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''dy'' ''dx'' ''d''2''y'' ''dx''2 In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just as and represent finite increments of and , respectively. Consider as a function of a variable , or = . If this is the case, then the derivative of with respect to , which later came to be viewed as the limit : was, according to Leibniz, the quotient of an infinitesimal increment of by an infinitesimal increment of , or : where the right hand side is Joseph-Louis Lagrange's notation for the derivative of at . From the point of view of modern infinitesimal theory, is an infinitesimal -increment, is the corresponding -increment, and the derivative is the standard part of the infinitesimal ratio: :. Then one sets , , so that by definition, is the ratio of ''dy'' by ''dx''. Similarly, although mathematicians sometimes now view an integral : as a limit : where is an interval containing , Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities . From the modern viewpoint, it is more correct to view the integral as the standard part of such an infinite sum. ==History== The Newton-Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the character. He based the character on the Latin word ''summa'' ("sum"), which he wrote ''ſumma'' with the elongated s commonly used in Germany at the time. This use first appeared publicly in his paper ''De Geometria'', published in ''Acta Eruditorum'' of June 1686,〔''Mathematics and its History'', John Stillwell, Springer 1989, p. 110〕 but he had been using it in private manuscripts at least since 1675.〔''Early Mathematical Manuscripts of Leibniz'', J. M. Child, Open Court Publishing Co., 1920, pp. 73–74, 80.〕 English mathematicians were encumbered by Newton's dot notation until 1803 when Robert Woodhouse published a description of the continental notation. Later the Analytical Society at Cambridge University promoted the adoption of Leibniz's notation. At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above, while Cauchy exploited both infinitesimals and limits (see Cours d'Analyse). Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard ''f''(''x'') as measured in meters per second, and d''x'' in seconds, so that ''f''(''x'') d''x'' is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis. In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed non-standard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based on Robinson's approach. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Leibniz's notation」の詳細全文を読む スポンサード リンク
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