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Preface | |
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Acknowledgements | |
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Notation | |
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Prologue | |
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The roots of science | |
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The quest for the forces that shape the world | |
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Mathematical truth | |
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Is Plato’s mathematical world ‘real’? | |
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Three worlds and three deep mysteries | |
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The Good, the True, and the Beautiful | |
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An ancient theorem and a modern question | |
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The Pythagorean theorem | |
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Euclid’s postulates | |
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Similar-areas proof of the Pythagorean theorem | |
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Hyperbolic geometry: conformal picture | |
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Other representations of hyperbolic geometry | |
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Historical aspects of hyperbolic geometry | |
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Relation to physical space | |
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Kinds of number in the physical world | |
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A Pythagorean catastrophe? | |
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The real-number system | |
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Real numbers in the physical world | |
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Do natural numbers need the physical world? | |
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Discrete numbers in the physical world | |
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Magical complex numbers | |
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The magic number ‘i’ | |
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Solving equations with complex numbers | |
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Convergence of power series | |
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Caspar Wessel’s complex plane | |
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How to construct the Mandelbrot set | |
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Geometry of logarithms, powers, and roots | |
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Geometry of complex algebra | |
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The idea of the complex logarithm | |
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Multiple valuedness, natural logarithms | |
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Complex powers | |
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Some relations to modern particle physics | |
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Real-number calculus | |
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What makes an honest function? | |
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Slopes of functions | |
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Higher derivatives; C1-smooth functions | |
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The ‘Eulerian’ notion of a function? | |
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The rules of differentiation | |
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Integration | |
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Complex-number calculus | |
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Complex smoothness; holomorphic functions | |
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Contour integration | |
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Power series from complex smoothness | |
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Analytic continuation | |
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Riemann surfaces and complex mappings | |
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The idea of a Riemann surface | |
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Conformal mappings | |
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The Riemann sphere | |
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The genus of a compact Riemann surface | |
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The Riemann mapping theorem | |
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Fourier decomposition and hyperfunctions | |
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Fourier series | |
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Functions on a circle | |
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Frequency splitting on the Riemann sphere | |
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The Fourier transform | |
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Frequency splitting from the Fourier transform | |
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What kind of function is appropriate? | |
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Hyperfunctions | |
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Surfaces | |
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Complex dimensions and real dimensions | |
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Smoothness, partial derivatives | |
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Vector Fields and 1-forms | |
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Components, scalar products | |
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The Cauchy–Riemann equations | |
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Hypercomplex numbers | |
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The algebra of quaternions | |
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The physical role of quaternions? | |
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Geometry of quaternions | |
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How to compose rotations | |
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Clifford algebras | |
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Grassmann algebras | |
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Manifolds of n dimensions | |
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Why study higher-dimensional manifolds? | |
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Manifolds and coordinate patches | |
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Scalars, vectors, and covectors | |
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Grassmann products | |
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Integrals of forms | |
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Exterior derivative | |
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Volume element; summation convention | |
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Tensors; abstract-index and diagrammatic notation | |
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Complex manifolds | |
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Symmetry groups | |
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Groups of transformations | |
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Subgroups and simple groups | |
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Linear transformations and matrices | |
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Determinants and traces | |
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Eigenvalues and eigenvectors | |
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Representation theory and Lie algebras | |
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Tensor representation spaces; reducibility | |
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Orthogonal groups | |
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Unitary groups | |
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Symplectic groups | |
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Calculus on manifolds | |
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Differentiation on a manifold? | |
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Parallel transport | |
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Covariant derivative | |
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Curvature and torsion | |
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Geodesics, parallelograms, and curvature | |
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Lie derivative | |
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What a metric can do for you | |
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Symplectic manifolds <br | |