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

Skew-symmetric differential forms. Invariants. Realization of invariant structures