Prove a rotationally symmetric central force field is conservative

[Ryker]

Homework Statement

Suppose g:(0, +∞) → ℝ is continuous, and consider F:ℝ^d\{0} → ℝ^d, where F(x) = xg(|x|). Prove F is conservative.

Homework Equations

F is conservative iff there exists a C¹ function f:ℝ^d\{0} → ℝ^d, s.t. F = grad(f). (edit: Or is the codomain of f actually ℝ, so that it's a scalar-valued function? I namely may have put it down wrong in class.)

F is conservative if, for every closed rectifiable path ∫Fdx = 0.

The Attempt at a Solution

I tried approaching this problem from both angles, i.e. "relevant equations", and came to the conclusion (might be wrong, of course) that if I'm to prove this the second one won't be of particular help. So I went for the first one and tried to construct the function f.

Here's where I run into problems, though. Since g is continuous, we know it has an anti-derivative, call it G. But...
$$\frac{\partial G}{\partial x_{i}} = \frac{x_{i} h(|x|)}{|x|} =: H$$
So the numerator is fine, but I can't get rid of |x|. I then also though of saying that H is conservative, and then showing |x|H is also such. But I can't get anywhere with that, either.

Any help here would be greatly appreciated!

 

[Dick]

If g(|x|) is continuous then so is |x|g(x). Call the antiderivative of that G(|x|).

 

[Ryker]

And then just differentiate G(|x|) to show that F is indeed grad(f), where f is just G? Is that it or am I missing something? Because if that IS it, then damn I'm slow, I can't believe I didn't arrive to that conclusion myself.

 

[Dick]

And then just differentiate G(|x|) to show that F is indeed grad(f), where f is just G? Is that it or am I missing something? Because if that IS it, then damn I'm slow, I can't believe I didn't arrive to that conclusion myself.

No, that's all there is to it. And f is a scalar. Force laws are usually written in terms of a unit vector times magnitude, (x/|x|)*m(|x|). Then you can just integrate m. Your form is missing that extra absolute value so you need to put it back in. x*g(|x|)=(x/|x|)*|x|*g(|x|).

 

[Ryker]

Thanks a lot!