Skew-symmetric differential forms in mathematics, mathematical physics and field theoryкнига

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[1] Petrova L. I. Skew-symmetric differential forms in mathematics, mathematical physics and field theory. — Skew-symmetric differential forms in mathematics, mathematical physics and field theory Moscow, URSS, 2013. — 234 p. 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. The closed exterior forms, which properties correspond to conservation laws, explicitly or implicitly manifest themselves essentially in all formalisms of field theory. In present paper, 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 proceses can be described since their description includes degenerate transformations and transitions from nonintegrable manifolds to integrable ones.

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