Transport Equation
Summary
This lecture introduces the transport equation $$ u_t + b\cdot Du = 0,\quad \text{in}~ \mathbb{R}^n\times\mathbb{R} $$ and gives a procedure based on (a very basic) implementation of the method of characteristics for the solution of the initial-value problem $$ \left\{ \begin{alignedat}{2} u_t + b\cdot Du &= 0\quad &&\text{in}~ \mathbb{R}^n\times\mathbb{R}\\ u&=g\quad &&\text{on}~ \mathbb{R}^n \end{alignedat} \right. $$ for an unknown function $u=u(x,t)$, $u \colon \mathbb{R}^n\times\mathbb{R} \to \mathbb{R}$ and a given function $g\colon \mathbb{R}^n\to \mathbb{R}$ with sufficient regularity.
Full Set of Lecture Notes
The notes for this lecture are available here (14 pages).