NationMaster - Encyclopedia: Del

In vector calculus, del is a vector differential operator represented by the nabla symbol: ∇. Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... Nabla is a symbol, shown as . ...

Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. Depending on the way del is applied, it can describe the gradient (slope), divergence (degree something converges or diverges) or curl. More intuitive descriptions of each of the many operations del performs can be found below. Convention has at least two very distinct but related meanings. ... Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ... This article is about equations in mathematics. ...

Mathematically, del can be viewed as the derivative in multi-dimensional space. When used in one dimension, it takes the form of the standard derivative of calculus. As an operator, it acts on vector fields and scalar fields with analogues of traditional multiplication. As with all operators, these analogues should not be confused with traditional multiplication; in particular, del does not commute. In mathematics, a derivative is the rate of change of a quantity. ... In mathematics, a derivative is the rate of change of a quantity. ... Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics and physics, a scalar field associates a single number (or scalar) to every point in space. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ...

1 Definition
2 Notational uses of del
3 Second derivatives
4 Precautions
5 See also
6 References
7 External links

Definition

In the three-dimensional Cartesian coordinate system R³ with coordinates (x, y, z), del is defined as Fig. ...

$nabla = mathbf{i}{partial over partial x} + mathbf{j}{partial over partial y} + mathbf{k}{partial over partial z}$

where (i,j,k) is the standard basis in R³. In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...

Though this page chiefly treats del in three dimensions, this definition can be generalized to the n-dimensional Euclidean space Rⁿ. In the Cartesian coordinate system with coordinates (x₁, x₂, …, x_n), del is: Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Fig. ...

$nabla = sum_{i=1}^n vec e_i {partial over partial x_i}$

where ${ vec e_i: 1leq ileq n}$ is the i^th standard basis in this space.

More compactly, using the Einstein summation notation, del is written as For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

$nabla = vec e_i partial_i.$

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 of general use in working with standard coordinate systems. ...

Notational uses of del

Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian. Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

Gradient

The vector derivative of a scalar field f is called the gradient, and it can be represented as: In mathematics and physics, a scalar field associates a scalar to every point in space. ... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ...

$mbox{grad},f = {partial f over partial x} mathbf{i} + {partial f over partial y} mathbf{j} + {partial f over partial z} mathbf{k} = nabla f$

It always points in the direction of greatest increase of f, and it has a magnitude 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 2d-gradient will be a vector in the xy-plane (sort of like an arrow on a map) pointing in the steepest direction. The magnitude of the gradient is the slope of the hill in that steepest direction. The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ...

In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:

$nabla(f g) = f nabla g + g nabla f$

Divergence

The divergence of a vector field v(x,y,z) = v_x i + v_y j + v_z k can be represented as: In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

$mbox{div},vec v = {partial v_x over partial x} + {partial v_y over partial y} + {partial v_z over partial z} = nabla cdot vec v$

The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately a measure of that field's tendency to converge on or repel from a point.

Curl

The curl of a vector field v(x,y,z) = v_x i + v_y j + v_z k can be represented as: In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ...

$mbox{curl};vec v = left( {partial v_z over partial y} - {partial v_y over partial z} right) mathbf{i} + left( {partial v_x over partial z} - {partial v_z over partial x} right) mathbf{j} + left( {partial v_y over partial x} - {partial v_x over partial y} right) mathbf{k} = nabla times vec v$

The curl at a point is proportional to the on-axis torque a tiny pinwheel would feel if it were centered at that point. It is easy to visualise it as a pseudo-determinant

$begin{bmatrix} mathbf{i} & mathbf{j} & mathbf{k} {frac{partial}{partial x}} & {frac{partial}{partial y}} & {frac{partial}{partial z}} F_x & F_y & F_z end{bmatrix}.$

Directional derivative

The directional derivative of a scalar field f(x,y,z) in the direction a(x,y,z) = a_x i + a_y j + a_z k is defined as: In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the...

