Is it correct to use two different variables when solving this integral?

[mlazos]

[SOLVED] I want a second opinion

We know the equation

F=-\frac{dV}{dr}

we want to find the integral from r_{0} to r.

I have seen someone doing this

\int^{r}_{r_{0}}Fdr'=-\int^{r}_{r_{0}}\frac{dV}{dr}dr'

I am a mathematician and the way I was doing at the university was

F=-\frac{dV}{dr}\RightarrowFdr=dV

and then I integrate

\int^{r}_{r_{0}}Fdr=-\int^{}_{r_{0}}dV

Since the potential depends on r we can integrate. So I would like someone who knows the subject to tell me if the first way is correct since I know the second is correct. Its difficult for me to accept the introduction of another variable r' while we have the r itself.

 

[arildno]

Well, the first one is incorrect; it should be:
\int_{r_{0}}^{r}Fdr'=-\int_{r_{0}}^{r}\frac{dV}{dr'}dr'

Now, the second integral simply equals V(r)-V(r_{0}), thus, we have identity between the expressions:
\int_{r_{0}}^{r}\frac{dV}{dr'}dr'=\int_{V(r_{0})}^{V(r)}dV

 

[mlazos]

ok the first way is what a I saw, exactly the way I wrote it. With your correction makes sense. Thank you very much for your answer.