Delta (letter)#Upper case

In Ancient Greek, delta represented a voiced dental plosive {{IPA|el|d|IPA}}. In Modern Greek, it represents a voiced dental fricative {{IPA|el|ð|IPA}}, like the "th" in "that" or "this" (while {{IPA|el|d|IPA}} in foreign words is instead commonly transcribed as ντ). Delta is romanized as d or dh.

The uppercase letter Δ is used to denote:

• Change of any changeable quantity, in mathematics and the sciences (in particular, the difference operator{{cite book

|author=Clarence H. Richardson

|title=An Introduction to the Calculus of Finite Differences

|url=https://archive.org/details/introductiontoca00rich

|url-access=registration

|year=1954

|publisher=Van Nostrand

|at=Chapter 1, pp. 1—3}}[http://babel.hathitrust.org/cgi/pt?id=mdp.39015000982945;view=1up;seq=5 online copy]{{cite book

|author=Michael Comenetz

|title=Calculus: The Elements

|year=2002

|publisher=World Scientific

|isbn=978-981-02-4904-5

|pages=73–74}}); for example, in $$

\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}

, the average change of y per unit x (i.e. the change of y over the change of x). Delta is the initial letter of the Greek word {{lang|el|διαφορά}}, diaphorá, "difference". (The small Latin letter d is used in much the same way for the notation of derivatives and differentials, which also describe change by infinitesimal amounts.)

• The Laplace operator:
• : $\backslash Delta\; f=\backslash sum\_\{i=1\}^n\{\backslash frac\{\backslash partial^2f\}\{\backslash partial\; x\_i^2\}\}$.
• The discriminant of a polynomial equation, especially the quadratic equation:{{cite book

|title=Solving polynomial equations: foundations, algorithms, and applications

|first1=Alicia

|last1=Dickenstein

|author1-link=Alicia Dickenstein

|first2=Ioannis Z.

|last2=Emiris

|publisher=Springer

|year=2005

|isbn=978-3-540-24326-7

|url=https://books.google.com/books?id=rSs-pQNrO_YC&pg=PA26

|at=Example 2.5.6, p. 120}}

{{cite book

|title=Integers, polynomials, and rings

|first1=Ronald S.

|last1=Irving

|publisher=Springer-Verlag New York, Inc.

|year=2004

|isbn=978-0-387-40397-7

|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154

|at=Ch. 10.1, pp. 145}}

• : $\backslash Delta=b^2-4ac$.
• The area of a triangle:
• : $\backslash Delta=\backslash tfrac\{1\}\{2\}ab\backslash sin\{C\}$.{{Cite web |last=Weisstein |first=Eric W. |title=Triangle Area |url=https://mathworld.wolfram.com/TriangleArea.html |access-date=2025-01-23 |website=mathworld.wolfram.com |language=en}}
• The symmetric difference of two sets.{{Cite web |last=Weisstein |first=Eric W. |title=Symmetric Difference |url=https://mathworld.wolfram.com/SymmetricDifference.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=The symmetric difference of sets A and B is variously written as A ⊖ B, A∇ B, A+B (Borowski and Borwein 1991) or AΔB (Harris and Stocker 1998, p. 3). All but the first notation should probably be deprecated since each of the other symbols has a common meaning in other areas of mathematics.}}
• A macroscopic change in the value of a variable in mathematics or science.
• Uncertainty in a physical variable as seen in the uncertainty principle.
• An interval of possible values for a given quantity.
• Any of the delta particles in particle physics.
• The determinant of the matrix of coefficients of a set of linear equations (see Cramer's rule).
• That an associated locant number represents the location of a covalent bond in an organic compound, the position of which is variant between isomeric forms.
• A simplex, simplicial complex, or convex hull.
• In chemistry, the addition of heat in a reaction.
• In legal shorthand, it represents a defendant.{{cite book |last1=Tepper |first1=Pamela |url=https://books.google.com/books?id=z9wbCgAAQBAJ |title=The Law of Contracts and the Uniform Commercial Code |publisher=Cengage Learning |year=2014 |page=32 |isbn=978-1285448947 |access-date=2018-04-30 }}
• In the financial markets, one of the Greeks, describes the rate of change of an option price for a given change in the underlying benchmark.{{Cite web |title=Black-Scholes Formulas (d1, d2, Call Price, Put Price, Greeks) - Macroption |url=https://www.macroption.com/black-scholes-formula/ |access-date=2025-01-23 |website=www.macroption.com |quote=Delta is the first derivative of option price with respect to underlying price S.}}
• A major seventh chord in jazz music notation.{{Cite web |title=Every chord symbol found on lead sheets |url=https://jazz-library.com/articles/chord-symbols/ |access-date=2025-01-23 |website=Jazz-Library |language=en-US |quote=Chord with a triangle Major 7th “C△”}}
• In genetics, it can stand for a gene deletion (e.g. the CCR5-Δ32, a 32 nucleotide/bp deletion within CCR5).
• The American Dental Association cites it (together with omicron for "odont") as the symbol of dentistry.{{cite web |url=http://www.ada.org/2517.aspx?currentTab=2 |title=Caduceus, the emblem of dentistry |website=American Dental Association |access-date=26 October 2017 |archive-url=https://web.archive.org/web/20121112073815/http://www.ada.org/2517.aspx?currentTab=2 |archive-date=12 November 2012 |url-status=dead}}
• The anonymous signature of James David Forbes.Proceedings of the Royal Society, vol. XIX, p. ii.
• Determinacy (having a definite truth-value) in philosophical logic.{{Cite journal |last1=Shramko |first1=Yaroslav |last2=Wansing |first2=Heinrich |date=2010-03-30 |title=Truth Values |url=https://plato.stanford.edu/archIves/fall2015/entries/truth-values/}}
• In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.{{cite web |url=https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol |title=Who first defined the "equal-delta" or "delta over equal" symbol? |access-date=2 October 2022 |archive-url=https://web.archive.org/web/20220306070901/https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol |archive-date=6 March 2022 |url-status=dead}}

File:NAMA Alphabet grec.jpg on a black figure vessel, with a D-shaped delta.]]

