I am having trouble doing this problem that has multiple parts to it and has applications to physics so I'm not sure if this belongs in here, but it is for a real analysis class in mathematics.
We are to assume that a tracer is being transported by a moving fluid in a one dimensional medium. Let $u(x,t)$ be the density of the tracer at position $x$ at time $t$. Also let $c(x,t)$ be the velocity of the fluid at $(x,t)$. The following conservation principle can be applied to any interval $a\le x\le b$. $d\over dt$ $($Total mass in $[a,b])$ $= ($Rate of flow past $x=a )$ - $($Rate of flow past $x=b )$
(a.) For any interval $[a,b]$, find an expression for the total mass of the tracer in $[a,b]$ at time $t$
(b.) Find an expression for the left side of the conservation principle stated above. It is an integral on $[a,b]$. Assume that the function $u$ satisfies the hypothesis of the theorem on differentiating under the integral.
There are other parts but I will not post them for now. So for (a.) I understand that. I am having a lot of problems just getting this question started on all parts, so any help to get started would be very helpful. I am familiar more so with math than physics so I am having trouble with the latter. Thank you for any help with this problem.
You can think of the interval as a vessel. The amount of something that is in the interval after a bit of time is the amount that was in there at the start plus what came in minus what went out. In your equation everything is flowing in the positive direction, so what comes in is what passes $a$ and what goes out is what passes $b$. That is the conservation principle you cite. It depends on the fact that there is no production or destruction in the interval.
You are expected to use the functions $u(x,t)$ and $c(x,t)$ to determine the amount of tracer coming in and leaving and write an equation for the amount of tracer in the interval based on this.