Del, or nabla, is an operator used in mathematics (particularly in

^{} with coordinates $(x\_1,\; \backslash dots,\; x\_n)$ and ^{3} with coordinates $(x,\; y,\; z)$ and standard basis or unit vectors of axes $\backslash $, del is written as
:$\backslash nabla\; =\; \backslash vec\; e\_x\; +\; \backslash vec\; e\_y\; +\; \backslash vec\; e\_z\; =\; \backslash left(,\; ,\; \backslash right)$
:Example:
:$f(x,\; y,\; z)\; =\; x\; +\; y\; +\; z$
:$\backslash nabla\; f\; =\; \backslash vec\; e\_x\; +\; \backslash vec\; e\_y\; +\; \backslash vec\; e\_z\; =\; \backslash left(1,\; 1,\; 1\; \backslash right)$
:
Del can also be expressed in other coordinate systems, see for example

A survey of the improper use of ∇ in vector analysis

(1994) Tai, Chen Vector calculus Mathematical notation

vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in ...

) as a vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

differential operator
300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator.
In mathematics, a differential operator is an Operator (mathe ...

, usually represented by the nabla symbol
∇
The nabla symbol
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, a landform at the mouth of a river
* D (NATO phonetic ...

∇. When applied to a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

defined on a one-dimensional
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...

domain, it denotes the standard derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the function as defined in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

. When applied to a ''field'' (a function defined on a multi-dimensional domain), it may denote any one of three operators depending on the way it is applied: the gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

or (locally) steepest slope of a scalar field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

(or sometimes of a vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

, as in the Navier–Stokes equations
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

); the divergence
In vector calculus
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 ...

of a vector field; or the curl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the p ...

(rotation) of a vector field.
Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation
Mathematical notation is a system of symbol
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or prag ...

for those three operators that makes many equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

easier to write and remember. The del symbol (or nabla) can be interpreted as a vector of partial derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

operators; and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

, and a cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

, respectively, of the "del operator" with the field. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
* Gradient: $\backslash operatornamef\; =\; \backslash nabla\; f$
* Divergence: $\backslash operatorname\backslash vec\; v\; =\; \backslash nabla\; \backslash cdot\; \backslash vec\; v$
* Curl: $\backslash operatorname\backslash vec\; v\; =\; \backslash nabla\; \backslash times\; \backslash vec\; v$
Definition

In theCartesian coordinate system
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early fly ...

Rstandard basis
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

$\backslash $, del is defined in terms of partial derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

operators as
:$\backslash nabla\; =\; \backslash sum\_^n\; \backslash vec\; e\_i\; =\; \backslash left(,\; \backslash ldots,\; \backslash right)$
In three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...

Cartesian coordinate system Rdel in cylindrical and spherical coordinatesThis is a list of some vector calculus
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 calculu ...

.
Notational uses

Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for thegradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

, divergence
In vector calculus
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 ...

, curl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the p ...

, directional derivative
In mathematics, the directional derivative of a multivariate differentiable function, differentiable (scalar) function along a given vector (mathematics), vector v at a given point x intuitively represents the instantaneous rate of change of the ...

, and Laplacian
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
Gradient

The vector derivative of ascalar field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

$f$ is called the gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

, and it can be represented as:
: $\backslash operatornamef\; =\; \backslash vec\; e\_x\; +\; \backslash vec\; e\_y\; +\; \backslash vec\; e\_z=\backslash nabla\; f$
It always points in the direction
Direction may refer to:
*Relative direction, for instance left, right, forward, backwards, up, and down
** Anatomical terms of location for those used in anatomy
*Cardinal direction
Mathematics and science
*Direction vector, a unit vector that ...

of greatest increase of $f$, and it has a magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...

equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane $h(x,y)$, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
: $\backslash nabla(f\; g)\; =\; f\; \backslash nabla\; g\; +\; g\; \backslash nabla\; f$
However, the rules for dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

s do not turn out to be simple, as illustrated by:
: $\backslash nabla\; (\backslash vec\; u\; \backslash cdot\; \backslash vec\; v)\; =\; (\backslash vec\; u\; \backslash cdot\; \backslash nabla)\; \backslash vec\; v\; +\; (\backslash vec\; v\; \backslash cdot\; \backslash nabla)\; \backslash vec\; u\; +\; \backslash vec\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; +\; \backslash vec\; v\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash vec\; u)$
Divergence

