Example 12.2.1.label We consider first the quasilinear case
\begin{align*}\sums{n }{i=1}a_{i}(x,u)\pd_{i}u=b(x,u)\end{align*}
where $x=(x_{1},...,x_{n})$. We consider the Cauchy problem $u|_{S}=\p$ for $S$ hypersurface in $\R^{n}$ and $\p$ is given (prescribes the graph of $u$ over $S$). Let $S^{*}=\curl{(x,\p(x):x\in S)}$ then this problem is reducible to an ODEs problem. Rewrite the graph of $u$ as $F=u(x)-y=0$ then we rewrite the original problem as
\begin{align*}\vec{A}\cd\nabla F=0&&\vec{A}=(a_{1},...,a_{n},b),\nabla F=(\pd_{1}u,...,\pd_{n}u,-1)\end{align*}
Thus $\vec{A}$ must be everywhere tangent to graph $F=0$. This suggests constructing graph of $u$ by ”flowing off” $S^{*}$ along integral curves of $\vec{A}$ for this to work it would be sensible to require that $\vec{A}$ to never be tangent to $S^{*}$.
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