I am analyzing a basic example of traffic flow presented here
http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf
and have a question to the last transition in the traffic flow equation derivation. (Example 1.7)
Given are:
density function $ \ u(x,t) \ $ expressed in cars/miles, $ \ \ $ velocity v in miles/hour, $ \ $ and $ \ \ \phi(x,t) \ $ expressed in cars per hour.
The relation can be described as
$$\phi(x,t)=uv$$
assuming constant relation between velocity and density
$$v=v_1- \frac{v_1}{u_1} u$$
substituting for v gives
$$\phi=uv=v_1 (u-\frac{u^2}{u_1})$$
Now there is the equation for the car density written as
$$0=u_t + \phi' u_x$$
which I don't understand. Where does it mathematically come from? where the $u_t$ comes from here?
should be the below equation applied here? How? $$\frac{\partial \phi}{\partial x}=\frac{d \phi}{d u} \frac{\partial u}{\partial x}$$