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" A primer of infinitesimal analysis / "
John L. Bell.
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BL
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| Record Number
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982665
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b737035
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| Main Entry
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Bell, J. L., (John Lane)
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| Title & Author
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A primer of infinitesimal analysis /\ John L. Bell.
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| Edition Statement
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2nd ed.
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| Publication Statement
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Cambridge ;New York :: Cambridge University Press,, ©2008.
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| Page. NO
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1 online resource (xi, 124 pages) :: illustrations
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| ISBN
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0511369956
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: 0511370962
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: 0511371438
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: 0521887186
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: 9780511369957
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: 9780511370960
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: 9780511371431
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: 9780521887182
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| Bibliographies/Indexes
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Includes bibliographical references (pages 121-122) and index.
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| Contents
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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.
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| Abstract
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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 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.
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| Subject
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Nonstandard mathematical analysis.
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MATHEMATICS-- Calculus.
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MATHEMATICS-- Mathematical Analysis.
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Nonstandard mathematical analysis.
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| Dewey Classification
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515
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| LC Classification
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QA299.82.B45 2008eb
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