Introduction ix

1. Exterior skewsymmetric differential forms  1

1.1. Definition of exterior differential forms  1

1.2. Properties and specific features of the closed exterior differential forms  4

1.2.1. Invariant properties of closed exterior differential forms  7

1.2.2. Conjugacy and duality of the exterior differential forms  7

1.3. Specific features of the mathematical apparatus of exterior differential forms  11

1.3.1. Operators of the theory of exterior differential forms  11

1.3.2. Identical relations of exterior differential forms  12

1.3.3. Nondegenerate transformations  15

1.3.4. Differentialgeometrical structure. Invariant structures  16

1.4. Connection between exterior differential forms and various branches of mathematics  17

2. Evolutionary skewsymmetric differential forms  21

2.1. Some properties of manifolds  21

2.2. Specific features of the evolutionary differential forms  23

2.2.1. Specific features of the evolutionary forms differential  23

2.2.2. Non closure of the evolutionary differential forms  25

2.3. Specific features of the mathematical apparatus of evolutionary differential forms  26

2.3.1. Nonidentical relations of evolutionary differential forms  27

2.3.2. Selfvariation of the evolutionary nonidentical relation  31

2.3.3. Realizations of pseudostructures and closed exterior differential forms. Degenerate transforms  33

2.3.4. Obtaining an identical relation from a nonidentical  34

2.3.5. Integration of a nonidentical evolutionary relation  36

2.3.6. Duality and unity of a closed inexact exterior and a dual form  37

2.4. Functional possibilities of evolutionary forms  38

2.4.1. Mechanism of realization of conjugated objects and operators  38

2.4.2. Realization of differentialgeometrical structures  40

2.4.3. Forming pseudometric and metric manifolds  42

3. The mathematical apparatus of exterior and evolutionary skewsymmetric differential forms  45

3.1. Identical and nonidentical relations in the theory of skewsymmetric differential forms  46

3.2. Nondegenerate and degenerate transforms in the theory of skewsymmetric differential forms  47

3.2.1. Conjugated and nonconjugated operators  48

3.3. Connection between the identity and nonidentity of relations, between the nondegeneracy and degeneracy of transformations  49

4. Role of skewsymmetric differential forms in mathematics  53

4.1. Qualitative investigation of the solutions to differential equations  54

4.2. On integrability of the partial differential equations. Analysis of the fieldtheory equations  58

4.3. Qualitative investigation of Hamiltonian systems by application of skewsymmetric differential forms  59

5. Role of skewsymmetric differential forms in mathematical physics: Conservation laws  65

5.1. Duality and unity of conservation laws  65

5.1.1. Closed exterior forms: Exact conservation laws  66

5.1.2. Evolutionary differential forms: Balance conservation laws  67

5.2. Connection of exact conservation laws with balanced conservation law  71

6. Hidden invariant and evolutionary properties of the equations of mathematical physics  73

6.1. Studying the integrability of the equations of mathematical physics. Evolutionary relation  74

6.1.1. Analysis of consistency of the conservation law equations. Evolutionary relation for the state functionals  75

6.1.2. Properties of evolutionary relation for the state functionals  77

6.2. Hidden properties and possibilities of the equations of mathematical physics  78

6.2.1. Double solutions of the equations of mathematical physics  78

6.2.2. Physical meaning of double solutions to the equations of mathematical physics  81

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

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 state  91

7.2.2. Origination of the physical structures  93

7.3. Emergence of observed formations of material media. Dark energy and dark matter  94

7.3.1. The nature and origins of dark energy and dark matter  95

7.4. Evolutionary processes in material media. The external and internal forces  98

7.5. Propagation of observable formation (?uctuations, pulsations, waves, vortices and so on) in material medium  100

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 formation  105

8.1. Characteristics physical structures  105

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 manifolds  110

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 spaces  112

9. The equations of mathematical physics as a foundation of the fieldtheory equations  117

9.1. The role of evolutionary forms in field theory  118

9.1.1. Conservation laws as a foundation of the equations of mathematical physics and the fieldtheory equations  119

9.2. Exact conservation laws as a basis of the fieldtheory equations  120

9.3. Properties of the balance conservation law equations made up the equations of mathematical physics for material media  121

9.3.1. Mathematical and physical properties of evolutionary relation. Realization of physical structures  125

9.3.2. Properties of solutions to the mathematical physics equations  125

9.3.3. Description of evolutionary processes in material media. The processes of physical structure emergence  127

9.3.4. State functionals of the equations of mathematical physics  130

9.4. Correspondence between the evolutionary relation and fieldtheory equations. The linkage between fieldtheory equations and equations of mathematical physics  131

9.4.1. Corection between fieldtheory equations and the equations of mathematical physics  132

9.5. Some foundations of field theory. Characteristics of physical structures  133

9.5.1. Some characteristics of physical structures  133

9.6. Some foundations of field theory  134

9.6.1. Foundations of unified and general field theories  134

10. The role of exterior and evolutionary skewsymmetric forms in field theory: Conservation laws as foundations of the unified and general field theory  137

10.1. Closed inexact exterior forms: Exact conservation laws as the basis of the unified field theories  137

10.2. Evolutionary differential forms: Balance conservation laws for material media as the basis of the general field theory  139

10.2.1. Connection of the equations of fieldtheory for physical field with the equations of the mathematical physics for material media  141

10.2.2. Role of nonidentical evolutionary relation as the equation of general field theory  143

10.2.3. The essence of postulates  145

Appendix 1. Thermodynamic and gasdynamic entropy. Entropy as a functional and as a state function  147

Appendix 2. Physical meaning of the principles of thermodynamics  157

Appendix 3. Hidden properties of the Euler and NavierStokes equations. Double solutions. Origination the vorticity and turbulence  165

Appendix 4. Spontaneous origination of physical structures and emergence observable formations  179

Appendix 5. Electromagnetic field  191

Appendix 6. Correspondence between Interpretation of the Einstein equation  195

References  205

Index  208
