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.