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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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Hi Uri, I added the tag "homework". Please tell us what you have tried and understand so far, and where you are stuck. (Sometimes it is also worthwhile to spell out the definitions you work with.) – Tim van Beek Aug 5 '11 at 17:48
Hi Tim. This is not homework but I added some explanations. Thanks. – uri Aug 5 '11 at 22:41
uri, I removed the homework tag @Tim: I think there is a consensus that one should not add the homework tag without the original user's consent. See e.g. here and here – t.b. Aug 5 '11 at 22:47

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