Explorations in Mathematical Physics: The Concepts Behind an Elegant LanguageHave you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature? This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and threedimensional rotations. You'll see how the accelerated frames of special relativity tell us about gravity. On the journey, you'll discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology. The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for oneforms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods. Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

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Contents
The Language of Physics  1 
A Trip Down Linear Lane  6 
21 Vector Spaces and Matrices  8 
22 Inner Products  10 
23 Crystallography and the Cobasis  12 
The Use of Determinants  16 
241 Deﬁnition and Properties of the Determinant  18 
242 Determinants Handedness and the nDimensional Cross Product  21 
Details of Setting Up Adams and Eves Coordinates  266 
The Elegance and Power of Tensor Notation  270 
811 Honing the Vector Idea  274 
812 Two Types of Vectors  276 
82 Vectors and Coordinate Changes  278 
83 Generalising the Idea of Vector Length  281 
831 Coordinate Transformation of the Metric  283 
84 A Natural Basis for Covectors  285 
243 Volume of a Parallelepiped in a HigherDimensional Space  24 
244 The Cobasis and the Wedge Product  25 
Changing Spaces  28 
251 Diagonalising a Matrix  31 
26 Diracs Bracket Notation  41 
27 Brackets and Hermitian Operators  43 
28 Frequency and Wavenumber  47 
29 Deriving the Fourier Transform Using Brackets  52 
210 Commutators and the Indeterminacy Principle  58 
211 Evolving Wave Functions in Time  65 
2111 Brackets and Wave Function Evolution  70 
212 The Transition to Quantum Mechanics  71 
The Natural Language of Random Processes  77 
32 The Privileged Sum of Squares  82 
321 Sums of Squares and the Random Walk  89 
33 Least Squares Analysis Bayes Theorem and the Matrix Pseudo Inverse  93 
331 Least Squares Analysis for Curve Fitting  95 
34 Time Constants to Describe Growth and Decay  105 
341 The Poisson Statistics of Radioactive Decay  106 
342 The Mean Life of the Decaying Nuclei  110 
343 The Notion of a Probability per Second  112 
35 Logarithms and Exponentials in Statistical Mechanics  114 
Chief Star of Statistical Mechanics  117 
353 Logarithms and Decibels  119 
36 Signal Processing and the zTransform  122 
361 Deriving the Fibonacci Sequence from the zTransform  123 
362 Convolving to Smoothen a Signal  124 
37 The Discrete Fourier Transform  128 
371 Sampling Using Nyquists Theorem  129 
372 Discretising the Fourier Transform  131 
373 Interpolating Real Data with the DFT  135 
Presenting Solutions to Problems  140 
381 Tailoring a Formula to a Given Set of Units  141 
382 Calculating a Nuclear Scattering Rate  142 
A Roundabout Route to Geometric Algebra  147 
41 Matrix Representation of an Orientation  148 
411 Describing an Orientation by a Rotation  150 
42 Calculating the Matrix for an Arbitrary Rotation  151 
421 Deriving the Rotation Matrix Rn9 via Diagonalisation  153 
422 Are Rotations Vectors?  155 
43 Combining Two Rotations  157 
44 Rotations Lead to Complex Numbers and Quaternions  158 
441 Tidying Up the Placeholders  163 
45 Producing a Geometric Algebra  166 
46 Rotations in Popular Usage  170 
461 Describing an Orientation by Using Three Rotations  171 
462 Confusing Euler Angle Orientation with Incremental Rotation  176 
463 Quaternions Used in Computer Graphics  182 
Special Relativity and the Lorentz Transform  185 
52 The Postulates of Special Relativity  186 
53 The Lorentz Transform  187 
531 Paradoxes or Conundrums?  189 
532 How Does Each Frame Measure the Other as Ageing Slowly?  192 
54 The Symmetry of the Lorentz Transform  195 
55 Using Radar to Derive Time Dilation  197 
56 Space Time Becomes Spacetime  200 
57 Spacetime Diagrams and Hyperbolic Geometry  202 
58 The Lorentz Transform in an Arbitrary Direction  204 
59 Energy and Momentum in Special Relativity  206 
591 Einsteins Relation of Mass and Energy  210 
FourVectors and the Road to Tensors  213 
62 Running Nonrelativistically  216 
63 Running Relativistically  218 
