| Preface | p. VII |
| The Language of Physics | p. 1 |
| A Trip Down Linear Lane | p. 7 |
| Vector Spaces and Matrices | p. 8 |
| Inner Products | p. 10 |
| Crystallography and the Cobasis | p. 12 |
| Finding Areas and Volumes: The Use of Determinants | p. 16 |
| Definition and Properties of the Determinant | p. 18 |
| Determinants, Handedness, and the n-Dimensional Cross Product | p. 21 |
| Volume of a Parallelepiped in a Higher-Dimensional Space | p. 24 |
| The Cobasis and the Wedge Product | p. 25 |
| Diagonalisation and Similar Matrices: Changing Spaces | p. 28 |
| Diagonalising a Matrix | p. 31 |
| Dirac's Bracket Notation | p. 41 |
| Brackets and Hermitian Operators | p. 43 |
| Frequency and Wavenumber | p. 47 |
| Deriving the Fourier Transform Using Brackets | p. 52 |
| Commutators and the Indeterminacy Principle | p. 58 |
| Evolving Wave Functions in Time | p. 65 |
| Brackets and Wave Function Evolution | p. 70 |
| The Transition to Quantum Mechanics | p. 71 |
| The Natural Language of Random Processes | p. 77 |
| From Bar Graphs to Histograms | p. 77 |
| The Privileged Sum of Squares | p. 82 |
| Sums of Squares and the Random Walk | p. 89 |
| Least Squares Analysis, Bayes' Theorem, and the Matrix Pseudo Inverse | p. 93 |
| Least Squares Analysis for Curve Fitting | p. 95 |
| Time Constants to Describe Growth and Decay | p. 105 |
| The Poisson Statistics of Radioactive Decay | p. 106 |
| The Mean Life of the Decaying Nuclei | p. 110 |
| The Notion of a "Probability per Second" | p. 112 |
| Logarithms and Exponentials in Statistical Mechanics | p. 114 |
| Entropy and Heat Flow Define Temperature | p. 114 |
| The Boltzmann Factor: Chief Star of Statistical Mechanics | p. 117 |
| Logarithms and Decibels | p. 119 |
| Signal Processing and the z-Transform | p. 122 |
| Deriving the Fibonacci Sequence from the z-Transform | p. 123 |
| Convolving to Smoothen a Signal | p. 124 |
| The Discrete Fourier Transform | p. 128 |
| Sampling Using Nyquist's Theorem | p. 129 |
| Discretising the Fourier Transform | p. 131 |
| Interpolating Real Data with the DFT | p. 135 |
| Correct and Convincing: Presenting Solutions to Problems | p. 140 |
| Tailoring a Formula to a Given Set of Units | p. 141 |
| Calculating a Nuclear Scattering Rate | p. 142 |
| A Roundabout Route to Geometric Algebra | p. 147 |
| Matrix Representation of an Orientation | p. 148 |
| Describing an Orientation by a Rotation | p. 150 |
| Calculating the Matrix for an Arbitrary Rotation | p. 151 |
| Deriving the Rotation Matrix R[subscript n] ([Theta]) via Diagonalisation | p. 153 |
| Are Rotations Vectors? | p. 155 |
| Combining Two Rotations | p. 157 |
| Rotations Lead to Complex Numbers and Quaternions | p. 158 |
| Tidying Up the Placeholders | p. 163 |
| Producing a "Geometric" Algebra | p. 166 |
| Rotations in Popular Usage | p. 170 |
| Describing an Orientation by Using Three Rotations | p. 171 |
| Confusing Euler Angle Orientation with Incremental Rotation | p. 176 |
| Quaternions Used in Computer Graphics | p. 182 |
| Special Relativity and the Lorentz Transform | p. 185 |
| Deriving the Doppler Shift from an Invariance | p. 185 |
| The Postulates of Special Relativity | p. 186 |
| The Lorentz Transform | p. 187 |
| Paradoxes or Conundrums? | p. 189 |
| How Does Each Frame Measure the Other as Ageing Slowly? | p. 192 |
| The Symmetry of the Lorentz Transform | p. 195 |
| Using Radar to Derive Time Dilation | p. 197 |
| Space-Time Becomes Spacetime | p. 200 |
| Spacetime Diagrams and Hyperbolic Geometry | p. 202 |
| The Lorentz Transform in an Arbitrary Direction | p. 204 |
| Energy and Momentum in Special Relativity | p. 206 |
| Einstein's Relation of Mass and Energy | p. 210 |
| Four-Vectors and the Road to Tensors | p. 213 |
| Number Density and Flux Density | p. 213 |
| Running Nonrelativistically | p. 216 |
| Running Relativistically | p. 218 |
| Combining Number and Flux Densities into Something New | p. 218 |
| The "Length" of the Four-Velocity | p. 223 |
| Examples of Other Four-Vectors | p. 223 |
| Introducing Covectors and Fully Covariant Notation | p. 231 |
| Accelerated Frames: Onward to the Principle of Covariance | p. 233 |
| The Clock Postulate | p. 235 |
| The Interval for Noninertial Observers | p. 238 |
| Coordinates for the Accelerated Frame | p. 240 |
| The Twin Conundrum | p. 252 |
| Making Eve Accelerate Uniformly | p. 256 |
| How the Twins Record Each Other's Trips | p. 258 |
| A Glance Ahead to Gauge Theory | p. 263 |
| Covariant Notation and Generalising the Clock Postulate | p. 264 |
| Appendix: Details of Setting Up Adam's and Eve's Coordinates | p. 266 |
| The Elegance and Power of Tensor Notation | p. 271 |
| Back to Vectors, in a More Generic Way | p. 271 |
