“infinitesimal”的概念、定义、翻译、参考文献-科学参考

infinitesimal

Physics

Vanishingly small but not zero. Infinitesimal changes are notionally made in the calculus, which is sometimes called the infinitesimal calculus.

Mathematics

A variable whose limit is zero, usually involving increments of functions, and of particular use in defining differentiation. The ratio $\frac{δ y}{δ x}$ tends to the differential $\frac{d y}{d x}$ as δ‎x → 0. See also hyperreals.

Chemical Engineering

A number that is not zero but is less than any finite number. While such numbers do not exist in the conventional system of real numbers, it allows calculations to adhere to certain restrictions and requires a good understanding of limits.

Philosophy

Both the infinitely large and its inverse, the infinitely small, cause headaches. The first philosophical explorations that turned disquiet into something more tangible were Zeno’s paradoxes. The 17th-century development of the calculus led to the consideration of magnitudes changing over ‘indefinitely’ smaller intervals of time, and the reaction on Zeno’s behalf came from Berkeley, who saw that the infinitesimals resulting (‘the vanished ghosts of departed quantities’) were treated inconsistently as finite quantities or as nothing, depending on the exigencies of the argument. A rigorous treatment of the notion of a limit was only provided by K. T. W. Weierstrass (1815–97) in the 19th century: it need only refer to finite numbers, and thus removes the place of the infinitesimal from the foundations of mathematics. However, a consistent non-standard analysis has been developed by the mathematician Abraham Robinson (1918–74): in it infinitesimals have the disturbing property that while they are not nothing, no sum of n of them is ever greater than any finite number, so the number sequence that includes them is not Archimedean.

单词	infinitesimal
释义	infinitesimal Physics Vanishingly small but not zero. Infinitesimal changes are notionally made in the calculus, which is sometimes called the infinitesimal calculus. Mathematics A variable whose limit is zero, usually involving increments of functions, and of particular use in defining differentiation. The ratio $\frac{δ y}{δ x}$ tends to the differential $\frac{d y}{d x}$ as δ‎x → 0. See also hyperreals. Chemical Engineering A number that is not zero but is less than any finite number. While such numbers do not exist in the conventional system of real numbers, it allows calculations to adhere to certain restrictions and requires a good understanding of limits. Philosophy Both the infinitely large and its inverse, the infinitely small, cause headaches. The first philosophical explorations that turned disquiet into something more tangible were Zeno’s paradoxes. The 17th-century development of the calculus led to the consideration of magnitudes changing over ‘indefinitely’ smaller intervals of time, and the reaction on Zeno’s behalf came from Berkeley, who saw that the infinitesimals resulting (‘the vanished ghosts of departed quantities’) were treated inconsistently as finite quantities or as nothing, depending on the exigencies of the argument. A rigorous treatment of the notion of a limit was only provided by K. T. W. Weierstrass (1815–97) in the 19th century: it need only refer to finite numbers, and thus removes the place of the infinitesimal from the foundations of mathematics. However, a consistent non-standard analysis has been developed by the mathematician Abraham Robinson (1918–74): in it infinitesimals have the disturbing property that while they are not nothing, no sum of n of them is ever greater than any finite number, so the number sequence that includes them is not Archimedean.
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