= Solution
For the <Cauchy problem for a partial differential equation>, suppose $a,b,c$ are $C^1$ near the lifted initial data, $\Gamma$ has a regular $C^1$ parametrization $(\xi(s),\eta(s))$, and the initial function $u_0(s)$ is $C^1$. The local existence and uniqueness theorem requires the <Noncharacteristic Cauchy data> condition
$$
\boxed{a(\xi,\eta,u_0)\eta'(s)-b(\xi,\eta,u_0)\xi'(s)\ne0}
$$
at the starting point. Then there is a neighbourhood of that point carrying a unique $C^1$ classical solution attaining the initial values.
The <method of characteristics> proves this by solving
$$
\frac{dx_1}{d\tau}=a(x_1,x_2,U),\quad\frac{dx_2}{d\tau}=b(x_1,x_2,U),\quad\frac{dU}{d\tau}=c(x_1,x_2,U),
$$
with initial values $(\xi(s),\eta(s),u_0(s))$ at $\tau=0$. Local existence for these ordinary differential equations gives a $C^1$ map. The noncharacteristic determinant is, up to sign, the Jacobian determinant of $(s,\tau)\mapsto(x_1,x_2)$ initially. The <inverse function theorem> makes this map locally invertible, defining $u(x_1,x_2)=U(s,\tau)$ and proving uniqueness along characteristics.
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