Semi-implicit differential-algebraic equations constitute a normal form

Continuously differentiable functions, the total derivative, or a partial derivative of which is of constant rank, play a part in many engineering problems. One usually exploits this property of constancy of rank by applying the Rank Theorem. However, in case only a partial derivative is of constant rank, which is the natural situation for functions involved in Differential-Algebraic Equations (DAE´s), this theorem does not apply immediately. The author generalizes known results to the latter case. More precisely, he gives a parameterized version of the Rank Theorem and results on functional dependence and presents a normal form for a class of nonlinear equations. Although these results are general in nature, the fundamental conclusion with respect to DAE´s is that here the normal form exactly corresponds to semi-implicit DAE´s. The author also generalizes results from the solution theory of DAE´s in case differential geometric techniques fail to apply. Such DAE´s occur, For example, in the analysis of certain circuits