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The road to reality : a complete guide to the laws of the universe

Roger Penrose

Adult Nonfiction QC20 .P366 2005

Roger Penrose

Adult Nonfiction QC20 .P366 2005

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Contents | Page |
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Preface | |

Acknowledgements | |

Notation | |

Prologue | |

1 - The roots of science | |

1.1 - The quest for the forces that shape the world | |

1.2 - Mathematical truth | |

1.3 - Is Plato's mathematical world 'real'? | |

1.4 - Three worlds and three deep mysteries | |

1.5 - The Good, the True, and the Beautiful | |

2 - An ancient theorem and a modern question | |

2.1 - The Pythagorean theorem | |

2.2 - Euclid's postulates | |

2.3 - Similar-areas proof of the Pythagorean theorem | |

2.4 - Hyperbolic geometry: conformal picture | |

2.5 - Other representations of hyperbolic geometry | |

2.6 - Historical aspects of hyperbolic geometry | |

2.7 - Relation to physical space | |

3 - Kinds of number in the physical world | |

3.1 - A Pythagorean catastrophe? | |

3.2 - The real-number system | |

3.3 - Real numbers in the physical world | |

3.4 - Do natural numbers need the physical world? | |

3.5 - Discrete numbers in the physical world | |

4 - Magical complex numbers | |

4.1 - The magic number 'i' | |

4.2 - Solving equations with complex numbers | |

4.3 - Convergence of power series | |

4.4 - Caspar Wessel's complex plane | |

4.5 - How to construct the Mandelbrot set | |

5 - Geometry of logarithms, powers, and roots | |

5.1 - Geometry of complex algebra | |

5.2 - The idea of the complex logarithm | |

5.3 - Multiple valuedness, natural logarithms | |

5.4 - Complex powers | |

5.5 - Some relations to modern particle physics | |

6 - Real-number calculus | |

6.1 - What makes an honest function? | |

6.2 - Slopes of functions | |

6.3 - Higher derivatives; C1-smooth functions | |

6.4 - The 'Eulerian' notion of a function? | |

6.5 - The rules of differentiation | |

6.6 - Integration | |

7 - Complex-number calculus | |

7.1 - Complex smoothness; holomorphic functions | |

7.2 - Contour integration | |

7.3 - Power series from complex smoothness | |

7.4 - Analytic continuation | |

8 - Riemann surfaces and complex mappings | |

8.1 - The idea of a Riemann surface | |

8.2 - Conformal mappings | |

8.3 - The Riemann sphere | |

8.4 - The genus of a compact Riemann surface | |

8.5 - The Riemann mapping theorem | |

9 - Fourier decomposition and hyperfunctions | |

9.1 - Fourier series | |

9.2 - Functions on a circle | |

9.3 - Frequency splitting on the Riemann sphere | |

9.4 - The Fourier transform | |

9.5 - Frequency splitting from the Fourier transform | |

9.6 - What kind of function is appropriate? | |

9.7 - Hyperfunctions | |

10 - Surfaces | |

10.1 - Complex dimensions and real dimensions | |

10.2 - Smoothness, partial derivatives | |

10.3 - Vector Fields and 1-forms | |

10.4 - Components, scalar products | |

10.5 - The Cauchy-Riemann equations | |

11 - Hypercomplex numbers | |

11.1 - The algebra of quaternions | |

11.2 - The physical role of quaternions? | |

11.3 - Geometry of quaternions | |

11.4 - How to compose rotations | |

11.5 - Clifford algebras | |

11.6 - Grassmann algebras | |

12 - Manifolds of n dimensions | |

12.1 - Why study higher-dimensional manifolds? | |

12.2 - Manifolds and coordinate patches | |

12.3 - Scalars, vectors, and covectors | |

12.4 - Grassmann products | |

12.5 - Integrals of forms | |

12.6 - Exterior derivative | |

12.7 - Volume element; summation convention | |

12.8 - Tensors; abstract-index and diagrammatic notation | |

12.9 - Complex manifolds | |

13 - Symmetry groups | |

13.1 - Groups of transformations | |

13.2 - Subgroups and simple groups | |

13.3 - Linear transformations and matrices | |

13.4 - Determinants and traces | |

13.5 - Eigenvalues and eigenvectors | |

13.6 - Representation theory and Lie algebras | |

13.7 - Tensor representation spaces; reducibility | |

13.8 - Orthogonal groups | |

13.9 - Unitary groups | |

13.10 - Symplectic groups | |

14 - Calculus on manifolds | |

14.1 - Differentiation on a manifold? | |

14.2 - Parallel transport | |

14.3 - Covariant derivative | |

14.4 - Curvature and torsion | |

14.5 - Geodesics, parallelograms, and curvature | |

14.6 - Lie derivative | |

14.7 - What a metric can do for you | |

14.8 - Symplectic manifolds |

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