Nabla symbol

∇

The nabla symbol

The nabla is a triangular symbol resembling an inverted Greek delta:^[1] $\nabla$ or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp,^[2]^[3] and was suggested by the encyclopedist William Robertson Smith in an 1870 letter to Peter Guthrie Tait.^[2]^[4]^[5]^[6]^[7]

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 an operator, it is often called del.

History

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

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

was introduced in 1831 by the Irish mathematician and physicist William Rowan Hamilton, who called it ◁.^[8] (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.^[9]^[10]

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):^[5]

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;^[2] the notes were published in Britain and the U.S. in 1904.^[11]

The name is acknowledged, and criticized, by Oliver Heaviside in 1891:^[12]

The fictitious vector ∇ given by

$\nabla =\mathbf {i} \nabla _{1}+\mathbf {j} \nabla _{2}+\mathbf {k} \nabla _{3}=\mathbf {i} {\frac {d}{dx}}+\mathbf {j} {\frac {d}{dy}}+\mathbf {k} {\frac {d}{dz}}$
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.^[13]

The influential 1901 text Vector Analysis, written by Edwin Bidwell Wilson and based on the lectures of Gibbs, advocates the name "del":^[14]

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.^[15]
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.

Footnotes

^ Indeed, it is called anadelta (ανάδελτα) in Modern Greek.
^ ^a ^b ^c "nabla". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
^ νάβλα. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project.
^ 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".
^ ^a ^b Cargill Gilston Knott (1911). Life and Scientific Work of Peter Guthrie Tait. Cambridge University Press.
^ "History of Nabla".
^ ^a ^b 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: "nable". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
^ W. R. Hamilton, "On Differences and Differentials of Functions of Zero," Trans. R. Irish Acad. XVII:235–236 esp. 236 (1831)
^ 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."
^ P. G. Tait (1890) An elementary treatise on quaternions, edition 3 via Internet Archive
^ William Thomson, Lord Kelvin (1904). Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. I took the liberty of asking Professor Ball two days ago whether he had a name for this symbol ∇², and he has mentioned to me nabla, a humorous suggestion of Maxwell's. It is the name of an Egyptian harp, which was of that shape. I do not know that it is a bad name for it. Laplacian I do not like for several reasons both historical and phonetic. [Jan. 22 1892. Since 1884 I have found nothing better, and I now call it Laplacian.] As this is written, he appears to be naming the Laplacian ∇² "nabla", but in the lecture was presumably referring to ∇ itself.
^ Heaviside (1891), On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field. Printed in Philosophical Transactions of the Royal Society, 1892.
^ Michael J. Crowe (1967). A History of Vector Analysis.
^ Gibbs; Wilson (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.
^ For example, in Anthony Everett (2013), The Nonexistent, p. 210:
We can represent cases of this form, cases where it is indeterminate whether in fiction f: a=b, as follows:
(A) ∇[^f a = b]^f.
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)."

External links

Arnold Neumaier (2004). "History of Nabla".
Arnold Neumaier (January 26, 1998). Cleve Moler (ed.). "History of Nabla". NA Digest, Volume 98, Issue 03. netlib.org.
Miller, Jeff. "Earliest Uses of Symbols of Calculus".
Tai, Chen. A survey of the improper use of ∇ in vector analysis (1994).

[anadelta-1] Indeed, it is called anadelta (ανάδελτα) in Modern Greek.

[OED-2] "nabla". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

[3] νάβλα. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project.

[Smith-4] 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".

[Knott-5] Cargill Gilston Knott (1911). Life and Scientific Work of Peter Guthrie Tait. Cambridge University Press.

[Neumaier-6] "History of Nabla".

[nabel-7] 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: "nable". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

[Hamilton-8] W. R. Hamilton, "On Differences and Differentials of Functions of Zero," Trans. R. Irish Acad. XVII:235–236 esp. 236 (1831)

[Knott2-9] 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."

[10] P. G. Tait (1890) An elementary treatise on quaternions, edition 3 via Internet Archive

[Kelvin-11] William Thomson, Lord Kelvin (1904). Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. I took the liberty of asking Professor Ball two days ago whether he had a name for this symbol ∇², and he has mentioned to me nabla, a humorous suggestion of Maxwell's. It is the name of an Egyptian harp, which was of that shape. I do not know that it is a bad name for it. Laplacian I do not like for several reasons both historical and phonetic. [Jan. 22 1892. Since 1884 I have found nothing better, and I now call it Laplacian.] As this is written, he appears to be naming the Laplacian ∇² "nabla", but in the lecture was presumably referring to ∇ itself.

[Heaviside-12] Heaviside (1891), On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field. Printed in Philosophical Transactions of the Royal Society, 1892.

[Crowe-13] Michael J. Crowe (1967). A History of Vector Analysis.

[14] Gibbs; Wilson (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.

[phil-15] For example, in Anthony Everett (2013), The Nonexistent, p. 210:
We can represent cases of this form, cases where it is indeterminate whether in fiction f: a=b, as follows:
(A) ∇[^f a = b]^f.
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)."

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History

Modern uses

See also

Footnotes

External links