The lowercase letter δ (or 𝛿) can be used to denote:

• A change in the value of a variable in calculus.
• A functional derivative in functional calculus.
• The (ε, δ)-definition of limits, in mathematics and more specifically in calculus.{{Cite web |last=Weisstein |first=Eric W. |title=Epsilon-Delta Proof |url=https://mathworld.wolfram.com/Epsilon-DeltaProof.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}
• The Kronecker delta in mathematics.{{Cite web |last=Weisstein |first=Eric W. |title=Kronecker Delta |url=https://mathworld.wolfram.com/KroneckerDelta.html |access-date=2025-01-23 |website=mathworld.wolfram.com |language=en}}
• The central difference for a function.{{Cite web |last=Weisstein |first=Eric W. |title=Central Difference |url=https://mathworld.wolfram.com/CentralDifference.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}
• The degree of a vertex in graph theory.
• The Dirac delta function in mathematics.{{Cite web |last=Weisstein |first=Eric W. |title=Delta Function |url=https://mathworld.wolfram.com/DeltaFunction.html |access-date=2025-01-23 |website=mathworld.wolfram.com |language=en}}
• The transition function in automata.
• Deflection in engineering mechanics.
• The force of interest in actuarial science.
• The chemical shift of nuclear magnetic resonance in chemistry.
• The relative electronegativity of different atoms in a molecule, δ⁻ being more electronegative than δ⁺.
• Text requiring deletion in proofreading; the usage is said to date back to classical times.
• In some of the manuscripts written by Dr. John Dee, the character of delta is used to represent Dee.{{Cite web |title=Greek Alphabet |url=https://www.ancient-symbols.com/greek-alphabet |access-date=2025-01-31 |website=Ancient Symbols |language=en-us |quote=Dr. John Dee, a mathematician, used the lowercase Delta symbol to represent himself in manuscripts.}}
• A subunit of the F1 sector of the F-ATPase.
• The declination of an object in the equatorial coordinate system of astronomy.{{Cite web |title=Greek Alphabet |url=https://www.ancient-symbols.com/greek-alphabet |access-date=2025-01-31 |website=Ancient Symbols |language=en-us |quote=In astronomy, the symbol is used to represent the declination of an object.}}
• The dividend yield in the Black–Scholes option pricing formula.
• Ratios of environmental isotopes, such as ¹⁸O/¹⁶O and D/¹H from water are displayed using delta notation – δ¹⁸O and δD, respectively.
• The rate of depreciation of the aggregate capital stock of an economy in an exogenous growth model in macroeconomics.{{cite web|url=http://econ.duke.edu/people?subpage=unit&Gurl=/aas/Economics&cname=Masters+Students|title=Faculty - Economics Department|website=econ.duke.edu|access-date=26 October 2017}}
• In a system that exhibits electrical reactance, the angle between voltage and current.
• Partial charge in chemistry.{{Citation |title=Declination |encyclopedia=The Encyclopedia of Astronomy and Astrophysics |date=2001 |url=https://doi.org/10.1888/0333750888/4541 |access-date=2025-01-31 |publisher=IOP Publishing Ltd |doi=10.1888/0333750888/4541 |isbn=0-333-75088-8|url-access=subscription }}
• The maximum birefringence of a crystal in optical mineralogy.{{Cite book|title=Mineralogia Óptica|last=MACHADO, Fábio Braz|first=NARDY, Antônio José Ranalli|publisher=Oficina de Textos|year=2018|isbn=9788579752452|location=São Paulo|pages=85}}
• An Old Irish voiced dental or alveolar fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish dh.
• Silver ratio{{Cite web |last=Weisstein |first=Eric W. |title=Silver Ratio |url=https://mathworld.wolfram.com/SilverRatio.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}

• {{unichar|018D|html=}}
• {{unichar|0394|html=}} ({{tt|\Delta}} in TeX)
• {{unichar|03B4|html=}} ({{tt|\delta}} in TeX)
• {{unichar|1D5F|html=}}
• {{unichar|1E9F|html=}}
• {{unichar|2207|html=}}
• {{unichar|225C|html=}}
• {{unichar|234B|html=}}
• {{unichar|234D|html=}}
• {{unichar|2359|html=}}
• {{unichar|2C86|html=}}
• {{unichar|2C87|html=}}
• {{unichar|1D6AB|html=}}{{efn|The {{sc|mathematical}} codes should only be used in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.}}
• {{unichar|1D6C5|html=}}
• {{unichar|1D6E5|html=}}
• {{unichar|1D6FF|html=}}
• {{unichar|1D71F|html=}}
• {{unichar|1D739|html=}}
• {{unichar|1D759|html=}}
• {{unichar|1D773|html=}}
• {{unichar|1D793|html=}}
• {{unichar|1D7AD|html=}}

{{notelist}}

{{Wiktionary|Δ|δ}}

• Arrow (symbol)
• Chevron (insignia)
• ∆ (disambiguation)
• D, d
• Д, д
• ẟ – Latin delta
• ∂ – the partial derivative symbol, a curved d, sometimes mistaken for a lowercase Greek letter Delta.
• ð – the small eth appears similar to a small delta and also represents a d sound in some contexts
• Th (digraph)
• Thorn (letter)
• Greek letters used in mathematics, science, and engineering
• ∇ – Nabla symbol
• Delta Air Lines
• SARS-CoV-2 Delta variant

Category:Greek letters