A primer of infinitesimal analysis
著者
書誌事項
A primer of infinitesimal analysis
Cambridge University Press, 1998
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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.
目次
- 1. Basic features of smooth worlds
- 2. Basic differential calculus
- 3. First applications of the differential calculus
- 4. Applications to physics
- 5. Multivariable calculus and applications
- 6. The definite integral: Higher order infinitesimals
- 7. Synthetic geometry
- 8. Smooth infinitesimal analysis as an axiomatic system
- Appendix
- Models for smooth infinitesimal analysis.
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