∂

Not to be confused with δ, ẟ, ꝺ, ð, σ, ə, ә, ǝ, d, or д.

The character ∂ (Unicode: U+2202) is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as ${\displaystyle {\partial z}/{\partial x}}$ (read as "the partial derivative of z with respect to x").^[1][2] It is also used for boundary of a set, 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

The symbol was introduced originally 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

The symbol is variously referred to as "partial", "curly d", "funky d", "rounded d", "curved d", "dabba", "number 6 mirrored",^[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] "dow" or "die".^[13]

The Unicode character U+2202 ∂ PARTIAL DIFFERENTIAL is accessed by HTML entities ∂ or ∂, and the equivalent LaTeX symbol (Computer Modern glyph: ${\displaystyle \partial }$) is accessed by \partial.

Uses

∂ is also used to denote the following:

• The Jacobian ${\displaystyle {\frac {\partial (x,y,z)}{\partial (u,v,w)}}}$.
• 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

• 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

1. Christopher, Essex (2013). Calculus : a complete course. Pearson. p. 682. ISBN 9780321781079. OCLC 872345701.
2. "Calculus III - Partial Derivatives". tutorial.math.lamar.edu. Retrieved 2020-09-16.
3. Adrien-Marie Legendre, "Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations," Histoire de l'Académie Royale des Sciences (1786), pp. 7–37.
4. Carl Gustav Jacob Jacobi, "De determinantibus Functionalibus," Crelle's Journal 22 (1841), pp. 319–352.
5. "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'Académie 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.

   However, the "curly d" was first used in the form ∂u/∂x by Adrien Marie Legendre in 1786 in his 'Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations,' Histoire de l'Académie Royale des Sciences, Annee M. DCCLXXXVI (1786), pp. 7-37, Paris, M. DCCXXXVIII (1788). On footnote of page 8, it reads:

   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.

   Legendre abandoned the symbol and it was re-introduced by Carl Gustav Jacob Jacobi in 1841. Jacobi used it extensively in his remarkable paper 'De determinantibus Functionalibus" Crelle's Journal, Band 22, pp. 319-352, 1841 (pp. 393-438 of vol. 1 of the Collected Works).

   Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristica d differentialia vulgaria, differentialia autem partialia characteristica ∂ denotare.

   The "curly d" symbol is sometimes called the "rounded d" or "curved d" or Jacobi's delta. It corresponds to the cursive "dey" (equivalent to our d) in the Cyrillic alphabet." Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 16 January 2014.
6. Gokhale, Mujumdar, Kulkarni, Singh, Atal, Engineering Mathematics I, p. 10.2, Nirali Prakashan ISBN 8190693549.
7. Bhardwaj, R.S. (2005), Mathematics for Economics & Business (2nd ed.), Excel Books India, p. 6.4, ISBN 9788174464507
8. Silverman, Richard A. (1989), Essential Calculus: With Applications, Courier Corporation, p. 216, ISBN 9780486660974
9. Pemberton, Malcolm; Rau, Nicholas (2011), Mathematics for Economists: An Introductory Textbook, University of Toronto Press, p. 271, ISBN 9781442612761
10. Munem, Mustafa; Foulis, David (1978). Calculus with Analytic Geometry. New York, NY: Worth Publishers, Inc. p. 828. ISBN 0-87901-087-8.
11. Bowman, Elizabeth (2014), Video Lecture for University of Alabama in Huntsville, archived from the original on 2021-12-22
12. Karmalkar, S., Department of Electrical Engineering, IIT Madras (2008), Lecture-25-PN Junction(Contd), 14 December 2007, archived from the original on 2021-12-22, retrieved 2020-04-22
13. Christopher, Essex; Adams, Robert Alexander (2014). Calculus : a complete course (Eighth ed.). Pearson. p. 682. ISBN 9780321781079. OCLC 872345701.