A Primer of Infinitesimal Analysis

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Cambridge University Press, Jul 28, 1998 - Mathematics - 122 pages
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One of the most remarkable recent occurrences in mathematics is the refounding, on a rigorous basis, of the idea of infinitesimal quantity, a notion that 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 the author shows, 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 text 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.
 

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Contents

Introduction
1
Basic features of smooth worlds
17
Basic differential calculus
26
22 Stationary points of functions
29
23 Areas under curves and the Constancy Principle
30
24 The special functions
32
First applications of the differential calculus
37
32 Volumes of revolution
42
53 Theory of surfaces
76
54 The heat equation
80
55 The basic equations of hydrodynamics
81
56 The CauchyRiemann equations for complex functions
84
The definite integral Higherorder infinitesimals
87
62 Higherorder infinitesimals and Taylors theorem
90
63 The three natural microneighbourhoods of zero
93
Synthetic differential geometry
94

33 Arc length surfaces of revolution curvature
45
Applications to physics
50
42 Centres of mass
55
43 Pappus theorems
56
44 Centres of pressure
59
45 Stretching a spring
61
47 The catenary the loaded chain and the bollardrope
64
48 The KeplerNewton areal law of motion under a central force
68
Multivariable calculus and applications
70
52 Stationary values of functions
73
72 Vector fields
96
Smooth infinitesimal analysis as an axiomatic system
100
81 Natural numbers in smooth worlds
106
82 Nonstandard analysis
108
Models for smooth infinitesimal analysis
111
Note on sources and further reading
117
References
119
Index
121
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