Differentiation index for a regular local semi-explicit DAE

Fix a semi-explicit presentation $x^{\mathrm{\prime }}=f(t,x,z)$, $0=g(t,x,z)$ as in Semi-explicit index-one DAEs and consistent initial data and a sufficiently differentiable consistent local solution. Differentiate the algebraic constraint along the differential equation, substituting $x^{\mathrm{\prime }}=f(t,x,z)$ and any previously obtained derivative relations at each stage. The differentiation index in this convention is the least positive integer $k$ for which the original constraint and its first $k$ differentiated equations locally determine $z$ and $z^{\mathrm{\prime }}$, with the equation for the highest derivative of $z$ having an invertible square Jacobian on a neighborhood of the solution. If no such finite $k$ exists, this local index is undefined. This convention counts through the first derivative of the algebraic variables and is local to the fixed presentation; it is not a universal index theory for arbitrary implicit systems.

If $g_z$ is invertible, one differentiation gives

$$g_t+g_{x}f+g_z{z}^{\mathrm{\prime }}=0,$$

which determines $z^{\mathrm{\prime }}$, so the index is one. If $g_z$ is singular, the index must be checked from the actual differentiated equations; no index-one conclusion follows from this definition alone.