Nabla symbol
Encyclopedia
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
(∇). The name comes from the GreekGreek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
word for a Hebrew harp
Harp
The harp is a multi-stringed instrument which has the plane of its strings positioned perpendicularly to the soundboard. Organologically, it is in the general category of chordophones and has its own sub category . All harps have a neck, resonator and strings...
, which had a similar shape. Related words also exist in Aramaic
Aramaic language
Aramaic is a group of languages belonging to the Afroasiatic language phylum. The name of the language is based on the name of Aram, an ancient region in central Syria. Within this family, Aramaic belongs to the Semitic family, and more specifically, is a part of the Northwest Semitic subfamily,...
and Hebrew
Hebrew language
Hebrew is a Semitic language of the Afroasiatic language family. Culturally, is it considered by Jews and other religious groups as the language of the Jewish people, though other Jewish languages had originated among diaspora Jews, and the Hebrew language is also used by non-Jewish groups, such...
. The symbol was first used by William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
in the form of a sideways wedge: ⊲. Another, less-common name for the symbol is atled (delta spelled backwards), because the nabla is an inverted Greek letter
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
delta
Delta (letter)
Delta 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...
. In actual Greek usage, the symbol is called ανάδελτα, anádelta, which means "upside-down delta".
The nabla symbol is available in standard HTML as
∇ and in LaTeXLaTeX
LaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as . The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to...
as
\nabla. In UnicodeUnicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
, it is the character at code point
Code point
In character encoding terminology, a code point or code position is any of the numerical values that make up the code space . For example, ASCII comprises 128 code points in the range 0hex to 7Fhex, Extended ASCII comprises 256 code points in the range 0hex to FFhex, and Unicode comprises 1,114,112...
U+2207, or 8711 in decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
notation.
Use in mathematics
Nabla is used in mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
to denote the del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
operator, a differential operator that indicates taking gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
, divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
, or curl
Curl
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl is represented by a vector...
. It also can refer to a connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
in differential geometry and to the backward difference operator in the calculus of finite differences, as well as the all relation (most commonly in lattice theory). It was introduced by the Irish
Ireland
Ireland is an island to the northwest of continental Europe. It is the third-largest island in Europe and the twentieth-largest island on Earth...
mathematician and physicist William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
in 1837. William Thomson
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
wrote in 1884: "I took the liberty of asking Professor Bell whether he had a name for this symbol
and he has mentioned to me nabla, a humorous suggestion of MaxwellJames Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
's. It is the name of an Egyptian harp, which was of that shape".
In 1901, Josiah Willard Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
and Edwin Bidwell Wilson
Edwin Bidwell Wilson
Edwin Bidwell Wilson was an American mathematician and polymath. He was the sole protégé of Yale's physicist Josiah Willard Gibbs and was mentor to MIT economist Paul Samuelson. He received his AB from Harvard College in 1899 and his PhD from Yale University in 1901, working under Gibbs.E.B...
wrote: "This symbolic operator
was introduced by Sir W. R. Hamilton and is now in universal employment. There seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated formulae in which 
occurs a number of times, no inconvenience to the speaker or listener arises from the repetition. 
V is read simply as 'del V ".Use in naval engineering (or naval architectureNaval architectureNaval architecture is an engineering discipline dealing with the design, construction, maintenance and operation of marine vessels and structures. Naval architecture involves basic and applied research, design, development, design evaluation and calculations during all stages of the life of a...
)
Nabla is used in naval engineering (ship design) to designate the volume displacementDisplacement (fluid)
In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place. The volume of the fluid displaced can then be measured, as in the illustration, and from this the volume of the immersed object can be deduced .An object that sinks...
of a ship or any other waterborne vessel. Where its counterpart, the Greek delta
Delta (letter)
Delta 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...
, is used to designate weight displacement (the total weight of water displaced by the ship), the nabla is used to designate volume displacement, or the total volume of water displaced by the ship in units of length cubed.
See also
- DelDelIn vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
, the vector differential operator - Del in cylindrical and spherical coordinatesDel in cylindrical and spherical coordinatesThis is a list of some vector calculus formulae of general use in working with various curvilinear coordinate systems.- Note :* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in...
- gradGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
, divDivergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
, and curl, differential operators defined using del - the Covariant derivativeCovariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
, a separate (and tensorial) differential operator defined for tensors
External links
- History of Nabla
- A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen