  For nonlinear differential-algebraic equations (DAEs), we define two kinds of
equivalences, namely, the external and internal equivalence. Roughly speaking,
the word "external" means that we consider a DAE (locally) everywhere and
"internal" means that we consider the DAE on its (locally) maximal invariant
submanifold (i.e., where its solutions exist) only. First, we revise the
geometric reduction method in DAEs solution theory and formulate an
implementable algorithm to realize that method. Then a procedure named
explicitation with driving variables is proposed to connect nonlinear DAEs with
nonlinear control systems and we show that the driving variables of an
explicitation system can be reduced under some involutivity conditions.
Finally, due to the explicitation procedure, we will use the notion of zero
dynamics from nonlinear control theory to derive a nonlinear generalization of
the Weierstrass form.
