无穷小演算

Isaac Newton
Gottfried Wilhelm von Leibniz
艾萨克·牛顿(左)和戈特弗里德·萊布尼茨(右),无穷小演算的发明者

无穷小演算(英語:Infinitesimal calculus)是微积分学的早期名称,在17世纪60年代由莱布尼茨牛顿基于巴罗笛卡尔等数学家的工作各自独立发展出来。它包括了微分演算积分演算,分别用来指微分学积分学的技术。无穷小演算以后发展为标准微积分非标准分析等形式不一但彼此等价的体系。

在早期微积分中,无穷小量的使用被认为是不严格的,被一些作者严厉地批评,其中最有名的包括米歇尔·罗尔贝克莱主教。后者在1734年的著作《分析家》中称它为“已死量的幽灵”。

一些数学家,如麦克劳林,试图去证明使用无穷小量在定理上的完备性。但直到150年后,才由柯西威尔斯特拉斯完成了这一证明,结束了只能使用模糊的概念去定义无穷小量的历史。

非標準分析

实无限分析它用严格定义的无穷小量的概念来构建分析学,使得无穷小可以直接参与运算。

衍生概念

参考资料

  • Baron, Margaret E.: The origins of the infinitesimal calculus. Dover Publications, Inc., New York, 1987.
  • Baron, Margaret E.: The origins of the infinitesimal calculus. Pergamon Press, Oxford-Edinburgh-New York 1969.(A new edition of Baron's book appeared in 2004)

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