# Partial derivative problem

1. Feb 9, 2016

### cmkluza

(Hope it's okay that I'm posting so much at the moment, I'm having quite a bit of trouble with something I'm doing)
1. The problem statement, all variables and given/known data
I'm having trouble with the simplification of the following equation. The answer is shown, but I can't figure out the process to get to it.
$$\frac{d}{dt} \int^b_a \rho (t,x)dx = \rho (t,a)v(\rho (t,a)) - \rho (t,b)v(\rho (t,b))$$
$$\int^b_a\frac{\partial }{\partial t}\rho dx = - \int^b_a\frac{\partial }{\partial x}[\rho v(\rho)]dx$$
$$\frac{\partial }{\partial t}\rho + \frac{\partial }{\partial x}[\rho v(\rho)] = 0$$
2. Relevant equations
Edit: As @SteamKing pointed out, I should probably give some information about the equation rather than just list it.

The above equation models traffic flow, namely a wave of density travelling throughout traffic from point b to a (as best I understand it). The variables are as follows:
$\rho (t,x)$ = density ($\frac{cars}{distance}$) as a function of time and distance
$v (\rho )$ = velocity ($\frac{distance}{time}$) as a function of density
$Q = \rho \times v$ = Flow rate ($\frac{cars}{time}$)

I've asked about the concept behind this equation in a previous thread located here which should elaborate on some questions you might have about it.
3. The attempt at a solution
I'm not entirely certain where to start. I've stared at this for quite some time, but I don't understand it. Are the left and right sides of the equation preserved from the first to the second step? I can see that the derivative with respect to time of the first bit becomes a partial with respect to time, though I can't necessarily understand that, but I don't see how the right side is combined and also becomes an integral with a partial inside of it. Sorry to post this without any real work, but I don't know what else to do at the moment since I have no one to explain this to me. If this silly explanation of my lack of understanding doesn't count as an attempt of course I'll remove this post.

Anyhow, thanks for any help!

Last edited: Feb 9, 2016
2. Feb 9, 2016

### SteamKing

Staff Emeritus
There are a couple of things in your post which aren't clear.

What is ρ(t,x) supposed to be? What is v(ρ) supposed to be?

You assume that these integrals exist in a vacuum. They do not. They describe something or some process, and it's difficult to help you without some context.

Ahh. I see now.