Nabla symbol

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 ${\displaystyle \{x,y,z\}}$ on three-dimensional Euclidean space by

$${\displaystyle \mathbf{i}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+\mathbf{j}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+\mathbf{k}\frac{\mathrm{\partial }}{\mathrm{\partial }z}}$$

was introduced in 1831 by the Irish mathematician and physicist William Rowan Hamilton, who called it ◁. (The unit vectors ${\displaystyle \{\mathbf{i},\mathbf{j},\mathbf{k}\}}$ 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):

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.

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

The fictitious vector ∇ given by

$${\displaystyle \mathrm{\nabla }=\mathbf{i}{\mathrm{\nabla }}_1+\mathbf{j}{\mathrm{\nabla }}_2+\mathbf{k}{\mathrm{\nabla }}_3=\mathbf{i}\frac{d}{dx}+\mathbf{j}\frac{d}{dy}+\mathbf{k}\frac{d}{dz}}$$

is 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.

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.