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Leibniz's notation
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==History== [[File:Leibniz Manuscript of integral and differential notation.png|thumb|Leibniz manuscript of integral and differential notation]] The Newton–Leibniz approach to [[infinitesimal calculus]] was introduced in the 17th century. While Newton worked with [[fluxion]]s and fluents, Leibniz based his approach on generalizations of sums and differences.<ref name="Katz524">{{harvnb|Katz|1993|loc=p. 524}}</ref> Leibniz adapted the [[integral symbol]] <math>\textstyle \int</math> from the initial [[long s|elongated s]] of the Latin word ''{{serif|ſ}}umma'' ("sum") as written at the time. Viewing differences as the inverse operation of summation,<ref>{{harvnb|Katz|1993|loc=p. 529}}</ref> he used the symbol {{mvar|d}}, the first letter of the Latin ''differentia'', to indicate this inverse operation.<ref name="Katz524" /> Leibniz was fastidious about notation, having spent years experimenting, adjusting, rejecting and corresponding with other mathematicians about them.<ref>{{harvnb|Mazur|2014|loc=p. 166}}</ref> Notations he used for the differential of {{mvar|y}} ranged successively from {{mvar|ω}}, {{mvar|l}}, and {{math|{{sfrac|''y''|''d''}}}} until he finally settled on {{mvar|dy}}.<ref>{{harvnb|Cajori|1993|loc=Vol. II, p. 203, footnote 4}}</ref> His [[integral sign]] first appeared publicly in the article "''De Geometria Recondita et analysi indivisibilium atque infinitorum''" ("On a hidden geometry and analysis of indivisibles and infinites"), published in ''[[Acta Eruditorum]]'' in June 1686,<ref>{{citation |first=Frank J. |last=Swetz |title=Mathematical Treasure: Leibniz's Papers on Calculus - Integral Calculus |publisher=[[Mathematical Association of America]] |url=http://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus |access-date=February 11, 2017 |series=Convergence}}</ref><ref>{{cite book |title=Mathematics and its History |url=https://archive.org/details/mathematicsitshi0000stil |url-access=registration |first=John |last=Stillwell |author-link=John Stillwell |publisher=Springer |year=1989 |page=[https://archive.org/details/mathematicsitshi0000stil/page/110 110] }}</ref> but he had been using it in private manuscripts at least since 1675.<ref>{{cite book |title=The Early Mathematical Manuscripts of Leibniz |first=G. W. |last=Leibniz |publisher=Dover |orig-year=1920 |year=2005 |pages=73–74, 80 |isbn=978-0-486-44596-0 |translator-first=J. M. |translator-last=Child}}</ref><ref>Leibniz, G. W., ''Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674-1676'', Berlin: Akademie Verlag, 2008, pp. [http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf 288–295] {{Webarchive|url=https://web.archive.org/web/20211009052830/https://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf |date=2021-10-09 }} ("''Analyseos tetragonisticae pars secunda''", October 29, 1675) and [http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf 321–331] ("''Methodi tangentium inversae exempla''", November 11, 1675).</ref><ref>{{cite web |url=http://jeff560.tripod.com/calculus.html |author=Aldrich, John |title=Earliest Uses of Symbols of Calculus |access-date=20 April 2017}}</ref> Leibniz first used {{mvar|dx}} in the article "''[[Nova Methodus pro Maximis et Minimis]]''" also published in ''Acta Eruditorum'' in 1684.<ref name="Cajori204">{{harvnb|Cajori|1993|loc=Vol. II, p. 204}}</ref> While the symbol {{math|{{sfrac|''dx''|''dy''}}}} does appear in private manuscripts of 1675,<ref>Leibniz, G. W., ''Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674-1676'', Berlin: Akademie Verlag, 2008, pp. [http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf 321–331 esp. 328] ("''Methodi tangentium inversae exempla''", November 11, 1675).</ref><ref>{{harvnb|Cajori|1993|loc=Vol. II, p. 186}}</ref> it does not appear in this form in either of the above-mentioned published works. Leibniz did, however, use forms such as {{mvar|dy ad dx}} and {{math|''dy'' : ''dx''}} in print.<ref name="Cajori204" /> 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 [[infinitesimal]]s 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]].
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