Vector Analysis Problem Involving Divergence

[MyName]

Homework Statement

[/B]
Let $f$ and $g$ be scalar functions of position. Show that:
$$\nabla f \cdot \nabla(\nabla ^2 g)-\nabla g \cdot \nabla(\nabla ^2f)$$
Can be written as the divergence of some vector function given in terms of $f$ and $g$.

Homework Equations

[/B]
All the identities given at https://en.wikipedia.org/wiki/Vector_calculus_identities, I suppose. Especially relevant would be the second derivative and divergence identites. Also, $\nabla ^2 =\nabla \cdot \nabla$

The Attempt at a Solution

[/B]
After considerable time messing around with various vector identites, I've been able to show the above is equivalent to:

$$\nabla \cdot (f \nabla (\nabla ^2 g)-g\nabla (\nabla ^2 f))+g(\nabla \cdot \nabla(\nabla ^2 f))-f(\nabla \cdot \nabla(\nabla ^2 g))$$

This is painfully close to the result I want, but I can't seem to show that the second and third terms either cancel or are themselves a divergence. I'd really like any hints, and can provide more detail as to the specific identities and manipulations I've used thus far if needed, thanks.

 

[jedishrfu]

Have you tried working it from the other end starting with the answer or is that an unknown?

 

[pasmith]

Homework Statement

[/B]
Let $f$ and $g$ be scalar functions of position. Show that:
$$\nabla f \cdot \nabla(\nabla ^2 g)-\nabla g \cdot \nabla(\nabla ^2f)$$
Can be written as the divergence of some vector function given in terms of $f$ and $g$.

Homework Equations

[/B]
All the identities given at https://en.wikipedia.org/wiki/Vector_calculus_identities, I suppose. Especially relevant would be the second derivative and divergence identites. Also, $\nabla ^2 =\nabla \cdot \nabla$

The Attempt at a Solution

[/B]
After considerable time messing around with various vector identites, I've been able to show the above is equivalent to:

$$\nabla \cdot (f \nabla (\nabla ^2 g)-g\nabla (\nabla ^2 f))+g(\nabla \cdot \nabla(\nabla ^2 f))-f(\nabla \cdot \nabla(\nabla ^2 g))$$

This is painfully close to the result I want, but I can't seem to show that the second and third terms either cancel or are themselves a divergence. I'd really like any hints, and can provide more detail as to the specific identities and manipulations I've used thus far if needed, thanks.

Those terms simplify to $g \nabla^4 f - f \nabla^4g$, which doesn't cancel.

Rather than building a vector field as a linear combination of $\nabla(\nabla^2 f)$ and $\nabla(\nabla^2 g)$, I would have started by building one as a linear combination of $\nabla f$ and $\nabla g$.

 

[MyName]

Have you tried working it from the other end starting with the answer or is that an unknown?

Reference https://www.physicsforums.com/threads/vector-analysis-problem-involving-divergence.909435/

Unfortuntely the end result is unknown, otherwise that'd be a great suggestion, thanks!

Those terms simplify to g∇4f−f∇4g, which doesn't cancel. Rather than building a vector field as a linear combination of ∇(∇2f) and ∇(∇2g), I would have started by building one as a linear combination of ∇f and ∇g.

Reference https://www.physicsforums.com/threads/vector-analysis-problem-involving-divergence.909435/

Yeah, I managed to get to that simplification, which like you said defnitely doesn't cancel, so I guess I must be able to somehow write the quantity as a divergence. I'm sorry, but I don't really understand what you mean by building a vector field?

Thanks for the help so far, I appreciate it.

 

[Fred Wright]

I would write,$$\nabla f \cdot\nabla\left ( \nabla^2 g\right )-\nabla g\cdot \nabla \left ( \nabla^2 f \right ) =\nabla \cdot \vec V $$
Then do the work of expanding the l.h.s. in it's spatial components and compare with the r.h.s.

 

[pasmith]

Unfortuntely the end result is unknown, otherwise that'd be a great suggestion, thanks!
Yeah, I managed to get to that simplification, which like you said defnitely doesn't cancel, so I guess I must be able to somehow write the quantity as a divergence. I'm sorry, but I don't really understand what you mean by building a vector field?

Thanks for the help so far, I appreciate it.

Consider the divergence of $D(g, \nabla^2 g)\nabla f - D(f, \nabla^2 f) \nabla g$ for some function $D$.

 

[MyName]

I would write,∇f⋅∇(∇2g)−∇g⋅∇(∇2f)=∇⋅⃗V Then do the work of expanding the l.h.s. in it's spatial components and compare with the r.h.s.

That is a great idea, thanks! I managed to solve it using this idea.