On geometric and differentiation index of nonlinear differential-algebraic equations

We discuss two notions of index, i.e., the geometric index and differentiation for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions DAEs by revising a reduction method (see e.g. Rabier Rheinboldt (2002), Riaza (2008)). Then show that although both serve as measure difficulties solving DAEs, they are actually related to existence uniqueness in different manner. It is claimed (Campbell Gear, 1995) indices coincide when sufficient smoothness assumptions satisfied, elaborate this claim indeed if only condition satisfied (under certain constant rank assumptions). Finally, an example pendulum system used illustrate our results on indices.