- #1

raving_lunatic

- 21

- 0

## Homework Statement

Hi, it's me again. I'm new to vector calculus so this might sound like a stupid question, but in relation to a specific problem, I was wondering when we could move the del operator under the integration sign - in relation to a specific problem, which is:

**A(r)**= integral over some volume v ∫

**∇**f(|

_{r'}**r**-

**r'**|)d

^{3}

**r'**

Where

**∇**represents the gradient with respect to

_{r'}**r'**

Now, we're supposed to show that

**∇**x

**A**= 0 by finding some function of which A is the gradient.

## Homework Equations

## The Attempt at a Solution

I'm not sure how to tackle the problem other than by saying that

**A(r)**=

**∇r'**∫ f(|

**r-r'**|) d

^{3}

**r**

I.e taking the del operator "outside" of the integral. We then have that A is the gradient of some kind of scalar function (it's given that f is a "well-behaved function of a single variable" and the volume is fixed, so I'm guessing this will be just a single function.) I've seen this done, but never been sure about the mathematics behind it.

I've looked online to see about "differentiation under the integral sign" but how it relates to the del operator, I haven't found anywhere.

If, as I suspect, this approach is flawed, please advise on where to go next. Would using Divergence or Stokes' Theorem (or corollaries) be helpful?

Thanks