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.

| Introduction |

Chapter 1. | Role of skew-symmetric differential forms in mathematics |

| 1. | Exterior differential forms |

| | 1.1. | Closed exterior differential forms |

| | 1.2. | Properties of the closed exterior forms |

| | 1.3. | Invariance as the result of conjugacy of elements of exterior or dual forms |

| | 1.4. | Specific features of the mathematical apparatus of exterior differential forms |

| | 1.5. | Connection between exterior differential forms and various branches of mathematics |

| | 1.6. | Qualitative investigation of the functional properties of the solutions to differential equations |

| 2. | Evolutionary skew-symmetric differential forms |

| | 2.1. | Some properties of manifolds |

| | 2.2. | Properties of the evolutionary differential forms |

| | 2.3. | Specific features of the mathematical apparatus of evolutionary differential forms. Generate closed external forms |

| | 2.4. | Functional possibilities of evolutionary forms |

| | Characteristics of the differential-geometrical structures realized |

| | Classification of differential-geometrical structures realized |

| *Bibliography* |

Chapter 2. | Role of skew-symmetric differential forms in Mathematical Physics and Field Theory |

| 1. | Role of closed exterior differential forms in mathematical physics and field theory |

| | 1.1. | Some specific features of closed exterior differential forms |

| | 1.2. | Closed exterior forms as the basis of field theories |

| 2. | Role of evolutionary skew-symmetric differential forms in mathematical physics and field theory |

| | 2.1. | Mechanism of generation of closed exterior forms corresponding to the conservation laws for physical fields |

| | 2.2. | Mechanism of evolutionary processes in material systems |

| | 2.3. | Connection between the field-theory equations and the equations of conservation laws for material systems |

| 3. | Specific features and physical meaning of the solutions to equations of mathematical physics and field theory |

| *Conclusion* |

| *Bibliography* |

Chapter 3. | Conservation laws. Generation of physical structures. Principles of field theories |

| *Introduction* |

| 1. | Conservation laws |

| | 1.1. | Exact conservation laws |

| | 1.2. | Balance conservation laws |

| 2. | Connection between physical fields and material systems. Generation of physical fields |

| 3. | Basic principles of existing field theories |

| 4. | On foundations of field theory |

| *Appendix* |

| *Bibliography* |

Chapter 4. | The effect of noncommutativity of the conservation laws on the development of thermodynamical and gas dynamical instability |

| 1. | Analysis of principles of thermodynamics |

| 2. | The development of the gas dynamic instability |

| *Bibliography* |

Chapter 5. | The development of gas dynamical instability. Mechanism of origination of vorticity and turbulence |

| *Bibliography* |

Chapter 6. | Qualitative investigation of Hamiltonian systems by application of skew-symmetric differential forms |

| *Bibliography* |

Chapter 7. | Electromagnetic field |

| *Bibliography* |

Chapter 8. | Formation of physical fields and manifolds |

| 1. | Classification of physical structures and physical fields (Parameters of the closed exterior and dual forms) |

| 2. | Formation of pseudometric and metric spaces |

| *Bibliography* |

Chapter 9. | On integrability of equations of mathematical physics |

| 1. | Analysis of partial differential equations that describe real processes |

| 2. | Integrability of the equations of mechanics and physics of continuous medium |

| 3. | Analysis of the field-theory equations |

| *Bibliography* |

Chapter 10. | Foundations of field theory |

| 1. | Closed inexact exterior forms is the basis of field theories |

| 2. | Mathematical apparatus of evolutionary differential forms as the basis of general field theory |

| 3. | Connection between the equations of field-theory and the equations for material systems |

| | 3.1. | Specific features of the equations of mathematical physics |

| | 3.2. | Peculiarities of nonidentical evolutionary relation |

| | 3.3. | Role of nonidentical evolutionary relation as the equation of general field theory |

| | *Bibliography* |

| Conclusion |

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
(whose existence has been established by the author)проверьте
правильность внесения правки. 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.

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 for 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.

The present paper is
a collection of separate completed articles (the repeated description of the
properties of skew-symmetric forms is sometimes possible).

Two first articles (Chapters 1, 2) contain general principles related to the
properties of skew-symmertic forms and specific features of their mathematical
apparatus. The properties of closed exterior forms (invariance, conjugacy,
duality, and so on), which manifest themselves in various branches of
mathematics such as algebra, differential geometry, the theory of functions of
complex variables, tensor analysis, differential and integral calculus, are
demonstrated. This discloses an internal connection between various
mathematical formalisms. It has been shown that there exist skew-symmetric
forms that are obtained from differential equations describing any processes.
Such skew-symmetric forms, which are evolutionary ones and are based on
a nontraditional mathematical apparatus (such as nonidentical relations,
degenerate transformations, the transition from nonintegrable manifold to
integrable one, and others), possess an unique property, namely, they generate
closed exterior forms whose invariant properties lie at the basis of many
mathematical formalisms. The process of obtaining closed exterior forms, which
discloses a realization of conjugacy and an emergence of invariant structures,
has been shown.

In the second article (Chapter 2) it has been shown that the properties of
skew-symmetric forms correspond to conservation laws. Thus, closed exterior
forms account for the properties of coservation laws for physical fields,
whereas evolutionary skew-symmetric forms disclose the properties of
conservation laws for energy, momentum, angular momentum, and mass, which are
conservation laws for material systems such as thermodynamic, gas-dynamic,
cosmological and other ones. It has been shown that from conservation laws for
material systems one derives the evolutionary skew-symmetric differential forms
from which the closed exterir forms corresponding to conservation laws for
physical fields are obtained. This discloses a relation between physical fields
and material systems and demonstrates a relation between the field-theory
equations and the equations of conservation laws for material systems. Such
a dependence discloses the properties and peculiarities of the field-theory
equations.

In Chapters (3--10) of the collected articles the below listed articles, in
which various problems and specific features of mathematical physics are
studied with the help of skew-symmetric forms, are presented.

Conservation laws. Generation of physical structures. Principles of field
theories.

The effect of noncommutativity of the conservation laws on
the development of thermodynamic and gas-dynamic instability.

The development of gas dynamical instability. Mechanism
of origination of vorticity and turbulence.

Qualitative investigation of Hamiltonian systems by application
of skew-symmetric differential forms.

Electromagnetic field.

Formation of physical fields and manifolds.

Skew-symmetric forms: On integrability of the equations of mathematical
physics.

Foundations of general field theory.

Specific features of differential equations of
mathematical physics.

It is shown that the skew-symmetric differential forms play an unique
role in mathematics and mathematical physics.

The invariant properties of 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.

The unique role of evolutionary skew-symmetric differential forms, which
were outlined in present work, relates to the fact that they generate
the closed exterior forms possessing invariant properties.
This is of fundamental importance for mathematical physics and field
theories. The process of generation of closed exterior forms
disclose the mechanism of evolutionary processes in material media,
emergence of physical structures formatting physical fields and
the determinacy of these processes.

Due their properties and peculiarities the closed exterior forms and
evolutionary forms enable one to see the internal connection between
various branches of mathematics.

Many foundations of the mathematical apparatus of evolutionary
forms may occur to be of great importance for development of
mathematics and mathematical physics. The nonidentical relations,
degenerate transformations, transitions from nonidentical
relations to identical ones, transitions from one frame of
reference to another (nonequivalent) frame, the generation of
closed inexact exterior forms and invariant structures, formatting
fields and manifolds, the transitions between closed inexact
exterior differential forms and exact forms and other phenomena
may find many applications in such branches of mathematics as the
qualitative theory of differential and integral equations,
differential geometry and topology, the theory of functions, the
theory of series, the theory of numbers, and others.

The evolutionary skew-symmetric differential forms may become a new
branch in mathematics. They possess the possibilities that are
contained in none of mathematical formalisms.

**Ludmila Ivanovna PETROVA**
Candidate of Physical and Mathematical Sciences,
Senior researcher at The Moscow State University, Department of Computational
Mathematics and Cybernetics, the Chair of Mathematial Physics.
The field of scientific interests: Mechanics of Continuous Media,
Differential Equations, Differential Geometry, Field Theory, etc. The main
contributions to the problems of turbulence, stability and integrability of
differential equations.
In the works the properties of skew-symmetric
differential forms, which lie at the basis of many formalisms of
mathematics and mathetical physics, are used.