$vec{a}cdotmbox{grad},f = a_x {partial f over partial x} + a_y {partial f over partial y} + a_z {partial f over partial z} = (vec a cdot nabla) f$

This gives the change of a field f in the direction of 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 — the 'moving' derivative of the fluid. Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... The convective derivative, also known as the Lagrangian derivative, is a derivative taken with a respect to a coordinate system moving with velocity u, and is often used in fluid mechanics. ...

Laplacian

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; it is defined as: In mathematics and physics, the Laplace operator or Laplacian, denoted by Î”, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

$Delta = {partial^2 over partial x^2} + {partial^2 over partial y^2} + {partial^2 over partial z^2} = nabla cdot nabla = nabla^2$

The Laplacian is ubiquitous throughout modern mathematical physics, appearing in Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation — to name a few. Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ...

Tensor derivative

Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field v is a 9-term second-rank tensor, but can be denoted simply as ∇ ⊗ v , where ⊗ represents the dyadic product. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics, in particular multilinear algebra, the dyadic product of a column vector and a row vector is the tensor product of the vectors. ...

For a small displacement δr, the change in the vector field is given by:

δv = (∇ ⊗ v)⋅δr

Second derivatives

When del operates on a scalar or vector, generally a scalar or vector is returned. Because of the diversity of vector products, one application of del already gives rise to three major derivatives — the divergence, gradient, and curl. 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 Laplacian gives two more:

$mbox{div},(mbox{grad},f ) = nabla cdot (nabla f)$

$mbox{curl},(mbox{grad},f ) = nabla times (nabla f)$

$Delta f = nabla^2 f$

$mbox{grad},(mbox{div}, vec v ) = nabla (nabla cdot vec v)$

$mbox{div},(mbox{curl},vec v ) = nabla cdot (nabla times vec v)$

$mbox{curl},(mbox{curl},vec v ) = nabla times (nabla times vec v)$

$Delta vec v = nabla^2 vec v$

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved, two of them are always zero: Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...

$mbox{curl},(mbox{grad},f ) = nabla times (nabla f) = 0$

$mbox{div},(mbox{curl},vec v ) = nabla cdot nabla times vec{v} = 0.$

Two of them are always equal:

$mbox{div},(mbox{grad},f ) = nabla cdot (nabla f) = nabla^2 f = Delta f$

The 3 remaining vector derivatives are related by the equation:

$nabla times nabla times vec{v} = nabla (nabla cdot vec{v}) - nabla^2 vec{v}$

And one of them can even be expressed with the tensor product, if the functions are well-behaved:

$nabla ( nabla cdot vec{v} ) = nabla cdot (nabla otimes vec{v})$

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 del is replaced by any other vector. This is part of the tremendous value gained in representing this operator as a vector in its own right.

Though you can often replace del with a vector and obtain a vector identity, making those identities intuitive, the reverse is not necessarily reliable, because del does not often commute.

A counterexample that relies on del's failure to commute:

$vec u cdot vec v = vec v cdot vec u$

$nabla cdot vec v ne vec v cdot nabla$

A counterexample that relies on del's differential properties:

$(nabla x) times (nabla y) = mathbf{k}$

$(vec u x )times (vec u y) = mathbf{0}$

Central to these distinctions is the fact that del is not a vector — it is a vector operator. Whereas a vector is an object with both a precise numerical magnitude and direction, del doesn't have a precise value for either until it is allowed to operate on something.

For that reason, identities involving del must be derived from scratch, not derived from pre-existing vector identities.

References

Div, Grad, Curl, and All That, H. M. Schey, ISBN 0-393-96997-5
Jeff Miller, Earliest Uses of Symbols of Calculus (Aug. 30, 2004).
Cleve Moler, ed., "History of Nabla", NA Digest 98 (Jan. 26, 1998).

External links

A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen

Categories: Vector calculus | Mathematical notation

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