| Honing the Vector Idea | p. 274 |
| Two Types of Vectors | p. 276 |
| Vectors and Coordinate Changes | p. 278 |
| Generalising the Idea of Vector Length | p. 281 |
| Coordinate Transformation of the Metric | p. 283 |
| A Natural Basis for Covectors | p. 285 |
| Raising and Lowering Indices | p. 290 |
| Tensor Components with More than Two Indices | p. 292 |
| Bases for More General Tensors | p. 294 |
| The Metric Tensor Versus the Metric Matrix | p. 296 |
| The Gradient Operator and the Cobasis | p. 297 |
| The Gradient Operator in Fully Covariant Notation | p. 301 |
| Is a Metric Needed? | p. 306 |
| Normalised Basis Vectors | p. 308 |
| The Normalised Polar Basis in Celestial Mechanics | p. 310 |
| An Example of Using Vectors to Calculate an Effective Potential | p. 312 |
| Some Final Remarks on Vector Terminology | p. 315 |
| Volume Elements, Determinants, and Cross Products Again | p. 315 |
| A Final Word: The Cross Product in General Coordinates | p. 320 |
| From Vector Calculus to Tensor Calculus | p. 322 |
| The Divergence in Tensor Notation | p. 323 |
| Christoffel Symbols for Cartesian Coordinates | p. 326 |
| Preparing to Make the Divergence Covariant | p. 329 |
| The Covariant Laplacian | p. 332 |
| The Covariant Curl | p. 333 |
| Exterior Calculus and the Theorems of Stokes and Gauss in Higher Dimensions | p. 335 |
| Curvature and Differential Geometry | p. 349 |
| Curvature in the Plane | p. 349 |
| Curves on Surfaces | p. 354 |
| Geodesics: Curves with No Geodesic Curvature | p. 359 |
| The Curvature of a Surface | p. 362 |
| The Method of Lagrange Multipliers | p. 365 |
| Gauss's Extraordinary Theorem | p. 369 |
| Translating Vectors by Parallel Transport | p. 373 |
| Relating Parallel Transport to Curvature | p. 378 |
| From Geometry to Topology: The Gauss-Bonnet Theorem in Euclidean 3-Space | p. 382 |
| Variational Calculus and Field Theory | p. 387 |
| The Story of the Fly and the Train | p. 387 |
| The Concept of a Field | p. 389 |
| The Idea of a Potential | p. 389 |
| The Lagrangian Formalism | p. 391 |
| Lagrange's Equation | p. 393 |
| Other Variational Approaches | p. 395 |
| Application to Mechanics: Hamilton's Principle | p. 396 |
| Nother's Theorem and Lagrangian Invariances | p. 398 |
| Continuous Systems: First Steps to a Field Theory | p. 399 |
| Nother's Theorem for a Scalar Field | p. 402 |
| Building a Lagrangian | p. 404 |
| A Relativistic Lagrangian for a Charge in an EM Field | p. 407 |
| Producing the Schrodinger Equation | p. 415 |
| Quantising Field Theory: Fields Describe Particles, Too! | p. 417 |
| First Steps: The Klein-Gordon Equation | p. 421 |
| A Route to the Dirac Equation | p. 422 |
| Gauge Theory and Quantum Electrodynamics | p. 427 |
| The Starting Point: Classical Gauge Theory | p. 427 |
| A Gauge Transformation for the Dirac Lagrangian | p. 429 |
| The Path-Integral Approach to Quantum Mechanics | p. 432 |
| Path Integrals Give the Schrodinger Equation | p. 435 |
| Density Matrices: The Language of Decoherence | p. 438 |
| The Green Function Approach to Solving Field Equations | p. 445 |
| The Idea of a Green Function | p. 445 |
| Deriving the Green Function for [nabla superscript 2] via Fourier Theory | p. 449 |
| The Other Way of Calculating the Integral (11.20) | p. 456 |
| Solving Maxwell's Equations via the Green Function Approach | p. 460 |
| Variations on the Green Function Solution of Maxwell's Equations | p. 468 |
| Fluctuation-Dissipation and Time's Arrow | p. 470 |
| Airliners, Black Holes, and Cosmology: The ABC of General Relativity | p. 471 |
| The Equivalence Principle | p. 471 |
| The Pound-Rebka-Snider Experiments | p. 474 |
| A Space or Spacetime Description of Gravity? | p. 476 |
| A Route to Curved Spacetime from Lagrangian Mechanics | p. 478 |
| Free Particles, Geodesics, and Locally Inertial Frames | p. 483 |
| Quantities That Are Conserved on Geodesics | p. 489 |
| A Path to Einstein's Equation | p. 490 |
| Solving Einstein's Equation for an Empty Spacetime: The Schwarzschild Metric | p. 495 |
| Deriving Gravitational Redshift Again | p. 498 |
| The Schwarzschild Black Hole | p. 500 |
| Tensor Components and Physical Measurements | p. 506 |
| Calculating Curvature More Efficiently: Cartan's Structural Equations | p. 508 |
| The Variational Approach to Einstein's Equation | p. 511 |
| Adding Extra Field Terms to the Lagrangian Density | p. 515 |
| Adding a Simple Field: The Cosmological Constant | p. 518 |
| Joining Electromagnetism to Gravity | p. 519 |
| Path Integrals in General Relativity | p. 523 |
| A Metric for the Universe: Proper Distances in Cosmology | p. 524 |
| Index | p. 531 |
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