Thedivergence
In vector calculus
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 ...

of a vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

$\backslash vec\; v(x,\; y,\; z)\; =\; v\_x\; \backslash vec\; e\_x\; +\; v\_y\; \backslash vec\; e\_y\; +\; v\_z\; \backslash vec\; e\_z$ is a scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

function that can be represented as:
:$\backslash operatorname\backslash vec\; v\; =\; +\; +\; =\; \backslash nabla\; \backslash cdot\; \backslash vec\; v$
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.
The power of the del notation is shown by the following product rule:
:$\backslash nabla\; \backslash cdot\; (f\; \backslash vec\; v)\; =\; (\backslash nabla\; f)\; \backslash cdot\; \backslash vec\; v\; +\; f\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)$
The formula for the vector product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two Euclidean vector, vectors in three-dimensional space \mathbb^3, and is denoted by the sym ...

is slightly less intuitive, because this product is not commutative:
:$\backslash nabla\; \backslash cdot\; (\backslash vec\; u\; \backslash times\; \backslash vec\; v)\; =\; (\backslash nabla\; \backslash times\; \backslash vec\; u)\; \backslash cdot\; \backslash vec\; v\; -\; \backslash vec\; u\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash vec\; v)$
Curl

Thecurl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the p ...

of a vector field $\backslash vec\; v(x,\; y,\; z)\; =\; v\_x\backslash vec\; e\_x\; +\; v\_y\backslash vec\; e\_y\; +\; v\_z\backslash vec\; e\_z$ is a vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

function that can be represented as:
:$\backslash operatorname\backslash vec\; v\; =\; \backslash left(\; -\; \backslash right)\; \backslash vec\; e\_x\; +\; \backslash left(\; -\; \backslash right)\; \backslash vec\; e\_y\; +\; \backslash left(\; -\; \backslash right)\; \backslash vec\; e\_z\; =\; \backslash nabla\; \backslash times\; \backslash vec\; v$
The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centred at that point.
The vector product operation can be visualized as a pseudo-determinant
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

:
:$\backslash nabla\; \backslash times\; \backslash vec\; v\; =\; \backslash left,\; \backslash begin\; \backslash vec\; e\_x\; \&\; \backslash vec\; e\_y\; \&\; \backslash vec\; e\_z\; \backslash \backslash $ & & \\v_x & v_y & v_z \end\
Again the power of the notation is shown by the product rule:
:$\backslash nabla\; \backslash times\; (f\; \backslash vec\; v)\; =\; (\backslash nabla\; f)\; \backslash times\; \backslash vec\; v\; +\; f\; (\backslash nabla\; \backslash times\; \backslash vec\; v)$
Unfortunately the rule for the vector product does not turn out to be simple:
:$\backslash nabla\; \backslash times\; (\backslash vec\; u\; \backslash times\; \backslash vec\; v)\; =\; \backslash vec\; u\; \backslash ,\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; -\; \backslash vec\; v\; \backslash ,\; (\backslash nabla\; \backslash cdot\; \backslash vec\; u)\; +\; (\backslash vec\; v\; \backslash cdot\; \backslash nabla)\; \backslash ,\; \backslash vec\; u\; -\; (\backslash vec\; u\; \backslash cdot\; \backslash nabla)\; \backslash ,\; \backslash vec\; v$
Directional derivative

Thedirectional derivative
In mathematics, the directional derivative of a multivariate differentiable function, differentiable (scalar) function along a given vector (mathematics), vector v at a given point x intuitively represents the instantaneous rate of change of the ...

of a scalar field $f(x,y,z)$ in the direction
$\backslash vec\; a(x,y,z)\; =\; a\_x\; \backslash vec\; e\_x\; +\; a\_y\; \backslash vec\; e\_y\; +\; a\_z\; \backslash vec\; e\_z$ is defined as:
:$\backslash vec\; a\backslash cdot\backslash operatornamef\; =\; a\_x\; +\; a\_y\; +\; a\_z\; =\; \backslash vec\; a\; \backslash cdot\; (\backslash nabla\; f)$
This gives the rate of change of a field $f$ in the direction of $\backslash vec\; a$, scaled by the magnitude of $\backslash vec\; a$. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

uses this convention extensively, terming it the convective derivative
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid
In physics, a fluid is a substance that continually Deformation (mechan ...

