∂

∂
Mathematical signs
arithmetic
Plus sign	+
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	∫
Concatenation characters	∘
Infinity symbol	∞
geometry
Angle sign	∠ , ∡ , ∢ , ∟
Vertical , parallel	⊥ , ∥
Triangle , square	△ , □
Diameter sign	⌀
Set theory
Union , cut	∪ , ∩
Difference , complement	∖ , ∁
Element character	∈
Subset , superset	⊂ , ⊆ , ⊇ , ⊃
Empty set	∅
logic
Follow arrow	⇒ , ⇔ , ⇐
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
character	position	designation	designation	hexadecimal	decimal	named	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`

Web link

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

swell

↑ Unicode character “∂” (U + 2202) , data on the symbol
↑ Introduction to partial derivatives , Khan Academy
↑ Prof. Stefan Kooths, Nicole Wägner: Formula overview , section operators and functions. Business and Information Technology School , 2014.
↑ 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'".
↑ 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)".
↑ ^a ^b ^c ^d John Aldrich: Earliest Uses of Symbols of Calculus , section partial derivative . Jeff Millers website, source for entire usage history.
^ Richard A. Silverman: Essential Calculus with Applications . Courier Corporation, 1977; second edition 1989, p. 216. Dover Publications Inc, New York. ISBN 0486660974
^ 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.
^ 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.
^ 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.
↑ 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«.
^ Will Robertson: Symbols defined by unicode-math , January 31, 2020

[1] Unicode character “∂” (U + 2202) , data on the symbol

[2] Introduction to partial derivatives , Khan Academy

[3] Prof. Stefan Kooths, Nicole Wägner: Formula overview , section operators and functions. Business and Information Technology School , 2014.

[4] 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'".

[5] 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)".

[Aldrich-6] John Aldrich: Earliest Uses of Symbols of Calculus , section partial derivative . Jeff Millers website, source for entire usage history.

[7] Richard A. Silverman: Essential Calculus with Applications . Courier Corporation, 1977; second edition 1989, p. 216. Dover Publications Inc, New York. ISBN 0486660974

[8] 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.

[9] 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.

[10] 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.

[11] 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«.

[12] Will Robertson: Symbols defined by unicode-math , January 31, 2020