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Full Description
This book documents an investigation aimed at translating field dynamics into a language that allows it to be further transferred from the differential geometrician to the wider scientific community. As such, it features 40 end-of-chapter problems -- with solutions free to lecturers -- to improve interdisciplinary problem-solving in engineering and applied sciences.
Contents
Preface IX
1 4DSpace 1
1.1 Convention 1
1.2 Cartesian Coordinates 2
1.3 Time as a Fourth Dimension 5
1.3.1 Images 5
1.3.2 Complex Numbers 6
1.3.3 The Temporal Coordinate x4 8
1.4 The Hypercube 9
1.5 The 4D Right-hand Rule 11
1.5.1 The Right-hand Indices 13
1.6 Exercises 14
2 Relativity 17
2.1 Transformations 17
2.2 Pure Rotations 19
2.3 The Lorentz Transformation 19
2.4 Historical Note about the Confusion Surrounding the Meaning of Time 22
2.5 Exercises 23
3 Energy 27
3.1 Energy Density Field A 27
3.2 Directional Derivative of A 28
3.3 Gradient Vector S 29
3.4 Energy Density Field Vector A 30
3.5 Exercises 31
4 Change 33
4.1 Component Gradient Vector Gi 33
4.2 Curvature Vector k 34
4.3 The Electrical and Mechanical Parts of dA/ds 35
4.4 3D Space 36
4.5 Summary 38
4.6 Exercises 38
5 Governing Equations 41
5.1 2D Field Lines 42
5.1.1 Comment on Bifurcations 43
5.1.2 The Differential Form 44
5.2 3D Field Lines 45
5.3 4D Field Lines 46
5.4 The Field Lines of Gi 47
5.5 Uniqueness 49
5.6 Exercises 51
6 Waves 53
6.1 Wave Superposition 53
6.2 Rectangular Waves 54
6.3 Spherical Waves 55
6.4 Initial Value Problem 56
6.4.1 Stiffness 57
6.4.2 Damping 59
7 Particles 61
7.1 The Stationary Field Particle 61
7.2 The Moving Field Particle 63
7.3 The Flux of a Field Particle 63
7.4 A Particle Exciting Another Field 65
7.5 Particle Motion at Nonrelativistic Speeds 67
7.6 Exercises 68
8 Electrodynamics 71
8.1 Maxwell's Equations in Differential Form 71
8.2 Maxwell's Equations in Integral Form 74
8.3 Polarization 78
8.3.1 3D Polarization 80
8.4 The Lorentz Force 80
8.5 Exercises 81
9 Mechanics 83
9.1 Noncoincident Observation 83
9.2 Particle Interaction 85
9.3 Newton's Second Law 87
9.4 Note About the Effect of Charge 88
9.5 Inverse Square Laws 89
9.5.1 Purely Electrical Behavior 89
9.5.2 Purely Mechanical Behavior 90
9.6 Relativistic Mass 91
10 Essay: An Evolving Science 93
10.1 The Past 93
10.1.1 Early Period 93
10.1.2 Force 94
10.1.3 Energy 96
10.2 The Present 96
10.2.1 The Starting Point 97
10.2.2 Four-dimensional Space 97
10.2.3 Energy in 4D Space is a Continuum 98
10.2.4 The Particle is Itself an Idealized Field 99
10.2.5 Mechanics and Electrodynamics are Parts of How Energy Changes 99
10.3 The Future 100
10.3.1 Scientific Impact 100
10.3.2 Human Impact 102
10.4 Closing Message about Scales of Observation 104
Appendix A: Integral Theorems 107
A.1 Longitudinal Integral Theorems 107
A.1.1 Circulation 107
A.1.2 Curl 109
A.1.3 Coil 111
A.2 Transverse Integral Theorems 112
A.2.1 Flux and Divergence 112
A.2.2 Strength and Dilatation 113
Appendix B: Curvilinear Coordinates 117
B.1 Principles 117
B.2 Integral Theorems 119
B.2.1 Circulation and Curl 119
B.2.2 Coil 119
B.2.3 Flux and Divergence 120
B.2.4 Strength and Dilatation 121
B.3 Continuity Equations 122
B.4 Circular Coordinates 123
B.4.1 Polar Coordinates 123
B.4.2 Cylindrical Coordinates 123
B.4.3 Spherical Coordinates 124
B.4.4 Hyperpolar Coordinates 125
B.4.5 Hypercylindrical Coordinates 125
B.4.6 Hyperspherical Coordinates 126
B.5 Exercises 127
Solutions to Exercises 135
References 177
Index 179
