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I am trying to learn a little about characteristics of PDEs. I think I understand how to find characteristic curves for an equation with 2 independent variables, but in case of 3 independent variables finding characteristic surfaces is less clear to me.

I want to know if this is the right way to find them:

(a) In general for a characteristic surface $P(x,y,z)$ we need: $A_{ij}P_iP_j=0$ where $A$ is a $3\times3$ matrix containing the coefficients of the second order derivatives. The subscripts of $P$ denote differentiation and summation is implied.

(b) So for example for $U_{yz}(x,y,z)=0$ assume a characteristic surface $z=N(x,y)$. Using (a) we get $A_{23}=A_{32}=1$ yielding $N_y=0$. Integrating we get $N(x,y)=F(x)=z$ with $F$ being an arbitrary function. So for example taking $F=0$ gives $z=0$ as a characteristic surface.

Is this correct?

Thanks Uri

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