Proving that central force is conservative and getting the energy

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

Hey, sorry in advance if something I write is unclear but I am not native English speaker.



I have a central force that is defined as [tex]F=-f(r) \frac{\vec{r}}{r}[/tex] where [tex]f(r)[/tex] is some function of distance and [tex]\vec{r}=(x,y,z)[/tex]. I have to calculate potential energy when [tex]f(r)=\frac{a}{r^{2}}[/tex] and I have to use [tex]\vec{r}d\vec{r}=\frac{dr^{2}}{2}[/tex]





2. Relevant equations







3. The attempt at a solution



I wrote Force F as [tex]\left( -\frac{f(r)x}{r}, -\frac{f(r)y}{r},-\frac{f(r)z}{r} \right)[/tex] can I or can I not do it?

Calculated the curl, it was 0 so it is conservative. Now I substituted f(r) and got [tex]\left( -\frac{ax}{r^{3}}, -\frac{ay}{r^{3}},-\frac{az}{r^{3}} \right)[/tex]



To calculate potential energy I have 3 equations:



[tex]-\frac{ax}{r^{3}} = \frac{dE}{dx}[/tex]

[tex]-\frac{ay}{r^{3}} = \frac{dE}{dy}[/tex]

[tex]-\frac{az}{r^{3}} = \frac{dE}{dz}[/tex]



and the result is [tex]-\frac{a}{2r^{3}}\left(x^{2}+y^{2}+z^{2}\right) + C[/tex]. I didn't use the [tex]\vec{r}d\vec{r}=\frac{dr^{2}}{2}[/tex] so I guess I did something wrong





When you reply please tell me where and why did I make a mistake and how to correct it.