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The
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
∂ (
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, whi ...
: U+2202) is a stylized
cursive Cursive (also known as script, among other names) is any style of penmanship in which characters are written joined in a flowing manner, generally for the purpose of making writing faster, in contrast to block letters. It varies in functionali ...
'' d'' mainly used as a
mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
, usually to denote a
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
such as / (read as "the partial derivative of ''z'' with respect to ''x''"). It is also used for the boundary operator in a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
, and the conjugate of the Dolbeault operator on smooth differential forms over a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
. It should be distinguished from other similar-looking symbols such as lowercase Greek letter delta (𝛿) or the lowercase Latin letter
eth (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , a ...
(ð).


History

The symbol was originally introduced in 1770 by
Nicolas de Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher and mathematician. His ideas, including support for a liberal economy, free and equal p ...
, who used it for a partial differential, and adopted for the partial derivative by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
in 1786. It represents a specialized cursive type of the letter ''d'', just as the
integral sign The integral symbol: : (Unicode), \displaystyle \int ( LaTeX) is used to denote integrals and antiderivatives in mathematics, especially in calculus. History The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in ...
originates as a specialized type of a
long s The long s , also known as the medial s or initial s, is an archaic form of the lowercase letter . It replaced the single ''s'', or one or both of the letters ''s'' in a 'double ''s sequence (e.g., "ſinfulneſs" for "sinfulness" and "poſ ...
(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 Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasion ...
in 1841, whose usage became widely adopted. "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.'' 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'Academie Royale des Sciences, Annee M. DCCLXXXVI (1786), pp. 7-37, Paris, M. DCCXXXVIII (1788). O
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."


Names and coding

The symbol is variously referred to as "curly d", "rounded d", "curved d", "dabba", or "Jacobi's delta", or as "del" (but this name is also used for the "nabla" symbol ). It may also be pronounced simply "dee", "partial dee", "doh", or "die". The Unicode character is accessed by
HTML entities In SGML, HTML and XML documents, the logical constructs known as ''character data'' and ''attribute values'' consist of sequences of characters, in which each character can manifest directly (representing itself), or can be represented by a series ...
∂ or ∂, and the equivalent
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperm ...
symbol (
Computer Modern Computer Modern is the original family of typefaces used by the typesetting program TeX. It was created by Donald Knuth with his Metafont program, and was most recently updated in 1992. Computer Modern, or variants of it, remains very widely u ...
glyph: \partial) is accessed by \partial.


Uses

∂ is also used to denote the following: * The Jacobian \frac. * The boundary of a set in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. * The boundary operator on a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
in
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
. * The boundary operator of a
differential graded algebra In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. __TOC__ Definition A differential graded alg ...
. * The conjugate of the Dolbeault operator on
complex differential form In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, t ...
s. * The boundary ''∂(S)'' of a set of vertices ''S'' in a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
is the set of edges leaving ''S'', which defines a cut.


See also

*
d'Alembert operator In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of M ...
*
Differentiable programming Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation. This allows for gradient-based optimization of parameters in the program, often via gradie ...
* *
List of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
*
Notation for differentiation In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with ...
* 𝒹 (Unicode ) * (lowercase ''d'' in Insular script) * '' δ'' (lowercase Greek Delta) * '' д'' (lowercase Cyrillic De, looks similar when italicized in some typefaces)


References

{{DEFAULTSORT:a Mathematical terminology Mathematical symbols