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 manifoldsġ3 Symmetry groups 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 The algebra of quaternions The physical role of quaternions? Geometry of quaternions How to compose rotations CliVord algebras Grassmann algebrasġ2 Manifolds of n dimensions 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 The idea of a Riemann surface Conformal mappings The Riemann sphere The genus of a compact Riemann surface The Riemann mapping theoremĩ Fourier decomposition and hyperfunctionsįourier 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? HyperfunctionsĬomplex dimensions and real dimensions Smoothness, partial derivatives Vector Welds and 1-forms Components, scalar products The Cauchy–Riemann equationsġ1 Hypercomplex numbers 11.1 11.2 11.3 11.4 11.5 11.6 What makes an honest function? Slopes of functions Higher derivatives C1 -smooth functions The ‘Eulerian’ notion of a function? The rules of diVerentiation Integrationħ Complex-number calculus 7.1 7.2 7.3 7.4Ĭomplex smoothness holomorphic functions Contour integration Power series from complex smoothness Analytic continuationĨ Riemann surfaces and complex mappings 8.1 8.2 8.3 8.4 8.5 Geometry of complex algebra The idea of the complex logarithm Multiple valuedness, natural logarithms Complex powers Some relations to modern particle physicsĦ Real-number calculus 6.1 6.2 6.3 6.4 6.5 6.6 The magic number ‘i’ Solving equations with complex numbersĬonvergence of power series Caspar Wessel’s complex plane How to construct the Mandelbrot setĥ Geometry of logarithms, powers, and roots 5.1 5.2 5.3 5.4 5.5 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 spaceģ Kinds of number in the physical world 3.1 3.2 3.3 3.4 3.5Ī 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 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 BeautifulĢ An ancient theorem and a modern question 2.1 2.2 2.3 2.4 2.5 2.6 2.7 954009 A CIP catalogue record for this book is available from the British Library ISBN 0–224–04447–8 Papers used by The Random House Group Limited are natural, recyclable products made from wood grown in sustainable forests the manufacturing processes conform to the environmental regulations of the country of origin Printed and bound in Great Britain by William Clowes, Beccles, Suffolk Published by Jonathan Cape 2004 2 4 6 8 10 9 7 5 3 1 Copyright ß Roger Penrose 2004 Roger Penrose has asserted his right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser First published in Great Britain in 2004 by Jonathan Cape Random House, 20 Vauxhall Bridge Road, London SW1V 2SA Random House Australia (Pty) Limited 20 Alfred Street, Milsons Point, Sydney, New South Wales 2061, Australia Random House New Zealand Limited 18 Poland Road, Glenfield, Auckland 10, New Zealand Random House South Africa (Pty) Limited Endulini, 5A Jubilee Road, Parktown 2193, South Africa The Random House Group Limited Reg. T H E R O A D TO REALITY A Complete Guide to the Laws of the Universe The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics Shadows of the Mind: A Search for the Missing Science of Consciousness
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