# ∂

Mathematical signs
arithmetic
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Minus sign - , ./.
Mark , ×
Divided sign : , ÷ , /
Plus minus sign ± ,
Comparison sign < , , = , , >
Root sign
Percent sign %
Analysis
Sum symbol Σ
Product mark Π
Difference sign , Nabla ,
Prime
Partial differential
Integral sign
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geometry
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Set theory
Union , cut ,
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logic
Universal quantifier
Existential quantifier
Conjunction , disjunction ,
Negation sign ¬

The (pronounced: Del ) is a mathematical symbol that is mainly used for partial derivative and partial differential . It's a stylized lowercase d or δ and has the Unicode number U + 2202 . ${\ displaystyle {\ tfrac {\ partial} {\ partial x}}}$ ${\ displaystyle \ partial f}$ ## Names

The most common name of the ∂ is Del , which, however, also describes the Nabla operator in English . Therefore there are other names for the symbol, including a. partial d , in English Dabba or Jacobi delta , as well as simply d . Then it can no longer be linguistically differentiated from the total derivative .

## Usage history

Just as the integral sign represents a special form of the long s , the ∂ is a special italic notation of the d s. It was first used in 1770 by the French mathematician Nicolas de Concordet as a symbol for the partial differential.

"Dans toute la suite de ce Memoire, dz & ∂z désigneront ou deux differences partielles de z, dont une par rapport ax, l'autre par rapport ay, ou bien dz sera une différentielle totale, & ∂z une difference partial."

"In the remainder of this paper, dz & ∂z denote either two partial differentials of z, one with respect to x, the other with respect to y, or dz is a total differential & ∂z a partial differential."

- Antoine-Nicolas Caritat, Marquis de Condorcet : Memoire sur les Equations aux différence partielles , 1773

Adrien-Marie Legendre first used it in 1786 for the partial derivative.

"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."

"To avoid ambiguity, I will use ∂u / ∂x to represent the coefficient of x in the differential of u & by du / dx to represent the total differential of u divided by dx."

- Adrien-Marie Legendre : Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations , 1786

Legendre later discontinued its use. Carl Gustav Jacob Jacobi took it up again in 1841 and spread the ∂ widely.

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

"However, since the accumulation of hooks for reading and writing is even more laborious, I prefer the usual d characteristic for ordinary differentials, for partial differentials characteristic ∂ is given."

- Carl Gustav Jacob Jacobi : De determinantibus Functionalibus , 1841

## Applications

${\ displaystyle {\ frac {\ partial f} {\ partial x}} = {\ frac {\ partial f (x, y)} {\ partial x}} = {\ frac {\ partial} {\ partial x} } f (x, y)}$ is the partial derivative of to${\ displaystyle f}$ ${\ displaystyle x}$ . You need it when a multivariable function is to be differentiated according to a variable in order to indicate which one.

${\ displaystyle {\ frac {\ partial \ mathbf {f}} {\ partial \ mathbf {x}}} = {\ frac {\ partial (f_ {1}, f_ {2}, \ cdots, f_ {m}} )} {\ partial (x_ {1}, x_ {2}, \ cdots, x_ {n})}} = {\ begin {pmatrix} {\ cfrac {\ partial f_ {1}} {\ partial x_ {1 }}} & \ cdots & {\ cfrac {\ partial f_ {1}} {\ partial x_ {n}}} \\\ vdots & \ ddots & \ vdots \\ {\ cfrac {\ partial f_ {m}} {\ partial x_ {1}}} & \ cdots & {\ cfrac {\ partial f_ {m}} {\ partial x_ {n}}} \ end {pmatrix}}}$ is called the m × n Jacobian matrix from to (matrix of the partial derivatives of the m-dimensional function dependent on n variables ). ${\ displaystyle f}$ ${\ displaystyle x}$ ${\ displaystyle f}$ In addition to partial derivative, partial differential and Jacobian matrix , the ∂ is also used in topology as the boundary of a set , in homological algebra as a limit operator in a chain complex or a DG algebra and in Dolbeault cohomology as the complex conjugate of the Dolbeault operator over a complex differential form is used. In linguistics , the ∂ is used for presuppositions of a sentence.

## Coding

Coding in Unicode, HTML and LaTeX
character Unicode designation HTML Latex
U+2202 partial differential Partial differential & # x2202; & # 8706; \partial
? U+1D6DB mathematical bold partial differential Mathematical bold partial derivative & # x1D6DB; & # 120539; \mbfpartial
? U+1D715 mathematical italic partial differential Mathematical italic partial derivative & # x1D715; & # 120597; \mitpartial
? U+1D74F mathematical bold italic partial differential Mathematical partial derivative in bold italics & # x1D74F; & # 120655; \mbfitpartial
? U+1D789 mathematical sans-serif bold partial differential Mathematical sans serif bold partial derivative & # x1D789; & # 120713; \mbfsanspartial
? U+1D7C3 mathematical sans-serif bold italic partial differential Mathematical sans-serif partial derivative in bold italics & # x1D7C3; & # 120771; \mbfitsanspartial

Wiktionary: del  - explanations of meanings, word origins, synonyms, translations

## swell

1. Unicode character “∂” (U + 2202) , data on the symbol
2. Prof. Stefan Kooths, Nicole Wägner: Formula overview , section operators and functions. Business and Information Technology School , 2014.
3. Malcolm Pemberton, Nicholas Rau: Mathematics for Economists: An Introductory Textbook . University of Toronto Press, 3rd Edition 2011. ISBN 1442612762 . Quote S, 270/271: "pronounced 'partial-dee-eff-by-dee-ex'".
4. MY Gokhale, NS Mujumdar, SS Kulkarni, AN Singh, Atal KR: Engineering Mathematics-i . Nirali Prakashan, 1981, section 10.5. ISBN 8190693549 . Quote p. 10.2: "we read it as dabba z by dabba x (or del z by del x)".
5. a b c d John Aldrich: Earliest Uses of Symbols of Calculus , section partial derivative . Jeff Millers website, source for entire usage history.
6. ^ Richard A. Silverman: Essential Calculus with Applications . Courier Corporation, 1977; second edition 1989, p. 216. Dover Publications Inc, New York. ISBN 0486660974
7. ^ Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet : Memoire sur les Equations aux différence partielles . In: Histoire de L'Academie Royale des Sciences, Annee M. DCCLXXIII (1773). Pp. 151-178, citation p. 152.
8. ^ Adrien-Marie Legendre : Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations . In: Histoire de l'Academie Royale des Sciences, Annee M. DCCLXXXVI (1786), Paris, M. DCCXXXVIII (1788). Pp. 7–37, citation footnote p. 8.
9. ^ Carl Gustav Jacob Jacobi : De determinantibus Functionalibus . In: Journal for pure and applied mathematics , Volume 22, 1841. P. 319–352, P. 393-438 in the 1st volume of the collected works.
10. Ljudmila Geist, Björn Rothstein: Copula verbs and copula sentences: interlingual and intralingual aspects . Linguistic work, Volume 512. Ed. Walter de Gruyter, 2012, first edition 2007. Max Niemeyer Verlag, Tübingen. ISBN 3110938839 . P. 154, quote: »" ∂ "serves as a marker for presuppositions«.
11. ^ Will Robertson: Symbols defined by unicode-math , January 31, 2020