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I'm reading a section of my text book that is deriving the virial theorm from the hydrostatic equilibrium equation. In part of the derivation it states that

$$-\int_0^M\frac{Gm(r)}{r}dm(r)=E_{GR}=-\frac{GmM}{r}$$

When I perform this integral I get the wrong answer. Here's my working.

$$-\int_0^M\frac{Gm(r)}{r}dm(r)=-Gr^{-1}\int_0^Mm(r)dm(r)=-Gr^{-1}\left[\frac12m^2\right]_0^M=-Gr^{-1}\left(\frac12M^2-0\right)=-\frac{GM^2}{2r}$$

I could just move on and take the text for granted but I like to understand how everything is derived so that I don't get tripped up in the future.

I'm not used to seeing $$m(r)$$ in the $$dm(r)$$ but surely the $$(r)$$ could be omitted right? Isn't this just stating that $$m$$ is a function of $$r$$?

Sorry for being a bit of a simpleton but having not done any studying over the summer I seem to have lost my flow somewhat. If someone could point me in the right direction I would be most grateful.

Thanks

Regards

Brian