Delta (letter)

Delta ( ; uppercase Δ, lowercase δ; , délta, ) 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. 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, this use of the word delta was not coined by Herodotus.

Pronunciation

In Ancient Greek, delta represented a voiced dental plosive . In Modern Greek, it represents a voiced dental fricative , like the "th" in "that", "this", or "though" (while in foreign words is instead commonly transcribed as ντ, nt). Delta is romanized as d or (in Modern Greek) 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">{{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}}online copy</ref><ref name="Comenetz2002">{{cite book

|author=Michael Comenetz

|title=Calculus: The Elements

|year=2002

|publisher=World Scientific

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

|pages=73–74}}</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 , 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:
.
The discriminant of a polynomial equation, especially the quadratic equation:<ref>{{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}}

</ref><ref>{{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}}</ref>

.
The area of a triangle:
.
The symmetric difference of two sets.
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.
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.
A major seventh chord in jazz music notation.
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.
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.
In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.

Lowercase

thumb|right|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.
The Kronecker delta in mathematics.
The central difference for a function.
The degree of a vertex in graph theory.
The Dirac delta function in mathematics.
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.
A subunit of the F1 sector of the F-ATPase.
The declination of an object in the equatorial coordinate system of astronomy.
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.
In a system that exhibits electrical reactance, the angle between voltage and current.
Partial charge in chemistry.
The maximum birefringence of a crystal in optical mineralogy.
An Old Irish voiced dental or alveolar fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish dh.
Silver ratio

Unicode

( in TeX)
( in TeX)

Delta (letter)

Pronunciation

Uppercase

Lowercase

Unicode

See also

References