Noncharacteristic Cauchy data ($a{\eta }^{\prime }-b{\xi }^{\prime }\ne 0$)

For $a{u}_{x_1}+b{u}_{x_2}=c$ with data along $(\xi (s),\eta (s))$, the initial curve is noncharacteristic when its tangent is linearly independent of the projected characteristic vector $(a,b)$ evaluated at the initial value of $u$. The determinant is $a{\eta }^{\prime }-b{\xi }^{\prime }$, up to sign. For $C^1$ coefficients and data this permits the inverse function theorem to invert the local characteristic parametrization and gives a unique local classical solution.