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The road to reality : a complete guide to the laws of the universe
Penrose, Roger.
Adult Nonfiction 530.1 P
Penrose, Roger.
Adult Nonfiction 530.1 P
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| Contents | Page |
|---|---|
| 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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