= Characteristic equations for a first-order quasilinear equation
{title2=$x_s=P(x,y,z),\quad y_s=Q(x,y,z),\quad z_s=R(x,y,z)$}
= Characteristic equations for a quasilinear equation
{synonym}
For the first-order <quasilinear partial differential equation> $P(x,y,z)z_x+Q(x,y,z)z_y=R(x,y,z)$, the displayed <ordinary differential equations> trace curves tangent to every classical <solution surface>. Indeed, if $z=u(x,y)$ solves the equation and $x_s=P$, $y_s=Q$, then the <chain rule> gives $z_s=u_xP+u_yQ=R$. Conversely a surface swept out by these curves solves the equation wherever its projection to $(x,y)$ is invertible and hence represents a differentiable graph. The <coefficients> can depend on the unknown $z$, which distinguishes these equations from characteristic equations for a prescribed linear <transport equation>.
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