| Introduction ix
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| 1. Exterior skew-symmetric differential forms | 1
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| 1.1. Definition of exterior differential forms | 1
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| 1.2. Properties and specific features of the closed exterior differential forms | 4
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| 1.2.1. Invariant properties of closed exterior differential forms | 7
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| 1.2.2. Conjugacy and duality of the exterior differential forms | 7
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| 1.3. Specific features of the mathematical apparatus of exterior differential forms | 11
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| 1.3.1. Operators of the theory of exterior differential forms | 11
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| 1.3.2. Identical relations of exterior differential forms | 12
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| 1.3.3. Nondegenerate transformations | 15
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| 1.3.4. Differential-geometrical structure. Invariant structures | 16
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| 1.4. Connection between exterior differential forms and various branches of mathematics | 17
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| 2. Evolutionary skew-symmetric differential forms | 21
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| 2.1. Some properties of manifolds | 21
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| 2.2. Specific features of the evolutionary differential forms | 23
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| 2.2.1. Specific features of the evolutionary forms differential | 23
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| 2.2.2. Non closure of the evolutionary differential forms | 25
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| 2.3. Specific features of the mathematical apparatus of evolutionary differential forms | 26
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| 2.3.1. Nonidentical relations of evolutionary differential forms | 27
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| 2.3.2. Selfvariation of the evolutionary nonidentical relation | 31
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| 2.3.3. Realizations of pseudostructures and closed exterior differential forms. Degenerate transforms | 33
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| 2.3.4. Obtaining an identical relation from a nonidentical | 34
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| 2.3.5. Integration of a nonidentical evolutionary relation | 36
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| 2.3.6. Duality and unity of a closed inexact exterior and a dual form | 37
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| 2.4. Functional possibilities of evolutionary forms | 38
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| 2.4.1. Mechanism of realization of conjugated objects and operators | 38
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| 2.4.2. Realization of differential-geometrical structures | 40
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| 2.4.3. Forming pseudometric and metric manifolds | 42
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| 3. The mathematical apparatus of exterior and evolutionary skew-symmetric differential forms | 45
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| 3.1. Identical and nonidentical relations in the theory of skew-symmetric differential forms | 46
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| 3.2. Nondegenerate and degenerate transforms in the theory of skew-symmetric differential forms | 47
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| 3.2.1. Conjugated and nonconjugated operators | 48
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| 3.3. Connection between the identity and nonidentity of relations, between the nondegeneracy and degeneracy of transformations | 49
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| 4. Role of skew-symmetric differential forms in mathematics | 53
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| 4.1. Qualitative investigation of the solutions to differential equations | 54
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| 4.2. On integrability of the partial differential equations. Analysis of the field-theory equations | 58
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| 4.3. Qualitative investigation of Hamiltonian systems by application of skew-symmetric differential forms | 59
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| 5. Role of skew-symmetric differential forms in mathematical physics: Conservation laws | 65
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| 5.1. Duality and unity of conservation laws | 65
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| 5.1.1. Closed exterior forms: Exact conservation laws | 66
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| 5.1.2. Evolutionary differential forms: Balance conservation laws | 67
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| 5.2. Connection of exact conservation laws with balanced conservation law | 71
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| 6. Hidden invariant and evolutionary properties of the equations of mathematical physics | 73
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| 6.1. Studying the integrability of the equations of mathematical physics. Evolutionary relation | 74
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| 6.1.1. Analysis of consistency of the conservation law equations. Evolutionary relation for the state functionals | 75
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| 6.1.2. Properties of evolutionary relation for the state functionals | 77
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| 6.2. Hidden properties and possibilities of the equations of mathematical physics | 78
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| 6.2.1. Double solutions of the equations of mathematical physics | 78
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| 6.2.2. Physical meaning of double solutions to the equations of mathematical physics | 81
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| 7. Mechanism of evolutionary processes in material media. Origination of the physical structures. Emergence of observed formations of material media. Dark energy and dark matter | 85
