The nabla is a triangular symbol resembling an inverted Greek
delta
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, at a river mouth
* D ( NATO phonetic alphabet: "Delta")
* Delta Air Lines, US
* Delta variant of SARS-CoV-2 that causes COVID-19
Delta may also ...
:
[Indeed, it is called ( ανάδελτα) in ]Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek language spoken in the modern era, including the official standardized form of the ...
. or ∇. The name comes, by reason of the symbol's shape, from the
Hellenistic Greek
Koine Greek (; Koine el, ἡ κοινὴ διάλεκτος, hē koinè diálektos, the common dialect; ), also known as Hellenistic Greek, common Attic, the Alexandrian dialect, Biblical Greek or New Testament Greek, was the common supra-reg ...
word for a
Phoenician harp,
and was suggested by the encyclopedist
William Robertson Smith
William Robertson Smith (8 November 184631 March 1894) was a Scottish orientalist, Old Testament scholar, professor of divinity, and minister of the Free Church of Scotland. He was an editor of the ''Encyclopædia Britannica'' and contributo ...
to
Peter Guthrie Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...
in correspondence.
[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".[Notably it is sometimes claimed to be from the ]Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...
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:
The nabla symbol is available in standard
HTML
The HyperText Markup Language or HTML is the standard markup language for documents designed to be displayed in a web browser. It can be assisted by technologies such as Cascading Style Sheets (CSS) and scripting languages such as JavaScri ...
as
∇
and in
LaTeX
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
as
\nabla
. In
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
, it is the character at
code point
In character encoding terminology, a code point, codepoint or code position is a numerical value that maps to a specific character. Code points usually represent a single grapheme—usually a letter, digit, punctuation mark, or whitespace—but ...
U+2207, or 8711 in
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
notation, in the
Mathematical Operators
Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation.
Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Basi ...
block.
It is also called
del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
.
History
The , the instrument after which the nabla symbol is named">harp, the instrument after which the nabla symbol is named
The
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
given in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
on three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
by
was introduced in 1837 by the
Irish
Irish may refer to:
Common meanings
* Someone or something of, from, or related to:
** Ireland, an island situated off the north-western coast of continental Europe
***Éire, Irish language name for the isle
** Northern Ireland, a constituent unit ...
mathematician and physicist
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
, who called it ◁.
[W. R. Hamilton,]
On Differences and Differentials of Functions of Zero
" ''Trans. R. Irish Acad.'' XVII:235–236 esp. 236 (1837) (The unit vectors
were originally
right versor
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction ve ...
s in Hamilton's
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s.) The mathematics of ∇ received its full exposition at the hands of
P. G. Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote w ...
.
[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."]
After receiving Smith's suggestion, Tait and
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
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.
[
As this is written, he appears to be naming the ]Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
∇2 "nabla", but in the lecture was presumably referring to ∇ itself.
The name is acknowledged, and criticized, by
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
in 1891:
[Heaviside (1891)]
''On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field.''
Printed in ''Philosophical Transactions of the Royal Society
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...
'', 1892.
The fictitious vector ∇ given by
is ''very'' important. Physical mathematics is very largely the mathematics of ∇. The name Nabla seems, therefore, ludicrously inefficient.
Heaviside and
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
(independently) are credited with the development of the version of vector calculus most popular today.
The influential 1901 text ''
Vector Analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
'', written by
Edwin Bidwell Wilson
Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...
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
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
as part of the names of three distinct differential operators: the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
(∇), the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
(∇⋅), and the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
(∇×). The last of these uses the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
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
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
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
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
.
* As the
backward difference operator
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
, in the
calculus of finite differences
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
.
* As the widening operator, an operator that permits static analysis of programs to terminate in finite time, in the
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
field of
abstract interpretation
In computer science, abstract interpretation is a theory of sound approximation of the semantics of computer programs, based on monotonic functions over ordered sets, especially lattices. It can be viewed as a partial execution of a computer prog ...
.
* As function definition marker and self-reference (
recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
) in the
APL programming language
APL (named after the book ''A Programming Language'') is a programming language developed in the 1960s by Kenneth E. Iverson. Its central datatype is the multidimensional array. It uses a large range of special graphic symbols to represent mos ...
* As an indicator of
indeterminacy in
philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
.
[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''">sup>''f'' ''a'' = ''b''sup>''f''.
Here, the brackets and superscript ''f''s together serve to denote fictitiousness; thus the nabla says "It is indeterminate whether", and the rest says "''a''=''b'' (fictively)."
* In
naval architecture
Naval architecture, or naval engineering, is an engineering discipline incorporating elements of mechanical, electrical, electronic, software and safety engineering as applied to the engineering design process, shipbuilding, maintenance, and o ...
(ship design), to designate the volume
displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
of a ship or any other waterborne vessel; the graphically similar
delta
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, at a river mouth
* D ( NATO phonetic alphabet: "Delta")
* Delta Air Lines, US
* Delta variant of SARS-CoV-2 that causes COVID-19
Delta may also ...
is used to designate weight displacement (the total weight of water displaced by the ship), thus
where
is the density of seawater.
See also
*
Del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
, treating the mathematics of the vector differential operator
*
Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae for working with common curvilinear coordinates, curvilinear coordinate systems.
Notes
* This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11#Coordinate systems, ISO 31- ...
*
grad,
div
Div or DIV may refer to:
Science and technology
* Division (mathematics), the mathematical operation that is the inverse of multiplication
* Span and div, HTML tags that implement generic elements
* div, a C mathematical function
* Divergence, ...
, and
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
, differential operators defined using nabla
*
History of quaternions
*
Notation for differentiation
In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with t ...
*
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
, also known as
connection
*
Nevel
Footnotes
External links
*
*
*
*Tai, Chen
A survey of the improper use of ∇ in vector analysis(1994).
{{DEFAULTSORT:Nabla Symbol
Mathematical symbols
Differential operators
William Rowan Hamilton