Delta (letter)

Template:Short description Template:RedirectTemplate:Distinguish Template:Refimprove Template:Greek AlphabetDelta (Template:IPAc-en Template:Respell;<ref>Template:OED</ref> uppercase Δ, lowercase δ; Template:Langx, délta, {{#invoke:IPA|main}})<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> is the fourth letter of the Greek alphabet. In the system of Greek numerals, it has a value of four. It was derived from the Phoenician letter dalet 𐤃.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Letters that come from delta include the Latin D and the Cyrillic Д.

A river delta (originally, the delta of the Nile River) is named so because its shape approximates the triangular uppercase letter delta. Contrary to a popular legend,Template:Vague this use of the word delta was not coined by Herodotus.<ref>Template:Cite journal</ref>

Pronunciation

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

Uppercase

The uppercase letter Δ is used to denote:

• Change of any changeable quantity, in mathematics and the sciences (in particular, the difference operator<ref name="Richardson1954">Template:Cite bookonline copy</ref><ref name="Comenetz2002">Template:Cite book</ref>); for example, in <math>

\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x} </math>, 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 {{#invoke:Lang|lang}}, 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:

  <math>\Delta f=\sum_{i=1}^n{\frac{\partial^2f}{\partial x_i^2}}</math>.

• The discriminant of a polynomial equation, especially the quadratic equation:<ref>Template:Cite book

</ref><ref>Template:Cite book</ref>

• <math>\Delta=b^2-4ac</math>.

• The area of a triangle:

  <math>\Delta=\tfrac{1}{2}ab\sin{C}</math>.<ref>{{#invoke:citation/CS1|citation

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• The symmetric difference of two sets.<ref>{{#invoke:citation/CS1|citation

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• 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.<ref>Template:Cite book</ref>
• 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.<ref>{{#invoke:citation/CS1|citation

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• A major seventh chord in jazz music notation.<ref>{{#invoke:citation/CS1|citation

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• 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.<ref>{{#invoke:citation/CS1|citation

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• The anonymous signature of James David Forbes.<ref>Proceedings of the Royal Society, vol. XIX, p. ii.</ref>
• Determinacy (having a definite truth-value) in philosophical logic.<ref>Template:Cite journal</ref>
• In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.<ref>{{#invoke:citation/CS1|citation

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Lowercase

File:NAMA Alphabet grec.jpg

The alphabet 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.<ref>{{#invoke:citation/CS1|citation

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• The Kronecker delta in mathematics.<ref>{{#invoke:citation/CS1|citation

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• The central difference for a function.<ref>{{#invoke:citation/CS1|citation

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• The degree of a vertex in graph theory.
• The Dirac delta function in mathematics.<ref>{{#invoke:citation/CS1|citation

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• 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.<ref>{{#invoke:citation/CS1|citation

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• A subunit of the F1 sector of the F-ATPase.
• The declination of an object in the equatorial coordinate system of astronomy.<ref>{{#invoke:citation/CS1|citation

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• 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.<ref>{{#invoke:citation/CS1|citation

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• In a system that exhibits electrical reactance, the angle between voltage and current.
• Partial charge in chemistry.<ref>Template:Citation</ref>
• The maximum birefringence of a crystal in optical mineralogy.<ref>Template:Cite book</ref>
• An Old Irish voiced dental or alveolar fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish dh.
• Silver ratio<ref>{{#invoke:citation/CS1|citation

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Unicode

• Template:Unichar
• Template:Unichar (Template:Tt in TeX)
• Template:Unichar (Template:Tt in TeX)
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See also

• 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

References

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