PDEMOC10

From Exampleproblems

If $S_1\,$ and $S_2\,$ are two graphs $\left[ S_i = u_i(x,y), i=1,2\right]\,$ that are integral surfaces of $V=<a,b,c>\,$ and intersect in a curve $\chi\,$ , show that $\chi\,$ is a characteristic curve.

From this problem, we know that given any point on an integral surface, the surface contains the unique characteristic curve passing through that point.

So let $p\isin \chi\,$ . Then $u_1\,$ contains the characteristic curve passing through $p\,$ and so does $u_2\,$ . Therefore $p\isin u_1 \cap u_2\,$ . But $u_1 \cap u_2=\chi\,$ , so $\chi\,$ is the characteristic curve passing through $p\,$ .