Nabla symbol

History

right|thumb|The [[harp]], the instrument after which the nabla symbol is named The differential operator given in Cartesian coordinates {x,y,z} on three-dimensional Euclidean space by } was introduced in 1831 by the Irish mathematician and physicist William Rowan Hamilton, who called it ◁. (The unit vectors {\mathbf{i},\mathbf{j},\mathbf{k}} were originally right versors in Hamilton's quaternions.) The mathematics of ∇ received its full exposition at the hands of P. G. Tait.

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):

William Thomson (Lord Kelvin) introduced the term to an American audience in an 1884 lecture; the notes were published in Britain and the U.S. in 1904.

As this is written, he appears to be naming the Laplacian ∇2 "nabla", but in the lecture was presumably referring to ∇ itself.

The name is acknowledged, and criticized, by Oliver Heaviside in 1891:

Heaviside and Josiah Willard Gibbs (independently) are credited with the development of the version of vector calculus most popular today.

The influential 1901 text Vector Analysis, written by Edwin Bidwell Wilson and based on the lectures of Gibbs, advocates the name "del":

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". 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.{{efn|1=For example, in Anthony Everett (2013), The Nonexistent, p. 210:

• 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 \nabla = \Delta/\rho where \rho is the density of seawater.
• In aerodynamics, in the application of fundamental concepts including vorticity.

Footnotes

References

References

1. {{cite OED. nabla
2. {{LSJ. na/bla. νάβλα. ref.
3. Letter from Smith to Tait, 10 November 1870:

   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).

   Quoted in Oxford English Dictionary entry "nabla".
4. Cargill Gilston Knott. (1911). "Life and Scientific Work of Peter Guthrie Tait". Cambridge University Press.
5. "History of Nabla".
6. W. R. Hamilton, "[http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/FunctZero/FunZero.pdf On Differences and Differentials of Functions of Zero]," ''Trans. R. Irish Acad.'' XVII:235–236 esp. 236 (1831)
7. 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."
8. [[P. G. Tait]] (1890) [https://archive.org/details/117770257/page/102 An elementary treatise on quaternions, edition 3] via [[Internet Archive]]
9. Heaviside (1891), [http://www.fisicafundamental.net/relicario/doc/heaviside.pdf ''On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field.''] Printed in ''[[Philosophical Transactions of the Royal Society]]'', 1892.
10. [[Michael J. Crowe]]. (1967). "A History of Vector Analysis".
11. (1901). "Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs by Edwin Bidwell Wilson".
12. John D. Anderson (1984), Fundamentals of Aerodynamics, Chapter 2. McGraw-Hill, ISBN 0-07-001656-9