# PDEMOC10

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 $V=\,$ and intersect in a curve $\chi \,$, show that $\chi \,$ is a characteristic curve.
So let $p\in \chi \,$. Then $u_{1}\,$ contains the characteristic curve passing through $p\,$ and so does $u_{2}\,$. Therefore $p\in u_{1}\cap u_{2}\,$. But $u_{1}\cap u_{2}=\chi \,$, so $\chi \,$ is the characteristic curve passing through $p\,$.