A Primer of Infinitesimal Analysis

by John L. Bell

• ISBN13: 9780521624015
• ISBN10: 0521624010
• Format: Hardcover
• Copyright: 1998-07-28
• Publisher: Cambridge University Press

Summary

One of the most remarkable recent occurrences in mathematics is the refounding, on a rigorous basis, of the idea of infinitesimal quantity, a notion which played an important role in the early development of the calculus and mathematical analysis. In this book, basic calculus, together with some of its applications to simple physical problems, are presented through the use of a straightforward, rigorous, axiomatically formulated concept of 'zero-square', or 'nilpotent' infinitesimal - that is, a quantity so small that its square and all higher powers can be set, literally, to zero. As we show, the systematic employment of these infinitesimals reduces the differential calculus to simple algebra and, at the same time, restores to use the 'infinitesimal' methods figuring in traditional applications of the calculus to physical problems - a number of which are discussed in this book. The book also contains a historical and philosophical introduction, a chapter describing the logical features of the infinitesimal framework, and an appendix sketching the developments in the mathematical discipline of category theory that have made the refounding of infinitesimals possible.

Table of Contents

Preface	xi	(2)
Acknowledgements	xiii
Introduction	1	(16)
1 Basic features of smooth worlds	17	(9)
2 Basic differential calculus	26	(11)
2.1 The derivative of a function	26	(3)
2.2 Stationary points of functions	29	(1)
2.3 Areas under curves and the Constancy Principle	30	(2)
2.4 The special functions	32	(4)
3 First applications of the differential calculus	37	(13)
3.1 Areas and volumes	37	(5)
3.2 Volumes of revolution	42	(3)
3.3 Arc length; surfaces of revolution; curvature	45	(5)
4 Applications to physics	50	(20)
4.1 Moments of inertia	50	(5)
4.2 Centres of mass	55	(1)
4.3 Pappus' theorems	56	(3)
4.4 Centres of pressure	59	(2)
4.5 Stretching a spring	61	(1)
4.6 Flexure of beams	61	(3)
4.7 The catenary, the loaded chain, and the bollard-rope	64	(4)
4.8 The Kepler-Newton areal law of motion under a central force	68	(2)
5 Multivariable calculus and applications	17	(70)
5.1 Partial derivatives	70	(3)
5.2 Stationary values of functions	73	(3)
5.3 Theory of surfaces	76	(4)
5.4 The heat equation	80	(1)
5.5 The basic equations of hydrodynamics	81	(3)
5.6 The Cauchy-Riemann equations for complex functions	84	(3)
6 The definite integral. Higher-order infinitesimals	87	(7)
6.1 The definite integral	87	(3)
6.2 Higher-order infinitesimals and Taylor's theorem	90	(3)
6.3 The three natural microneighbourhoods of zero	93	(1)
7 Synthetic differential geometry	94	(6)
7.1 Tangent vectors and tangent spaces	94	(2)
7.2 Vector fields	96	(1)
7.3 Differentials and directional derivatives	96	(4)
8 Smooth infinitesimal analysis as an axiomatic system	100	(11)
8.1 Natural numbers in smooth worlds	106	(2)
8.2 Nonstandard analysis	108	(3)
Appendix. Models for smooth infinitesimal analysis	111	(6)
Note on sources and further reading	117	(2)
References	119	(2)
Index	121