The Road to Reality

Produktbeschreibung

One of todays most accomplished scientists presents the only comprehensive and comprehensible account of the physics of the universe. Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing. 

Inhaltsverzeichnis

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  

Autoreninfo

Roger Penrose is Emeritus Rouse Ball Professor of Mathematics at Oxford University. He has received a number of prizes and awards, including the 2020 Nobel Prize in Physics for his work on black hole formation, as well as the 1988 Wolf Prize for physics, which he shared with Stephen Hawking for their joint contribution to our understanding of the universe. His books include Cycles of Time, The Road to Reality, The Nature of Space and Time, which he wrote with Hawking, and The Emperor's New Mind. He has lectured extensively at universities throughout America. He lives in Oxford.