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| 7.1. Nonequilibrium of the material media. (Nonidentical of the evolutionary relation) | 86
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| 7.1.1. Selfvariation of nonequilibrium state of material medium. (Selfvariation of the evolutionary relation) | 88
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| 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
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| 7.2.1. Transition of the material media into a locally equilibrium state | 91
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| 7.2.2. Origination of the physical structures | 93
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| 7.3. Emergence of observed formations of material media. Dark energy and dark matter | 94
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| 7.3.1. The nature and origins of dark energy and dark matter | 95
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| 7.4. Evolutionary processes in material media. The external and internal forces | 98
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| 7.5. Propagation of observable formation (?uctuations, pulsations, waves, vortices and so on) in material medium | 100
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| 7.6. Potential forces. (Duality of closed exterior forms as conserved quantities and as potential forces) | 101
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| 8. Evolutionary forms: Characteristics of physical structures and observed formation | 105
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| 8.1. Characteristics physical structures | 105
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| 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
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| 8.3. Evolutionary forms: Formation of physical fields and manifolds | 110
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| 8.4. Classification of physical structures and physical fields (Parameters of the closed exterior and dual forms) | 111
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| 8.5. Formation of pseudometric and metric spaces | 112
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| 9. The equations of mathematical physics as a foundation of the field-theory equations | 117
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| 9.1. The role of evolutionary forms in field theory | 118
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| 9.1.1. Conservation laws as a foundation of the equations of mathematical physics and the field-theory equations | 119
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| 9.2. Exact conservation laws as a basis of the field-theory equations | 120
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| 9.3. Properties of the balance conservation law equations made up the equations of mathematical physics for material media | 121
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| 9.3.1. Mathematical and physical properties of evolutionary relation. Realization of physical structures | 125
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| 9.3.2. Properties of solutions to the mathematical physics equations | 125
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| 9.3.3. Description of evolutionary processes in material media. The processes of physical structure emergence | 127
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| 9.3.4. State functionals of the equations of mathematical physics | 130
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| 9.4. Correspondence between the evolutionary relation and field-theory equations. The linkage between field-theory equations and equations of mathematical physics | 131
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| 9.4.1. Corection between field-theory equations and the equations of mathematical physics | 132
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| 9.5. Some foundations of field theory. Characteristics of physical structures | 133
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| 9.5.1. Some characteristics of physical structures | 133
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| 9.6. Some foundations of field theory | 134
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| 9.6.1. Foundations of unified and general field theories | 134
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| 10. The role of exterior and evolutionary skew-symmetric forms in field theory: Conservation laws as foundations of the unified and general field theory | 137
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| 10.1. Closed inexact exterior forms: Exact conservation laws as the basis of the unified field theories | 137
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| 10.2. Evolutionary differential forms: Balance conservation laws for material media as the basis of the general field theory | 139
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| 10.2.1. Connection of the equations of field-theory for physical field with the equations of the mathematical physics for material media | 141
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| 10.2.2. Role of nonidentical evolutionary relation as the equation of general field theory | 143
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| 10.2.3. The essence of postulates | 145
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| Appendix 1. Thermodynamic and gas-dynamic entropy. Entropy as a functional and as a state function | 147
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| Appendix 2. Physical meaning of the principles of thermodynamics | 157
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| Appendix 3. Hidden properties of the Euler and Navier-Stokes equations. Double solutions. Origination the vorticity and turbulence | 165
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| Appendix 4. Spontaneous origination of physical structures and emergence observable formations | 179
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| Appendix 5. Electromagnetic field | 191
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| Appendix 6. Correspondence between Interpretation of the Einstein equation | 195
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| References | 205
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| Index | 208
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Page 23, line 28: An evolutionary form differential of degree. Read as: An evolutionary differential form of degree
Lyudmila Ivanovna PETROVA