Heythere,
I am confused by the terms characteristic lines and integral lines. The book introduces characteristic lines as the curves along which a function is constant. Now in the notes integral lines are curves to which the vector field is tangential, i.e. in the case of the gradient vector field the lines along which the function changes most (in abs. value).
So I thought these two should be orthogonal in the case of $au_t+bu_x=0$.
So are integral lines the same as characteristic lines?
If we consider 1-st order PDEs in the form
\begin{equation}
a_0\partial_t u + a_1\partial_x u + a_2\partial_y u=0
\label{eq-1}
\end{equation}
then characteristics of the equation (\ref{eq-1}) are integral lines of the vector field $(a_0,a_1,a_2)$ i.e. curves
\begin{equation}
\frac{dt}{a_0}=\frac{dx}{a_1}=\frac{dy}{a_2}.
\label{eq-2}
\end{equation}
There could be just two variables $(t,x)$ or more ... and coefficients are not necessary constant. Yes, for (\ref{eq-1}) characteristics are lines along which $u$ is constant.
But we preserve the same definition of characteristics as integral lines for more general equation f.e.
\begin{equation}
a_0\partial_t u + a_1\partial_x u + a_2\partial_y u =f (t,x,y,u)
\label{eq-3}
\end{equation}
and here $u$ is no more constant along characteristics but solves ODE
\begin{equation}
\frac{dt}{a_0}=\frac{dx}{a_1}=\frac{dy}{a_2}=\frac{du}{f}.
\label{eq-4}
\end{equation}
Further, notion of characteristics generalizes to higher order equations. Definition
curves along which solution is constant goes to the garbage bin almost immediately.
