There seems to be a major inconsistency (perhaps due to my lack of understanding) between what Folland calls a "characteristic" and what I had previously thought was a characteristic.

For example, Folland says that the characteristics of the equation $\partial_x u = 0$ are $\{ \xi : \xi_1 = 0\}$ (in $\mathbb{R}^2$). This confuses me to no end since I thought the characteristic hypersurface was the $x_1$ axis here? According to this definition it's the orthogonal compliment to this. The same issue arises with the heat operator $L=u_t-u_{xx}$ where he says the characteristics are $\{\xi \neq 0 : \xi_x = 0\}$. Aren't the characteristics $t=$ cosntant which are orthogonal to these?

Sorry if this question is elementary but it's given me a real headache and I'm not sure what it is I'm missing here.

characteristicfor $L$ at $x \in S$ if the normal vector $\nu(x)$ to $S$ at $x$ is in $\mathrm{char}_x(L)$". – Hans Lundmark Sep 26 '10 at 17:31isconfusing. I also recommend spending the time to learn the more modern language of cotangent and conormal vectors. It is also confusing at first, but makes things much clearer in the long run. – Deane Yang Sep 26 '10 at 17:43withouthaving any hypersurface that realizes it. – Willie Wong Sep 29 '10 at 17:48