∂
The character ∂ (Unicode: U+2202) is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as (read as "the partial derivative of z with respect to x").[1][2] It is also used for the boundary operator in a chain complex, and the conjugate of the Dolbeault operator on smooth differential forms over a complex manifold. It should be distinguished from other similar-looking symbols such as lowercase Greek letter delta (𝛿) or the lowercase Latin letter eth (ð).
History[edit]
The symbol was originally introduced in 1770 by Nicolas de Condorcet, who used it for a partial differential, and adopted for the partial derivative by Adrien-Marie Legendre in 1786.[3] It represents a specialized cursive type of the letter d, just as the integral sign originates as a specialized type of a long s (first used in print by Leibniz in 1686). Use of the symbol was discontinued by Legendre, but it was taken up again by Carl Gustav Jacob Jacobi in 1841,[4] whose usage became widely adopted.[5]
Names and coding[edit]
The symbol is variously referred to as "curly d", "rounded d", "curved d", "dabba",[6] or "Jacobi's delta",[5] or as "del"[7] (but this name is also used for the "nabla" symbol ∇). It may also be pronounced simply "dee",[8] "partial dee",[9][10] "doh",[11][12] or "die".[13]
The Unicode character U+2202 ∂ PARTIAL DIFFERENTIAL is accessed by HTML entities ∂
or ∂
, and the equivalent LaTeX symbol (Computer Modern glyph: ) is accessed by \partial
.
Uses[edit]
∂ is also used to denote the following:
- The Jacobian .
- The boundary of a set in topology.
- The boundary operator on a chain complex in homological algebra.
- The boundary operator of a differential graded algebra.
- The conjugate of the Dolbeault operator on complex differential forms.
- The boundary ∂(S) of a set of vertices S in a graph is the set of edges leaving S, which defines a cut.
See also[edit]
- d'Alembert operator
- Differentiable programming
- Differential operator § Notations
- List of mathematical symbols
- Notation for differentiation
- 𝒹 (Unicode MATHEMATICAL SCRIPT SMALL D)
- ꝺ (lowercase d in Insular script)
- δ (lowercase Greek Delta)
- д (lowercase Cyrillic De, looks similar when italicized in some typefaces)
References[edit]
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Look up ∂ in Wiktionary, the free dictionary. |
- ^ Christopher, Essex (2013). Calculus : a complete course. p. 682. ISBN 9780321781079. OCLC 872345701.
- ^ "Calculus III - Partial Derivatives". tutorial.math.lamar.edu. Retrieved 2020-09-16.
- ^ Adrien-Marie Legendre, "Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations," Histoire de l'Academie Royale des Sciences (1786), pp. 7–37.
- ^ Carl Gustav Jacob Jacobi, "De determinantibus Functionalibus," Crelle's Journal 22 (1841), pp. 319–352.
- ^ a b
"The 'curly d' was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in 'Memoire sur les Equations aux différence partielles,' which was published in Histoire de L'Academie Royale des Sciences, pp. 151-178, Annee M. DCCLXXIII (1773). On page 152, Condorcet says:
- Dans toute la suite de ce Memoire, dz & ∂z désigneront ou deux differences partielles de z, dont une par rapport a x, l'autre par rapport a y, ou bien dz sera une différentielle totale, & ∂z une difference partielle.
- Pour éviter toute ambiguité, je représenterai par ∂u/∂x le coefficient de x dans la différence de u, & par du/dx la différence complète de u divisée par dx.
- Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristica d differentialia vulgaria, differentialia autem partialia characteristica ∂ denotare.
- ^ Gokhale, Mujumdar, Kulkarni, Singh, Atal, Engineering Mathematics I, p. 10.2, Nirali Prakashan ISBN 8190693549.
- ^ Bhardwaj, R.S. (2005), Mathematics for Economics & Business (2nd ed.), p. 6.4, ISBN 9788174464507
- ^ Silverman, Richard A. (1989), Essential Calculus: With Applications, p. 216, ISBN 9780486660974
- ^ Pemberton, Malcolm; Rau, Nicholas (2011), Mathematics for Economists: An Introductory Textbook, p. 271, ISBN 9781442612761
- ^ Munem, Mustafa; Foulis, David (1978). Calculus with Analytic Geometry. New York, NY: Worth Publishers, Inc. p. 828. ISBN 0-87901-087-8.
- ^ Bowman, Elizabeth (2014), Video Lecture for University of Alabama in Huntsville, archived from the original on 2021-12-22
- ^ Karmalkar, S., Department of Electrical Engineering, IIT Madras (2008), Lecture-25-PN Junction(Contd), archived from the original on 2021-12-22, retrieved 2020-04-22
- ^ Christopher, Essex; Adams, Robert Alexander (2014). Calculus : a complete course (Eighth ed.). p. 682. ISBN 9780321781079. OCLC 872345701.