## 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. |

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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 |

119 | |

121 | |

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### Common terms and phrases

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