I don't understand the 'idea' behind the method of characteristics Below is an image of my lecture notes explaining the idea behind the method of characteristics for quasilinear first order PDEs.





However I don't understand how the curve $C_s$ is defined and how it translates into the graph $Gr_u$. How does $C_s$ 'start' on the initial curve if there is no $f(s)$ or $g(s)$ in the equation for $C_s$? What does $x(t;s)$ mean, does that mean (say for the $x$ term) that for fixed $s$ one starts at $x(s-\epsilon_s)$ and finishes at $x(s+\epsilon_s)$? If so what is this 'function' $x$? 

Thanks, I would really appreciate someone explaining this to me! :
To clarify what your notes imply, the curve ${\cal C}_s$ contains points $(x(t;s),y(t;s),z(t;s))$ defined by solving the ODE $\frac{d}{dt}(x,y,z)=(a(x,y,z),b(x,y,z),c(x,y,z))$ for $-\varepsilon_s&lt;t&lt;\varepsilon_s$, and condition $(x(0;s),y(0;s),z(0;s))=(f(s),g(s),h(s))$ on $\Gamma$ at $t=0$.
To clarify what your notes imply, the curve ${\cal C}_s$ contains points $(x(t;s),y(t;s),z(t;s))$ defined by solving the ODE $\frac{d}{dt}(x,y,z)=(a(x,y,z),b(x,y,z),c(x,y,z))$ for $-\varepsilon_s&lt;t&lt;\varepsilon_s$, and condition $(x(0;s),y(0;s),z(0;s))=(f(s),g(s),h(s))$ on $\Gamma$ at $t=0$.