A Primer of Infinitesimal AnalysisOne 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. |
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 |
Common terms and phrases
a₁ abscissa arbitrary assertion assume axioms axis beam BSIA called cancelling category theory Cauchy-Riemann equations centre of mass Chapter circle codomain coincide concept of infinitesimal Constancy Principle constant construction continuum coordinates corresponding curvature curve defined derivative determine differential calculus equation excluded middle exercise follows function f given inertia intuitionistic logic inverse law of excluded length manifolds map f mathematical microcancellation microneighbourhood micropolynomiality microquantities Microstraightness models of smooth Moerdijk moment of inertia n-microvectors natural numbers nilsquare infinitesimal nonstandard analysis object obtain pair plane presheaf principle of microaffineness proof quantity radius real line real numbers satisfies segment set theory Show smooth infinitesimal analysis smooth map smooth worlds space stationary point straight line straight microsegment suppose surface synthetic differential geometry tangent bundle tangent vector Taylor's theorem theorem topos toposes unique V₁ variables vector field volume x-axis zero ε₁