The nabla symbol
The nabla is a triangular symbol resembling an inverted Greek delta:[1]
\nabla
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.
It is also called del.
\{x,y,z\}
\{i,j,k\}
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):
It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his humorous Tyndallic Ode, which is dedicated to the "Chief Musician upon Nabla", that is, Tait.
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.[10]
The name is acknowledged, and criticized, by Oliver Heaviside in 1891:[11]
The fictitious vector ∇ given byis very important. Physical mathematics is very largely the mathematics of ∇. The name Nabla seems, therefore, ludicrously inefficient.
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.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".
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.:
\nabla=\Delta/\rho
\rho
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".
As this is written, he appears to be naming the Laplacian ∇2 "nabla", but in the lecture was presumably referring to ∇ itself.
We can represent cases of this form, cases where it is indeterminate whether in fiction f: a=b, as follows:Here, the brackets and superscript fs together serve to denote fictitiousness; thus the nabla says "It is indeterminate whether", and the rest says "a=b (fictively)."(A) ∇[<sup>''f''</sup> ''a'' = ''b'']f.