Delta (letter)

Delta (uppercase Δ, lowercase δ; δέλτα, délta, ) is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter dalet 𐤃. Letters that come from delta include Latin D and Cyrillic Д.

A river delta (originally, the delta of the Nile River) is so named 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" or "this" (while 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: \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}, $$ the average change of y per unit x (i.e. the change of y over the change of x). Delta is the initial letter of the Greek word διαφορά diaphorá, "difference". (The small Latin letter d is used in much the same way for the notation of derivatives and differentials, which also describe change by infinitesimal amounts.)
 * Change of any changeable quantity, in mathematics and the sciences (in particular, the difference operator ); for example, in $$
 * The Laplace operator:
 * $$\Delta f = \sum_{i=1}^n {\frac{\partial^2 f}{\partial x_i^2}}.$$
 * The discriminant of a polynomial equation, especially the quadratic equation:
 * $$\Delta = b^2 - 4ac.$$
 * The area of a triangle
 * $$\Delta = \tfrac{1}{2} a b \sin{C}.$$
 * 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, describing 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.
 * 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


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
 * An auxiliary function in calculus, used to rigorously define the limit or continuity of a given function
 * The Kronecker delta in mathematics
 * The degree of a vertex (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 birrefringence 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

Computer encodings

 * Greek Delta / Coptic Dalda


 * Latin Delta


 * Technical and Mathematical symbols


 * Mathematical Delta

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.