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Del, or nabla, is an operator used in mathematics (particularly 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 subjec ...
) as a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
differential operator, usually represented by the
nabla symbol The nabla symbol The nabla is a triangular symbol resembling an inverted Greek delta:Indeed, it is called ( ανάδελτα) in Modern Greek. \nabla or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word ...
∇. When applied to a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
defined on a
one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...
domain, it denotes the standard
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the function as defined in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. 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, 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 gr ...
or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
); 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 t ...
of a vector field; or the curl (rotation) of a vector field. Strictly speaking, del is not a specific operator, but rather a convenient
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
for those three operators that makes many equations easier to write and remember. The del symbol (or nabla) can be interpreted as a vector of partial derivative operators; and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
with a scalar, a
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, and a cross product, 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: \operatornamef = \nabla f * Divergence: \operatorname\vec v = \nabla \cdot \vec v * Curl: \operatorname\vec v = \nabla \times \vec v


Definition

In the Cartesian coordinate system R with coordinates (x_1, \dots, x_n) and standard basis \, del is defined in terms of partial derivative operators as : \nabla = \sum_^n \vec e_i = \left(, \ldots, \right) Where the expression in parentheses is a row vector. In
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
Cartesian coordinate system R3 with coordinates (x, y, z) and standard basis or unit vectors of axes \, del is written as :\nabla = \vec e_x + \vec e_y + \vec e_z = \left(, , \right) :Example: :f(x, y, z) = x + y + z :\nabla f = \vec e_x + \vec e_y + \vec e_z = \left(1, 1, 1 \right) : Del can also be expressed in other coordinate systems, see for example
del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reve ...
.


Notational uses

Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for 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 gr ...
,
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 t ...
, curl,
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
, and Laplacian.


Gradient

The vector derivative of a scalar field f is called 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 gr ...
, and it can be represented as: : \operatornamef = \vec e_x + \vec e_y + \vec e_z=\nabla f It always points in the direction 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: : \nabla(f g) = f \nabla g + g \nabla f However, the rules for
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
s do not turn out to be simple, as illustrated by: : \nabla (\vec u \cdot \vec v) = (\vec u \cdot \nabla) \vec v + (\vec v \cdot \nabla) \vec u + \vec u \times (\nabla \times \vec v) + \vec v \times (\nabla \times \vec u)


Divergence

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 t ...
of a vector field \vec v(x, y, z) = v_x \vec e_x + v_y \vec e_y + v_z \vec e_z is a scalar field that can be represented as: :\operatorname\vec v = + + = \nabla \cdot \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: : \nabla \cdot (f \vec v) = (\nabla f) \cdot \vec v + f (\nabla \cdot \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 vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
is slightly less intuitive, because this product is not commutative: : \nabla \cdot (\vec u \times \vec v) = (\nabla \times \vec u) \cdot \vec v - \vec u \cdot (\nabla \times \vec v)


Curl

The curl of a vector field \vec v(x, y, z) = v_x\vec e_x + v_y\vec e_y + v_z\vec e_z is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
function that can be represented as: :\operatorname\vec v = \left( - \right) \vec e_x + \left( - \right) \vec e_y + \left( - \right) \vec e_z = \nabla \times \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, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
: :\nabla \times \vec v = \left, \begin \vec e_x & \vec e_y & \vec e_z \\ pt & & \\ ptv_x & v_y & v_z \end\ Again the power of the notation is shown by the product rule: :\nabla \times (f \vec v) = (\nabla f) \times \vec v + f (\nabla \times \vec v) Unfortunately the rule for the vector product does not turn out to be simple: :\nabla \times (\vec u \times \vec v) = \vec u \, (\nabla \cdot \vec v) - \vec v \, (\nabla \cdot \vec u) + (\vec v \cdot \nabla) \, \vec u - (\vec u \cdot \nabla) \, \vec v


Directional derivative

The
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
of a scalar field f(x,y,z) in the direction \vec a(x,y,z) = a_x \vec e_x + a_y \vec e_y + a_z \vec e_z is defined as: :\vec a\cdot\operatornamef = a_x + a_y + a_z = \vec a \cdot (\nabla f) This gives the rate of change of a field f in the direction of \vec a, scaled by the magnitude of \vec a. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics 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, most commonly density and gravity (see buoyancy). When the cause of the convecti ...
—the "moving" derivative of the fluid. Note that (\vec a \cdot \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

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as: : \Delta = + + = \nabla \cdot \nabla = \nabla^2 and the definition for more general coordinate systems is given in
vector 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 ...
. 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,
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, the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
, and the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
.


Hessian matrix

While \nabla^2 usually represents the Laplacian, sometimes \nabla^2 also represents the
Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
. The former refers to the inner product of \nabla, while the latter refers to the dyadic product of \nabla: : \nabla^2 = \nabla \cdot \nabla^T. So whether \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 a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
. The
tensor 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 ...
of a vector field \vec (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as \nabla \otimes \vec, where \otimes represents the
dyadic product In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two v ...
. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix. For a small displacement \delta \vec, the change in the vector field is given by: : \delta \vec = (\nabla \otimes \vec)^T \sdot \delta \vec


Product rules

For
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 subjec ...
: :\begin \nabla (fg) &= f\nabla g + g\nabla f \\ \nabla(\vec u \cdot \vec v) &= \vec u \times (\nabla \times \vec v) + \vec v \times (\nabla \times \vec u) + (\vec u \cdot \nabla) \vec v + (\vec v \cdot \nabla)\vec u \\ \nabla \cdot (f \vec v) &= f (\nabla \cdot \vec v) + \vec v \cdot (\nabla f) \\ \nabla \cdot (\vec u \times \vec v) &= \vec v \cdot (\nabla \times \vec u) - \vec u \cdot (\nabla \times \vec v) \\ \nabla \times (f \vec v) &= (\nabla f) \times \vec v + f (\nabla \times \vec v) \\ \nabla \times (\vec u \times \vec v) &= \vec u \, (\nabla \cdot \vec v) - \vec v \, (\nabla \cdot \vec u) + (\vec v \cdot \nabla) \, \vec u - (\vec u \cdot \nabla) \, \vec v \end For
matrix calculus In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a ...
(for which \vec u \cdot \vec v can be written \vec u^\text \vec v): :\begin \left(\mathbf\nabla\right)^\text \vec u &= \nabla^\text \left(\mathbf^\text\vec u\right) - \left(\nabla^\text \mathbf^\text\right) \vec u \end Another relation of interest (see e.g. ''
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
'') is the following, where \vec u \otimes \vec v is the outer product tensor: :\begin \nabla \cdot (\vec u \otimes \vec v) = (\nabla \cdot \vec u) \vec v + (\vec u \cdot \nabla) \vec v \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 scalar Laplacian and
vector 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 ...
gives two more: : \begin \operatorname(\operatornamef) &= \nabla \cdot (\nabla f) \\ \operatorname(\operatornamef) &= \nabla \times (\nabla f) \\ \Delta f &= \nabla^2 f \\ \operatorname(\operatorname\vec v) &= \nabla (\nabla \cdot \vec v) \\ \operatorname(\operatorname\vec v) &= \nabla \cdot (\nabla \times \vec v) \\ \operatorname(\operatorname\vec v) &= \nabla \times (\nabla \times \vec v) \\ \Delta \vec v &= \nabla^2 \vec v \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 ( C^\infty in most cases), two of them are always zero: : \begin \operatorname(\operatornamef) &= \nabla \times (\nabla f) = 0 \\ \operatorname(\operatorname\vec v) &= \nabla \cdot (\nabla \times \vec v) = 0 \end Two of them are always equal: : \operatorname(\operatornamef) = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f The 3 remaining vector derivatives are related by the equation: :\nabla \times \left(\nabla \times \vec v\right) = \nabla (\nabla \cdot \vec v) - \nabla^2 \vec And one of them can even be expressed with the tensor product, if the functions are well-behaved: : \nabla (\nabla \cdot \vec v) = \nabla \cdot (\vec v \otimes \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: :\begin (\vec u \cdot \vec v) f &\equiv (\vec v \cdot \vec u) f \\ (\nabla \cdot \vec v) f &= \left (\frac + \frac + \frac \right)f = \fracf + \fracf + \fracf \\ (\vec v \cdot \nabla) f &= \left (v_x \frac + v_y \frac + v_z \frac \right)f = v_x \frac + v_y \frac + v_z \frac \\ \Rightarrow (\nabla \cdot \vec v) f &\ne (\vec v \cdot \nabla) f \\ \end A counterexample that relies on del's differential properties: : \begin (\nabla x) \times (\nabla y) &= \left (\vec e_x \frac+\vec e_y \frac+\vec e_z \frac \right) \times \left (\vec e_x \frac+\vec e_y \frac+\vec e_z \frac \right) \\ &= (\vec e_x \cdot 1 +\vec e_y \cdot 0+\vec e_z \cdot 0) \times (\vec e_x \cdot 0+\vec e_y \cdot 1+\vec e_z \cdot 0) \\ &= \vec e_x \times \vec e_y \\ &= \vec e_z \\ (\vec u x)\times (\vec u y) &= x y (\vec u \times \vec u) \\ &= x y \vec 0 \\ &= \vec 0 \end Central to these distinctions is the fact that del is not simply a vector; it is a
vector operator A vector operator is a differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl: :\begin \operatorname &\equiv \nabla \\ \operatorname &\equiv \nabla \cdot \\ \op ...
. 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 coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reve ...
* Notation for differentiation *
Vector calculus identities The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: \o ...
*
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
*
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
*
Table of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula ...
* 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 was instrumental in ...
&
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 ...
(1901)
Vector Analysis 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 subjec ...
,
Yale University Press Yale University Press is the university press of Yale University. It was founded in 1908 by George Parmly Day, and became an official department of Yale University in 1961, but it remains financially and operationally autonomous. , Yale Universi ...
, 1960:
Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...
. * * * {{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