What is included with this book?
Preface | |
Acknowledgements | |
Notation | |
Prologue | |
The roots of science | |
The quest for the forces that shape the world | |
Mathematical truth | |
Is Plato’s mathematical world ‘real’? | |
Three worlds and three deep mysteries | |
The Good, the True, and the Beautiful | |
An ancient theorem and a modern question | |
The Pythagorean theorem | |
Euclid’s postulates | |
Similar-areas proof of the Pythagorean theorem | |
Hyperbolic geometry: conformal picture | |
Other representations of hyperbolic geometry | |
Historical aspects of hyperbolic geometry | |
Relation to physical space | |
Kinds of number in the physical world | |
A Pythagorean catastrophe? | |
The real-number system | |
Real numbers in the physical world | |
Do natural numbers need the physical world? | |
Discrete numbers in the physical world | |
Magical complex numbers | |
The magic number ‘i’ | |
Solving equations with complex numbers | |
Convergence of power series | |
Caspar Wessel’s complex plane | |
How to construct the Mandelbrot set | |
Geometry of logarithms, powers, and roots | |
Geometry of complex algebra | |
The idea of the complex logarithm | |
Multiple valuedness, natural logarithms | |
Complex powers | |
Some relations to modern particle physics | |
Real-number calculus | |
What makes an honest function? | |
Slopes of functions | |
Higher derivatives; C1-smooth functions | |
The ‘Eulerian’ notion of a function? | |
The rules of differentiation | |
Integration | |
Complex-number calculus | |
Complex smoothness; holomorphic functions | |
Contour integration | |
Power series from complex smoothness | |
Analytic continuation | |
Riemann surfaces and complex mappings | |
The idea of a Riemann surface | |
Conformal mappings | |
The Riemann sphere | |
The genus of a compact Riemann surface | |
The Riemann mapping theorem | |
Fourier decomposition and hyperfunctions | |
Fourier series | |
Functions on a circle | |
Frequency splitting on the Riemann sphere | |
The Fourier transform | |
Frequency splitting from the Fourier transform | |
What kind of function is appropriate? | |
Hyperfunctions | |
Surfaces | |
Complex dimensions and real dimensions | |
Smoothness, partial derivatives | |
Vector Fields and 1-forms | |
Components, scalar products | |
The Cauchy–Riemann equations | |
Hypercomplex numbers | |
The algebra of quaternions | |
The physical role of quaternions? | |
Geometry of quaternions | |
How to compose rotations | |
Clifford algebras | |
Grassmann algebras | |
Manifolds of n dimensions | |
Why study higher-dimensional manifolds? | |
Manifolds and coordinate patches | |
Scalars, vectors, and covectors | |
Grassmann products | |
Integrals of forms | |
Exterior derivative | |
Volume element; summation convention | |
Tensors; abstract-index and diagrammatic notation | |
Complex manifolds | |
Symmetry groups | |
Groups of transformations | |
Subgroups and simple groups | |
Linear transformations and matrices | |
Determinants and traces | |
Eigenvalues and eigenvectors | |
Representation theory and Lie algebras | |
Tensor representation spaces; reducibility | |
Orthogonal groups | |
Unitary groups | |
Symplectic groups | |
Calculus on manifolds | |
Differentiation on a manifold? | |
Parallel transport | |
Covariant derivative | |
Curvature and torsion | |
Geodesics, parallelograms, and curvature | |
Lie derivative | |
What a metric can do for you | |
Symplectic manifolds | |
Table of Contents provided by Publisher. All Rights Reserved. |
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
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Excerpted from The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose, Roger Penrose
All rights reserved by the original copyright owners. Excerpts are provided for display purposes only and may not be reproduced, reprinted or distributed without the written permission of the publisher.