Nabla symbol
The nabla is a triangular symbol resembling an inverted Greek delta:Template:Efn or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word Template:Lang for a Phoenician harp,[1][2] and was suggested by the encyclopedist William Robertson Smith in an 1870 letter to Peter Guthrie Tait.[1][3][4][5][6]
The nabla symbol is available in standard HTML as ∇ and in LaTeX as \nabla. In Unicode, it is the character at code point U+2207, or 8711 in decimal notation, in the Mathematical Operators block.
As a mathematical operator, it is often called del.
History
The differential operator given in Cartesian coordinates on three-dimensional Euclidean space by
was introduced in 1831 by the Irish mathematician and physicist William Rowan Hamilton, who called it ◁.[7] (The unit vectors were originally right versors in Hamilton's quaternions.) The mathematics of ∇ received its full exposition at the hands of P. G. Tait.[8][9]
After receiving Smith's suggestion, Tait and James Clerk Maxwell referred to the operator as nabla in their extensive private correspondence; most of these references are of a humorous character. C. G. Knott's Life and Scientific Work of Peter Guthrie Tait (p. 145):[4]
William Thomson (Lord Kelvin) introduced the term to an American audience in an 1884 lecture;[1] the notes were published in Britain and the U.S. in 1904.[10]
The name is acknowledged, and criticized, by Oliver Heaviside in 1891:[11]
Heaviside and Josiah Willard Gibbs (independently) are credited with the development of the version of vector calculus most popular today.[12]
The influential 1901 text Vector Analysis, written by Edwin Bidwell Wilson and based on the lectures of Gibbs, advocates the name "del":[13]
This book is responsible for the form in which the mathematics of the operator in question is now usually expressed—most notably in undergraduate physics, and especially electrodynamics, textbooks.
Modern uses
The nabla is used in vector calculus as part of three distinct differential operators: the gradient (∇), the divergence (∇⋅), and the curl (∇×). The last of these uses the cross product and thus makes sense only in three dimensions; the first two are fully general. They were all originally studied in the context of the classical theory of electromagnetism, and contemporary university physics curricula typically treat the material using approximately the concepts and notation found in Gibbs and Wilson's Vector Analysis.
The symbol is also used in differential geometry to denote a connection.
A symbol of the same form, though presumably not genealogically related, appears in other areas, e.g.:
- As the all relation, particularly in lattice theory.
- As the backward difference operator, in the calculus of finite differences.
- As the widening operator, an operator that permits static analysis of programs to terminate in finite time, in the computer science field of abstract interpretation.
- As function definition marker and self-reference (recursion) in the APL programming language.
- As an indicator of indeterminacy in philosophical logic.Template:Efn
- In naval architecture (ship design), to designate the volume displacement of a ship or any other waterborne vessel; the graphically similar delta is used to designate weight displacement (the total weight of water displaced by the ship), thus where is the density of seawater.
- In London Underground Working Timetables,[14] to denote that the operator of the train carries out the Stepping Back procedure.
See also
- Covariant derivative, also known as connection
- Del, treating the mathematics of the vector differential operator
- Del in cylindrical and spherical coordinates
- Dirac operator
- grad, div, and curl, differential operators defined using nabla
- History of quaternions
- Nevel[6]
- Notation for differentiation
Footnotes
References
External links
- Template:Cite web
- Template:Cite web
- Template:Cite web
- Tai, Chen. A survey of the improper use of ∇ in vector analysis (1994).
- ↑ 1.0 1.1 1.2 Template:Cite OED
- ↑ Template:LSJ.
- ↑ Letter from Smith to Tait, 10 November 1870:
Quoted in Oxford English Dictionary entry "nabla".My dear Sir, The name I propose for ∇ is, as you will remember, Nabla... In Greek the leading form is ναβλᾰ... As to the thing it is a sort of harp and is said by Hieronymus and other authorities to have had the figure of ∇ (an inverted Δ).
- ↑ 4.0 4.1 Template:Cite book
- ↑ Template:Cite web
- ↑ 6.0 6.1 Notably it is sometimes claimed to be from the Hebrew nevel (נֶבֶל)—as in the Book of Isaiah, 5th chapter, 12th sentence: "וְהָיָה כִנּוֹר וָנֶבֶל תֹּף וְחָלִיל וָיַיִן מִשְׁתֵּיהֶם וְאֵת פֹּעַל יְהוָה לֹא יַבִּיטוּ וּמַעֲשֵׂה יָדָיו לֹא רָאוּ"—, but this etymology is mistaken; the Greek νάβλα comes from the Phoenician to which נֶבֶל is cognate. See: Template:Cite OED
- ↑ Template:Cite journal
- ↑ Knott, pp. 142–143: "Unquestionably, however, Tait's great work was his development of the powerful operator ∇. Hamilton introduced this differential operator in its semi-Cartesian trinomial form on page 610 of his Lectures and pointed out its effects on both a scalar and a vector quantity. ... Neither in the Lectures nor in the Elements, however, is the theory developed. This was done by Tait in the second edition of his book (∇ is little more than mentioned in the first edition) and much more fully in the third and last edition."
- ↑ Template:Cite book
- ↑ Template:Cite book As this is written, he appears to be naming the Laplacian ∇2 "nabla", but in the lecture was presumably referring to ∇ itself.
- ↑ Template:Cite journal
- ↑ Template:Cite book
- ↑ Template:Cite book
- ↑ Template:Cite web