- #1

Ryker

- 1,086

- 2

## 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 C

^{1}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...

[tex]\frac{\partial G}{\partial x_{i}} = \frac{x_{i} h(|x|)}{|x|} =: H[/tex]

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!

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