Differential of a function (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Differential of a function" in English language version.

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Courant 1937a, II, §9: "Here we remark merely in passing that it is possible to use this approximate representation of the increment $\Delta y$ by the linear expression $hf(x)$ to construct a logically satisfactory definition of a "differential", as was done by Cauchy in particular." Courant, Richard (1937a), Differential and integral calculus. Vol. I, Wiley Classics Library, New York: John Wiley & Sons (published 1988), ISBN 978-0-471-60842-4, MR 1009558.
See, for instance, the influential treatises of Courant 1937a, Kline 1977, Goursat 1904, and Hardy 1908. Tertiary sources for this definition include also Tolstov 2001 and Itô 1993, §106. Courant, Richard (1937a), Differential and integral calculus. Vol. I, Wiley Classics Library, New York: John Wiley & Sons (published 1988), ISBN 978-0-471-60842-4, MR 1009558. Kline, Morris (1977), "Chapter 13: Differentials and the law of the mean", Calculus: An intuitive and physical approach, John Wiley and Sons. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155. Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, Cambridge University Press, ISBN 978-0-521-09227-2. Tolstov, G.P. (2001) [1994], "Differential", Encyclopedia of Mathematics, EMS Press. Itô, Kiyosi (1993), Encyclopedic Dictionary of Mathematics (2nd ed.), MIT Press, ISBN 978-0-262-59020-4.
Cauchy 1823. See also, for instance, Goursat 1904, I, §14. Cauchy, Augustin-Louis (1823), Résumé des Leçons données à l'Ecole royale polytechnique sur les applications du calcul infinitésimal, archived from the original on 2007-07-08, retrieved 2009-08-19. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.
Goursat 1904, I, §14 Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.
In particular to infinite dimensional holomorphy (Hille & Phillips 1974) and numerical analysis via the calculus of finite differences. Hille, Einar; Phillips, Ralph S. (1974), Functional analysis and semi-groups, Providence, R.I.: American Mathematical Society, MR 0423094.
Goursat 1904, I, §17 Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.
Goursat 1904, I, §§14,16 Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.

archive.org

For a detailed historical account of the differential, see Boyer 1959, especially page 275 for Cauchy's contribution on the subject. An abbreviated account appears in Kline 1972, Chapter 40. Kline, Morris (1972), Mathematical thought from ancient to modern times (3rd ed.), Oxford University Press (published 1990), ISBN 978-0-19-506136-9
See, for instance, the influential treatises of Courant 1937a, Kline 1977, Goursat 1904, and Hardy 1908. Tertiary sources for this definition include also Tolstov 2001 and Itô 1993, §106. Courant, Richard (1937a), Differential and integral calculus. Vol. I, Wiley Classics Library, New York: John Wiley & Sons (published 1988), ISBN 978-0-471-60842-4, MR 1009558. Kline, Morris (1977), "Chapter 13: Differentials and the law of the mean", Calculus: An intuitive and physical approach, John Wiley and Sons. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155. Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, Cambridge University Press, ISBN 978-0-521-09227-2. Tolstov, G.P. (2001) [1994], "Differential", Encyclopedia of Mathematics, EMS Press. Itô, Kiyosi (1993), Encyclopedic Dictionary of Mathematics (2nd ed.), MIT Press, ISBN 978-0-262-59020-4.
Cauchy 1823. See also, for instance, Goursat 1904, I, §14. Cauchy, Augustin-Louis (1823), Résumé des Leçons données à l'Ecole royale polytechnique sur les applications du calcul infinitésimal, archived from the original on 2007-07-08, retrieved 2009-08-19. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.
Goursat 1904, I, §14 Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.
Goursat 1904, I, §17 Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.
Goursat 1904, I, §§14,16 Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.

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See Kock 2006 and Moerdijk & Reyes 1991. Kock, Anders (2006), Synthetic Differential Geometry (PDF) (2nd ed.), Cambridge University Press. Moerdijk, I.; Reyes, G.E. (1991), Models for Smooth Infinitesimal Analysis, Springer-Verlag.

encyclopediaofmath.org

See, for instance, the influential treatises of Courant 1937a, Kline 1977, Goursat 1904, and Hardy 1908. Tertiary sources for this definition include also Tolstov 2001 and Itô 1993, §106. Courant, Richard (1937a), Differential and integral calculus. Vol. I, Wiley Classics Library, New York: John Wiley & Sons (published 1988), ISBN 978-0-471-60842-4, MR 1009558. Kline, Morris (1977), "Chapter 13: Differentials and the law of the mean", Calculus: An intuitive and physical approach, John Wiley and Sons. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155. Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, Cambridge University Press, ISBN 978-0-521-09227-2. Tolstov, G.P. (2001) [1994], "Differential", Encyclopedia of Mathematics, EMS Press. Itô, Kiyosi (1993), Encyclopedic Dictionary of Mathematics (2nd ed.), MIT Press, ISBN 978-0-262-59020-4.

ujf-grenoble.fr

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Cauchy explicitly denied the possibility of actual infinitesimal and infinite quantities (Boyer 1959, pp. 273–275), and took the radically different point of view that "a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge to zero" (Cauchy 1823, p. 12; translation from Boyer 1959, p. 273). Cauchy, Augustin-Louis (1823), Résumé des Leçons données à l'Ecole royale polytechnique sur les applications du calcul infinitésimal, archived from the original on 2007-07-08, retrieved 2009-08-19.
Cauchy 1823. See also, for instance, Goursat 1904, I, §14. Cauchy, Augustin-Louis (1823), Résumé des Leçons données à l'Ecole royale polytechnique sur les applications du calcul infinitésimal, archived from the original on 2007-07-08, retrieved 2009-08-19. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.

web.archive.org

Cauchy explicitly denied the possibility of actual infinitesimal and infinite quantities (Boyer 1959, pp. 273–275), and took the radically different point of view that "a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge to zero" (Cauchy 1823, p. 12; translation from Boyer 1959, p. 273). Cauchy, Augustin-Louis (1823), Résumé des Leçons données à l'Ecole royale polytechnique sur les applications du calcul infinitésimal, archived from the original on 2007-07-08, retrieved 2009-08-19.
Cauchy 1823. See also, for instance, Goursat 1904, I, §14. Cauchy, Augustin-Louis (1823), Résumé des Leçons données à l'Ecole royale polytechnique sur les applications du calcul infinitésimal, archived from the original on 2007-07-08, retrieved 2009-08-19. Goursat, Édouard (1904), A course in mathematical analysis: Vol 1: Derivatives and differentials, definite integrals, expansion in series, applications to geometry, E. R. Hedrick, New York: Dover Publications (published 1959), MR 0106155.

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See Robinson 1996 and Keisler 1986. Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3. Keisler, H. Jerome (1986), Elementary Calculus: An Infinitesimal Approach (2nd ed.).