vec{B}, \vec{C}$?

4. Mar 15, 2015

### Flotensia

Scalar triple product means det(a,b,c) or volume of a parallelepiped. Is it a key to solve this problem?

5. Mar 15, 2015

### Brian T

When you wrote the components out, what did you get?

6. Mar 15, 2015

### Flotensia

I wrote

by using levi-civita symbol

7. Mar 16, 2015

### Dick

One way is easy. If f(u,v,w)=0 then to show the triple product is zero you just have to show that the gradients of u, v and w are linearly dependent. Use the chain rule. The other way is harder, you need to have some sort of integrability theorem like Frobenius. I'm a little fuzzy on that. What have you got?

Last edited: Mar 16, 2015
8. Mar 16, 2015

### Staff: Mentor

Sorry, am I missing something? There is always such a function f(u,v,w) - just define it to be zero everywhere. Are we looking for a non-trivial (!) linear function f?

9. Mar 16, 2015

### Dick

Good point. I'm guessing the problem should state that f has at least one nonzero partial derivative at points where you want to show the triple product is zero. Just a guess.

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