Is this integral a convolution ?

[Mentz114]

I'm struggling to find a function [itex]E(t)[/itex] which is the energy inside a sphere with energy density [itex]\rho(t,r)[/itex] where the radius [itex]r \equiv r(t)[/itex] is itself a function of time. This
[tex]
E(t) = 8\pi \int_0^{r(t)} \rho(r,t) dr
[/tex]
doesn't make sense, does it ? Is the thing I'm looking for some kind of convolution of r and [itex]\rho[/itex] ?

 

[HallsofIvy]

Why do you say it doesn't make sense? It looks perfectly reasonable to me.

 

[mathman]

I'm struggling to find a function [itex]E(t)[/itex] which is the energy inside a sphere with energy density [itex]\rho(t,r)[/itex] where the radius [itex]r \equiv r(t)[/itex] is itself a function of time. This
[tex]
E(t) = 8\pi \int_0^{r(t)} \rho(r,t) dr
[/tex]
doesn't make sense, does it ? Is the thing I'm looking for some kind of convolution of r and [itex]\rho[/itex] ?

You should try to be more careful in your notation. You have ρ(t,r) in one place and ρ(r,t) someplace else. More important if r is the upper limit of the integral, it should not be used as the dummy for integration.

 

[Mentz114]

Thanks for the replies. Sorry about the sloppiness.

More important if r is the upper limit of the integral, it should not be used as the dummy for integration.

Yes, I thought there was something wrong but I'm still baffled.