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The road to reality : a complete guide to the laws of the universe
Roger Penrose
Adult Nonfiction QC20 .P366 2005

Comments  Summary  Contents  Excerpt  Reviews  Author Notes

ContentsPage
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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