632 The Length of the FourVelocity  223 
65 Introducing Covectors and Fully Covariant Notation  231 
Accelerated Frames Onward to the Principle of Covariance  233 
71 The Clock Postulate  235 
711 The Interval for Noninertial Observers  238 
72 Coordinates for the Accelerated Frame  240 
73 The Twin Conundrum  252 
731 Making Eve Accelerate Uniformly  256 
732 How the Twins Record Each Others Trips  258 
74 A Glance Ahead to Gauge Theory  263 
75 Covariant Notation and Generalising the Clock Postulate  264 
841 Raising and Lowering Indices  290 
85 Tensor Components with More than Two Indices  292 
851 Bases for More General Tensors  294 
852 The Metric Tensor Versus the Metric Matrix  296 
86 The Gradient Operator and the Cobasis  297 
861 The Gradient Operator in Fully Covariant Notation  301 
862 Is a Metric Needed?  306 
87 Normalised Basis Vectors  308 
871 The Normalised Polar Basis in Celestial Mechanics  310 
872 An Example of Using Vectors to Calculate an Effective Potential  312 
873 Some Final Remarks on Vector Terminology  315 
The Cross Product in General Coordinates  320 
89 From Vector Calculus to Tensor Calculus  322 
891 The Divergence in Tensor Notation  323 
892 Christoffel Symbols for Cartesian Coordinates  326 
893 Preparing to Make the Divergence Covariant  329 
894 The Covariant Laplacian  332 
895 The Covariant Curl  333 
810 Exterior Calculus and the Theorems of Stokes and Gauss in Higher Dimensions  335 
Curvature and Differential Geometry  349 
911 Curves on Surfaces  354 
Curves with No Geodesic Curvature  359 
93 The Curvature of a Surface  362 
931 The Method of Lagrange Multipliers  365 
94 Gausss Extraordinary Theorem  369 
95 Translating Vectors by Parallel Transport  373 
96 Relating Parallel Transport to Curvature  378 
The GaussBonnet Theorem in Euclidean 3Space  382 
Variational Calculus and Field Theory  387 
102 The Concept of a Field  389 
103 The Lagrangian Formalism  391 
1031 Lagranges Equation  393 
1032 Other Variational Approaches  395 
Hamiltons Principle  396 
1034 Nothers Theorem and Lagrangian Invariances  398 
First Steps to a Field Theory  399 
1036 Nothers Theorem for a Scalar Field  402 
104 Building a Lagrangian  404 
1041 A Relativistic Lagrangian for a Charge in an EM Field  407 
105 Producing the Schrodinger Equation  415 
Fields Describe Particles Too  417 
The Klein Gordon Equation  421 
1062 A Route to the Dirac Equation  422 
107 Gauge Theory and Quantum Electrodynamics  427 
1072 A Gauge Transformation for the Dirac Lagrangian  429 
108 The PathIntegral Approach to Quantum Mechanics  432 
1081 Path Integrals Give the Schrodinger Equation  435 
The Language of Decoherence  438 
The Green Function Approach to Solving Field Equations  445 
112 Deriving the Green Function for V2 via Fourier Theory  449 
1121 The Other Way of Calculating the Integral 1120  456 
113 Solving Maxwells Equations via the Green Function Approach  460 
114 Variations on the Green Function Solution of Maxwells Equations  468 
115 Fluctuation Dissipation and Times Arrow  470 
Airliners Black Holes and Cosmology The ABC of General Relativity  471 
122 The Pound Rebka Snider Experiments  474 
123 A Space or Spacetime Description of Gravity?  476 
1231 A Route to Curved Spacetime from Lagrangian Mechanics  478 
1232 Free Particles Geodesics and Locally Inertial Frames  483 
1233 Quantities That Are Conserved on Geodesics  489 
124 A Path to Einsteins Equation  490 
The Schwarzschild Metric  495 
1251 Deriving Gravitational Redshift Again  498 
126 The Schwarzschild Black Hole  500 
1261 Tensor Components and Physical Measurements  506 
Cartans Structural Equations  508 
128 The Variational Approach to Einsteins Equation  511 
1281 Adding Extra Field Terms to the Lagrangian Density  515 
The Cosmological Constant  518 
1283 Joining Electromagnetism to Gravity  519 
1284 Path Integrals in General Relativity  523 
Proper Distances in Cosmology  524 
531  
Other editions  View all
Explorations in Mathematical Physics: The Concepts Behind an Elegant Language Don Koks Limited preview  2006 
Explorations in Mathematical Physics: The Concepts Behind an Elegant Language Don Koks No preview available  2010 