# Vector Analysis Problem Involving Divergence

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1. Mar 29, 2017

### MyName

1. The problem statement, all variables and given/known data

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 $g$.

2. Relevant equations

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, $\nabla ^2 =\nabla \cdot \nabla$

3. The attempt at a solution

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.

2. Mar 29, 2017

### Staff: Mentor

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

3. Mar 29, 2017

### pasmith

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 $\nabla g$.

4. Mar 30, 2017

### MyName

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.

5. Mar 30, 2017

### 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 spacial components and compare with the r.h.s.

6. Mar 30, 2017

### pasmith

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

7. Mar 30, 2017

### MyName

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