Skew-symmetric differential forms possess unique capabilities that manifest themselves in various branches of mathematics and mathematical physics. The invariant properties of closed exterior skew-symmetric differential forms lie at the basis of practically all invariant mathematical and physical formalisms. Closed exterior forms, whose properties correspond to conservation laws, explicitly or implicitly manifest themselves essentially in all formalisms of field theory. In the present work, firstly, the role of closed exterior skew-symmetric differential forms in mathematics, mathematical physics and field theory is illustrated, and, secondly, it is shown that there exist skew-symmetric differential forms that generate closed exterior differential forms. These skew-symmetric forms are derived from differential equations and possess evolutionary properties. The process of extracting closed exterior forms from evolutionary forms enables one to describe discrete transitions, quantum jumps, the generation of various structures, origination of such formations as waves, vortices and so on. In none of other mathematical formalisms such processes can be described since their description includes degenerate transformations and transitions from nonintegrable manifolds to integrable ones.
The unique role played by skew-symmetric differential forms in mathematics and mathematical physics, firstly, is due to the fact that they are differentials and differential expressions, and, therefore, they are suitable for describing invariants and invariant structures. And, secondly, skew-symmetric forms have a structure that combines objects of different nature, namely, the algebraic nature of the form coefficients and the geometric nature of the base. The interaction between these objects enables to describe evolutionary processes, discrete transitions, the realization of conjugacy of operators or objects, the emergence of structures and observable formations, and so on.
Other Books Mecánica Cuántica: