Encuadernación Petrova L. I. Skew-symmetric differential forms. Conservation laws: The foundation of equations of mathematical physics and field theory
Id: 262747
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Skew-symmetric differential forms.
Conservation laws: The foundation of equations of mathematical physics and field theory

URSS. 222 pp. (English). ISBN 978-5-396-01023-9.
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Resumen del libro

Skew-symmetric differential forms possess unique capabilities that manifest themselves in various branches of mathematics, mathematical physics and field theory. The invariant properties of closed exterior skew-symmetric differential forms lie at the basis of practically all invariant mathematical and physical formalisms.

In present paper, firstly, the role of closed exterior skew-symmetric differential forms in mathematics, mathematical physics ...(Información más detallada)and field theory is illustrated, and, secondly, it is shown that there exist skew-symmetric differential forms that are derived from differential equations and possess evolutionary properties. Such evolutionary forms (which contain an unconventional mathematical apparatus that includes such basic concepts as degenerate transformations and nonidentical relations) can describe evolutionary processes and generate closed exterior forms corresponding to invariant structures. The process of implementation 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. 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. The unique role of skew-symmetric differential forms in mathematics and mathematical physics is due to the fact that they reflect the properties of conservation laws, which are the basis of equations of mathematical physics and field theory


Содержание
Introduction ix
1. Exterior skew-symmetric differential forms1
1.1. Definition of exterior differential forms1
1.2. Properties and specific features of the closed exterior differential forms4
1.2.1. Invariant properties of closed exterior differential forms7
1.2.2. Conjugacy and duality of the exterior differential forms7
1.3. Specific features of the mathematical apparatus of exterior differential forms11
1.3.1. Operators of the theory of exterior differential forms11
1.3.2. Identical relations of exterior differential forms12
1.3.3. Nondegenerate transformations15
1.3.4. Differential-geometrical structure. Invariant structures16
1.4. Connection between exterior differential forms and various branches of mathematics17
2. Evolutionary skew-symmetric differential forms21
2.1. Some properties of manifolds21
2.2. Specific features of the evolutionary differential forms23
2.2.1. Specific features of the evolutionary forms differential23
2.2.2. Non closure of the evolutionary differential forms25
2.3. Specific features of the mathematical apparatus of evolutionary differential forms26
2.3.1. Nonidentical relations of evolutionary differential forms27
2.3.2. Selfvariation of the evolutionary nonidentical relation31
2.3.3. Realizations of pseudostructures and closed exterior differential forms. Degenerate transforms33
2.3.4. Obtaining an identical relation from a nonidentical34
2.3.5. Integration of a nonidentical evolutionary relation36
2.3.6. Duality and unity of a closed inexact exterior and a dual form37
2.4. Functional possibilities of evolutionary forms38
2.4.1. Mechanism of realization of conjugated objects and operators38
2.4.2. Realization of differential-geometrical structures40
2.4.3. Forming pseudometric and metric manifolds42
3. The mathematical apparatus of exterior and evolutionary skew-symmetric differential forms45
3.1. Identical and nonidentical relations in the theory of skew-symmetric differential forms46
3.2. Nondegenerate and degenerate transforms in the theory of skew-symmetric differential forms47
3.2.1. Conjugated and nonconjugated operators48
3.3. Connection between the identity and nonidentity of relations, between the nondegeneracy and degeneracy of transformations49
4. Role of skew-symmetric differential forms in mathematics53
4.1. Qualitative investigation of the solutions to differential equations54
4.2. On integrability of the partial differential equations. Analysis of the field-theory equations58
4.3. Qualitative investigation of Hamiltonian systems by application of skew-symmetric differential forms59
5. Role of skew-symmetric differential forms in mathematical physics: Conservation laws65
5.1. Duality and unity of conservation laws65
5.1.1. Closed exterior forms: Exact conservation laws66
5.1.2. Evolutionary differential forms: Balance conservation laws67
5.2. Connection of exact conservation laws with balanced conservation law71
6. Hidden invariant and evolutionary properties of the equations of mathematical physics73
6.1. Studying the integrability of the equations of mathematical physics. Evolutionary relation74
6.1.1. Analysis of consistency of the conservation law equations. Evolutionary relation for the state functionals75
6.1.2. Properties of evolutionary relation for the state functionals77
6.2. Hidden properties and possibilities of the equations of mathematical physics78
6.2.1. Double solutions of the equations of mathematical physics78
6.2.2. Physical meaning of double solutions to the equations of mathematical physics81
7. Mechanism of evolutionary processes in material media. Origination of the physical structures. Emergence of observed formations of material media. Dark energy and dark matter85
7.1. Nonequilibrium of the material media. (Nonidentical of the evolutionary relation)86
7.1.1. Selfvariation of nonequilibrium state of material medium. (Selfvariation of the evolutionary relation)88
7.2. Transition of the material medium into a locally equilibrium state. Origination of the physical structures. (Degenerate transform. Emergence of closed exterior forms. Realization of identical relation)90
7.2.1. Transition of the material media into a locally equilibrium state91
7.2.2. Origination of the physical structures93
7.3. Emergence of observed formations of material media. Dark energy and dark matter94
7.3.1. The nature and origins of dark energy and dark matter95
7.4. Evolutionary processes in material media. The external and internal forces98
7.5. Propagation of observable formation (?uctuations, pulsations, waves, vortices and so on) in material medium100
7.6. Potential forces. (Duality of closed exterior forms as conserved quantities and as potential forces)101
8. Evolutionary forms: Characteristics of physical structures and observed formation105
8.1. Characteristics physical structures105
8.2. Characteristics of a observed formation: intensity, vorticity, absolute and relative speeds of propagation of the formation. (Value of the evolutionary form commutator, the properties of the material medium)107
8.3. Evolutionary forms: Formation of physical fields and manifolds110
8.4. Classification of physical structures and physical fields (Parameters of the closed exterior and dual forms)111
8.5. Formation of pseudometric and metric spaces112
9. The equations of mathematical physics as a foundation of the field-theory equations117
9.1. The role of evolutionary forms in field theory118
9.1.1. Conservation laws as a foundation of the equations of mathematical physics and the field-theory equations119
9.2. Exact conservation laws as a basis of the field-theory equations120
9.3. Properties of the balance conservation law equations made up the equations of mathematical physics for material media121
9.3.1. Mathematical and physical properties of evolutionary relation. Realization of physical structures125
9.3.2. Properties of solutions to the mathematical physics equations125
9.3.3. Description of evolutionary processes in material media. The processes of physical structure emergence127
9.3.4. State functionals of the equations of mathematical physics130
9.4. Correspondence between the evolutionary relation and field-theory equations. The linkage between field-theory equations and equations of mathematical physics131
9.4.1. Corection between field-theory equations and the equations of mathematical physics132
9.5. Some foundations of field theory. Characteristics of physical structures133
9.5.1. Some characteristics of physical structures133
9.6. Some foundations of field theory134
9.6.1. Foundations of unified and general field theories134
10. The role of exterior and evolutionary skew-symmetric forms in field theory: Conservation laws as foundations of the unified and general field theory137
10.1. Closed inexact exterior forms: Exact conservation laws as the basis of the unified field theories137
10.2. Evolutionary differential forms: Balance conservation laws for material media as the basis of the general field theory139
10.2.1. Connection of the equations of field-theory for physical field with the equations of the mathematical physics for material media141
10.2.2. Role of nonidentical evolutionary relation as the equation of general field theory143
10.2.3. The essence of postulates145
Appendix 1. Thermodynamic and gas-dynamic entropy. Entropy as a functional and as a state function147
Appendix 2. Physical meaning of the principles of thermodynamics157
Appendix 3. Hidden properties of the Euler and Navier-Stokes equations. Double solutions. Origination the vorticity and turbulence165
Appendix 4. Spontaneous origination of physical structures and emergence observable formations179
Appendix 5. Electromagnetic field191
Appendix 6. Correspondence between Interpretation of the Einstein equation195
References205
Index208

Corrections
Page 23, line 28: An evolutionary form differential of degree. Read as: An evolutionary differential form of degree
El autor
Lyudmila Ivanovna PETROVA
PhD in Mathematical Physics, Senior Fellow of the Department of Computational Mathematics and Cybernetics at MSU, Honoured Fellow of Moscow State University. Fields of scientific interest: Mechanics of Continuous Media, Differential Equations, Differential Geometry, Field Theory, etc. Has made an important contribution to the turbulence problem, and the problem of stability and integrability of differential equations. In her research, she widely applies the formalism of skew-symmetric differential forms. She is the author of more than 100 publications.