= Solution
For a <quasilinear partial differential equation>, the <characteristic equations for a quasilinear equation> are the ordinary <differential equations>
$$
\boxed{\frac{dx}{ds}=P(x,y,z),\qquad\frac{dy}{ds}=Q(x,y,z),\qquad\frac{dz}{ds}=R(x,y,z).}
$$
A <solution surface> is locally a differentiable graph $z=u(x,y)$ whose tangent plane contains the <vector> $(P,Q,R)$ at every point. Its normal is $(-u_x,-u_y,1)$, so the tangency condition is precisely $Pu_x+Qu_y=R$. Thus every characteristic lying on that graph follows the <vector> field without leaving the graph.
Parameterize the given initial curve by $\gamma(t)=(x_0(t),y_0(t),z_0(t))$. For each $t$, solve the <characteristic equations for a quasilinear equation> with initial value $\gamma(t)$ at $s=0$. The resulting map $(s,t)\mapsto(x(s,t),y(s,t),z(s,t))$ sweeps out a candidate solution surface. If the <coefficients> and initial data are continuously differentiable, the <Picard-Lindelöf theorem> supplies this local family. To make it a graph we also need the projection $(s,t)\mapsto(x,y)$ to be locally invertible. At $s=0$ its <Jacobian determinant> is
$$
J(0,t)=P(\gamma(t))y_0'(t)-Q(\gamma(t))x_0'(t).
$$
The <Noncharacteristic Cauchy data> condition is $J(0,t)\ne0$. The <inverse function theorem> then gives $s=s(x,y)$ and $t=t(x,y)$, and hence $u(x,y)=z(s(x,y),t(x,y))$. Its <derivatives> obey the PDE by the <chain rule>. The same characteristic initial-value uniqueness gives local uniqueness of this solution wherever the projection remains invertible and the relevant characteristic meets the prescribed data.
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