A primer of infinitesimal analysis / John L. Bell.

Format:

Book

Author/Creator:

Bell, J. L. (John Lane), author.

Language:

English

Subjects (All):

Nonstandard mathematical analysis.

Physical Description:

1 online resource (xi, 124 pages) : digital, PDF file(s).

Edition:

Second edition.

Place of Publication:

Cambridge : Cambridge University Press, 2008.

Language Note:

English

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 new edition 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. 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. This edition also contains an expanded historical and philosophical introduction.

Contents:

Basic features of smooth worlds

Basic differential calculus

The derivative of a function

Stationary points of functions

Areas under curves and the constancy principle

The special functions

First applications of the differential calculus

Areas and volumes

Volumes of revolution

Arc length; surfaces of revolution; curvature

Application to physics

Moments of inertia

Centres of mass

Pappus' theorems

Centres of pressure

Stretching a spring

Flexure of beams

The catenary, the loaded chain, and the bollard-rope

The Kepler-Newton areal law of motion under a central force

Multivariable calculus and applications

Partial derivatives

Stationary values of functions

Theory of surfaces. Spacetime metrics

The heat equation

The basic equations of hydrodynamics

The wave equation

The Cauchy-Riemann equations for complex functions

The definite integral. Higher-order infinitesimals

The definite integral

Higher-order infinitesimals and Taylor's theorem

The three natural microneighbourhoods of zero

Synthetic differential geometry

Tangent vectors and tangent spaces

Vector fields

Differentials and directional derivatives

Smooth infinitesimal analysis as an axiomatic system

Natural numbers in smooth worlds

Nonstandard analysis.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 121-122) and index.

ISBN:

1-107-18697-8

1-281-94443-2

9786611944438

0-511-37096-2

0-511-36995-6

0-511-37143-8

0-511-36943-3

0-511-61962-6

0-511-37045-8

OCLC:

437240800