—the "moving" derivative of the fluid.
Note that $(\backslash vec\; a\; \backslash cdot\; \backslash nabla)$ is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
Laplacian

TheLaplace operator
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
: $\backslash Delta\; =\; +\; +\; =\; \backslash nabla\; \backslash cdot\; \backslash nabla\; =\; \backslash nabla^2$
and the definition for more general coordinate systems is given in vector Laplacian.
The Laplacian is ubiquitous throughout modern mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

, appearing for example in Laplace's equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

, the heat equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, the wave equation
The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and .
Historically, the problem of a such as that of a was studied by , , , and ...

, and the Schrödinger equation
The Schrödinger equation is a linear
Linearity is the property of a mathematical relationship (''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a ma ...

.
Hessian matrix

While $\backslash nabla^2$ usually represents theLaplacian
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, sometimes $\backslash nabla^2$ also represents the Hessian matrix
In mathematic
Mathematics (from Greek: ) includes the study of such topics as quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in te ...

. The former refers to the inner product of $\backslash nabla$, while the latter refers to the dyadic product of $\backslash nabla$:
: $\backslash nabla^2\; =\; \backslash nabla\; \backslash cdot\; \backslash nabla^T$.
So whether $\backslash nabla^2$ refers to a Laplacian or a Hessian matrix depends on the context.
Tensor derivative

Del can also be applied to a vector field with the result being atensor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. The tensor derivative of a vector field $\backslash vec$ (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as $\backslash nabla\; \backslash otimes\; \backslash vec$, where $\backslash otimes$ represents the dyadic productIn mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second Tensor (intrinsic definition)#Definition via tensor products of vector spaces, order tensor, written in a notation that fits in with vector algebra.
There are nu ...

. This quantity is equivalent to the transpose of the Jacobian matrix
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Produ ...

of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band)
Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...

of this matrix.
For a small displacement $\backslash delta\; \backslash vec$, the change in the vector field is given by:
: $\backslash delta\; \backslash vec\; =\; (\backslash nabla\; \backslash otimes\; \backslash vec)^T\; \backslash sdot\; \backslash delta\; \backslash vec$
Product rules

Forvector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in ...

:
:$\backslash begin\; \backslash nabla\; (fg)\; \&=\; f\backslash nabla\; g\; +\; g\backslash nabla\; f\; \backslash \backslash \; \backslash nabla(\backslash vec\; u\; \backslash cdot\; \backslash vec\; v)\; \&=\; \backslash vec\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; +\; \backslash vec\; v\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash vec\; u)\; +\; (\backslash vec\; u\; \backslash cdot\; \backslash nabla)\; \backslash vec\; v\; +\; (\backslash vec\; v\; \backslash cdot\; \backslash nabla)\backslash vec\; u\; \backslash \backslash \; \backslash nabla\; \backslash cdot\; (f\; \backslash vec\; v)\; \&=\; f\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; +\; \backslash vec\; v\; \backslash cdot\; (\backslash nabla\; f)\; \backslash \backslash \; \backslash nabla\; \backslash cdot\; (\backslash vec\; u\; \backslash times\; \backslash vec\; v)\; \&=\; \backslash vec\; v\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash vec\; u)\; -\; \backslash vec\; u\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; \backslash \backslash \; \backslash nabla\; \backslash times\; (f\; \backslash vec\; v)\; \&=\; (\backslash nabla\; f)\; \backslash times\; \backslash vec\; v\; +\; f\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; \backslash \backslash \; \backslash nabla\; \backslash times\; (\backslash vec\; u\; \backslash times\; \backslash vec\; v)\; \&=\; \backslash vec\; u\; \backslash ,\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; -\; \backslash vec\; v\; \backslash ,\; (\backslash nabla\; \backslash cdot\; \backslash vec\; u)\; +\; (\backslash vec\; v\; \backslash cdot\; \backslash nabla)\; \backslash ,\; \backslash vec\; u\; -\; (\backslash vec\; u\; \backslash cdot\; \backslash nabla)\; \backslash ,\; \backslash vec\; v\; \backslash end$
For matrix calculus
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

