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## Homework Statement

"Let F be a central vector field, that is, [tex]F(x,y,z) = h(x,y,z)(x,y,z)[/tex] for some smooth function [tex]h[/tex].

Prove that its potential [tex]V[/tex] is radial, i.e., [tex]V(x,y,z) = g(\sqrt{x^2 + y^2 + z^2})[/tex] for some smooth function [tex]g[/tex]"

## Homework Equations

Since we know that a central vector field must be conservative, we have that [tex]F(x,y,z)=-\nabla{V(x,y,z)}[/tex]

## The Attempt at a Solution

The original problem statement is slightly different, in fact, I have to prove that given a central vector field F, there exists a radial function [tex]g[/tex] such that

[tex]F=g(\sqrt{x^2+y^2+z^2})(x,y,z)[/tex], but the existence of this function follows directly once the first problem statement is proved.

My first attempt to prove it has been using Taylor, using the [tex]V(x,y,z)[/tex] smoothness and the fact that [tex]\nabla{V(x,y,z)} = -h(x,y,z)(x,y,z)[/tex].

By Taylor polynomial we have for any point [tex]a = (a_1,a_2,a_3)[/tex] in the domain of [tex]V[/tex],

[tex]V(x,y,z) = V(a_1,a_2,a_3) + \nabla{V(x-a_1,y-a_2,z-a_3)}^t(x-a_1,y-a_2,z-a_3) + R_{2,a} (x,y,z) [/tex]

whence,

[tex]V(x,y,z) = V(a_1,a_2,a_3) - h(x-a_1,y-a_2,z-a_3)(x-a_1,y-a_2,z-a_3)^t(x-a_1,y-a_2,z-a_3) + R_{2,a} (x,y,z) = [/tex]

[tex]V(a_1,a_2,a_3) - h(x-a_1,y-a_2,z-a_3)||(x-a_1,y-a_2,z-a_3)||^2 + R_{2,a} (x,y,z) [/tex]

But that does not show anything.

It would be interesting to have a hint of how should I proceed to prove this statement.

Thanks for your help.

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