  Skew-symmetric differential forms play an unique role in mathematics and
mathematical physics. This relates to the fact that closed exterior
skew-symmetric differential forms are invariants. The concept of "Exterior
differential forms" was introduced by E.Cartan for a notation of integrand
expressions, which can create the integral invariants.(The existence of
integral invariants was recognized by A.Poincare while studying the general
equations of dynamics.)
  All invariant mathematical formalisms are based on invariant properties of
closed exterior forms. The invariant properties of closed exterior forms
explicitly or implicitly manifest themselves essentially in all formalisms of
field theory, such as the Hamilton formalism, tensor approaches, group methods,
quantum mechanics equations, the Yang-Mills theory and others. They lie at the
basis of field theory.
  However, in this case the question of how the closed exterior forms are
obtained arises. In present work it is shown that closed exterior forms, which
possess the invariant properties, are obtained from skew-symmetric differential
forms, which, as contrasted to exterior forms, are defined on nonintegrable
manifolds. The process of generating closed exterior forms describes the
mechanism of realization of invariants and invariant structures.