(for which $\backslash vec\; u\; \backslash cdot\; \backslash vec\; v$ can be written $\backslash vec\; u^\backslash text\; \backslash vec\; v$):
:$\backslash begin\; \backslash left(\backslash mathbf\backslash nabla\backslash right)^\backslash text\; \backslash vec\; u\; \&=\; \backslash nabla^\backslash text\; \backslash left(\backslash mathbf^\backslash text\backslash vec\; u\backslash right)\; -\; \backslash left(\backslash nabla^\backslash text\; \backslash mathbf^\backslash text\backslash right)\; \backslash vec\; u\; \backslash end$
Another relation of interest (see e.g. '' Euler equations'') is the following, where $\backslash vec\; u\; \backslash otimes\; \backslash vec\; v$ is the outer product
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

tensor:
:$\backslash begin\; \backslash nabla\; \backslash cdot\; (\backslash vec\; u\; \backslash otimes\; \backslash vec\; v)\; =\; (\backslash nabla\; \backslash cdot\; \backslash vec\; u)\; \backslash vec\; v\; +\; (\backslash vec\; u\; \backslash cdot\; \backslash nabla)\; \backslash vec\; v\; \backslash end$
Second derivatives

When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field ''f'' or a vector field ''v''; the use of the scalarLaplacian
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

and vector Laplacian gives two more:
: $\backslash begin\; \backslash operatorname(\backslash operatornamef)\; \&=\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; f)\; \backslash \backslash \; \backslash operatorname(\backslash operatornamef)\; \&=\; \backslash nabla\; \backslash times\; (\backslash nabla\; f)\; \backslash \backslash \; \backslash Delta\; f\; \&=\; \backslash nabla^2\; f\; \backslash \backslash \; \backslash operatorname(\backslash operatorname\backslash vec\; v)\; \&=\; \backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; \backslash \backslash \; \backslash operatorname(\backslash operatorname\backslash vec\; v)\; \&=\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; \backslash \backslash \; \backslash operatorname(\backslash operatorname\backslash vec\; v)\; \&=\; \backslash nabla\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; \backslash \backslash \; \backslash Delta\; \backslash vec\; v\; \&=\; \backslash nabla^2\; \backslash vec\; v\; \backslash end$
These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved
In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is ...

, two of them are always zero:
: $\backslash begin\; \backslash operatorname(\backslash operatornamef)\; \&=\; \backslash nabla\; \backslash times\; (\backslash nabla\; f)\; =\; 0\; \backslash \backslash \; \backslash operatorname(\backslash operatorname\backslash vec\; v)\; \&=\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash vec\; v)\; =\; 0\; \backslash end$
Two of them are always equal:
: $\backslash operatorname(\backslash operatornamef)\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; f)\; =\; \backslash nabla^2\; f\; =\; \backslash Delta\; f$
The 3 remaining vector derivatives are related by the equation:
:$\backslash nabla\; \backslash times\; \backslash left(\backslash nabla\; \backslash times\; \backslash vec\; v\backslash right)\; =\; \backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; -\; \backslash nabla^2\; \backslash vec$
And one of them can even be expressed with the tensor product, if the functions are well-behaved:
: $\backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; =\; \backslash nabla\; \backslash cdot\; (\backslash vec\; v\; \backslash otimes\; \backslash nabla\; )$
Precautions

Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector. Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is ''not'' necessarily reliable, because del does not commute in general. A counterexample that relies on del's failure to commute: :$\backslash begin\; (\backslash vec\; u\; \backslash cdot\; \backslash vec\; v)\; f\; \&\backslash equiv\; (\backslash vec\; v\; \backslash cdot\; \backslash vec\; u)\; f\; \backslash \backslash \; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; f\; \&=\; \backslash left\; (\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; \backslash right)f\; =\; \backslash fracf\; +\; \backslash fracf\; +\; \backslash fracf\; \backslash \backslash \; (\backslash vec\; v\; \backslash cdot\; \backslash nabla)\; f\; \&=\; \backslash left\; (v\_x\; \backslash frac\; +\; v\_y\; \backslash frac\; +\; v\_z\; \backslash frac\; \backslash right)f\; =\; v\_x\; \backslash frac\; +\; v\_y\; \backslash frac\; +\; v\_z\; \backslash frac\; \backslash \backslash \; \backslash Rightarrow\; (\backslash nabla\; \backslash cdot\; \backslash vec\; v)\; f\; \&\backslash ne\; (\backslash vec\; v\; \backslash cdot\; \backslash nabla)\; f\; \backslash \backslash \; \backslash end$ A counterexample that relies on del's differential properties: : $\backslash begin\; (\backslash nabla\; x)\; \backslash times\; (\backslash nabla\; y)\; \&=\; \backslash left\; (\backslash vec\; e\_x\; \backslash frac+\backslash vec\; e\_y\; \backslash frac+\backslash vec\; e\_z\; \backslash frac\; \backslash right)\; \backslash times\; \backslash left\; (\backslash vec\; e\_x\; \backslash frac+\backslash vec\; e\_y\; \backslash frac+\backslash vec\; e\_z\; \backslash frac\; \backslash right)\; \backslash \backslash \; \&=\; (\backslash vec\; e\_x\; \backslash cdot\; 1\; +\backslash vec\; e\_y\; \backslash cdot\; 0+\backslash vec\; e\_z\; \backslash cdot\; 0)\; \backslash times\; (\backslash vec\; e\_x\; \backslash cdot\; 0+\backslash vec\; e\_y\; \backslash cdot\; 1+\backslash vec\; e\_z\; \backslash cdot\; 0)\; \backslash \backslash \; \&=\; \backslash vec\; e\_x\; \backslash times\; \backslash vec\; e\_y\; \backslash \backslash \; \&=\; \backslash vec\; e\_z\; \backslash \backslash \; (\backslash vec\; u\; x)\backslash times\; (\backslash vec\; u\; y)\; \&=\; x\; y\; (\backslash vec\; u\; \backslash times\; \backslash vec\; u)\; \backslash \backslash \; \&=\; x\; y\; \backslash vec\; 0\; \backslash \backslash \; \&=\; \backslash vec\; 0\; \backslash end$ Central to these distinctions is the fact that del is not simply a vector; it is avector operator
A vector operator is a differential operator used in vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \m ...

. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.
For that reason, identities involving del must be derived with care, using both vector identities and ''differentiation'' identities such as the product rule.
See also

*Del in cylindrical and spherical coordinatesThis is a list of some vector calculus
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 calculu ...

* Notation for differentiation
In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function (mathematics), function or dependent variable, variable have been proposed by different mathem ...

* Vector calculus identities
The following are important identities involving derivatives and integrals in vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), th ...

* Maxwell's equations
Maxwell's equations are a set of coupled partial differential equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

* Navier–Stokes equations
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

* Table of mathematical symbols
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that ...

* Quabla operator
References

*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
Thermodynamics is ...

& Edwin Bidwell Wilson
Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...

(1901) 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 ...

, Yale University Press
Yale University Press is a university press
A university press is an academic publishing
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditional ...

, 1960: Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term ...

.
*
*
* {{cite web , author=Arnold Neumaier , editor=Cleve Moler , url=http://www.netlib.org/na-digest-html/98/v98n03.html#2 , title=History of Nabla , series=NA Digest, Volume 98, Issue 03 , publisher=netlib.org , date=January 26, 1998
External links

A survey of the improper use of ∇ in vector analysis

(1994) Tai, Chen Vector calculus Mathematical notation

Differential operators
{{Cat main, Differential operator
In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider Derivative, differentiation as an abstr ...