Hello all:
I have been studying a little traffic flow because of having been in road construction I never dreamed it was so involved, mathematically.
Here's a problem:
Suppose we are interested in the change in the number of cars N(t) between two observers, one fixed at x=a and the other moving in some prescribed manner, x=b(t).
$\displaystyle N(t)=\int_{a}^{b(t)}{\rho}(x,t)dx$
The derivative of an integral with a variable limit is
$\displaystyle \frac{dN}{dt}=\frac{db}{dt}{\rho}(b,t)+\int_{a}^{b (t)}\frac{{\partial}{\rho}}{{\partial}t}dx$
Show this result using either considering:
$\displaystyle \lim_{h\to 0}\frac{N(t+h)-N(t)}{h}$ or by using the chain rule for derivatives.
Also, Using $\displaystyle \frac{{\partial}{\rho}}{{\partial}t}=\frac{{-\partial}}{{\partial}x}({\rho}u)$, show that
$\displaystyle \frac{dN}{dt}={-\rho}(b,t)\left[u(b,t)-\frac{db}{dt}\right]+{\rho}(a,t)u(a,t)$
I must admit, I am rather stymied. I ahve been going thorugh a rough time and I am having a difficult time concentrating.