Del

For other uses, see Del (disambiguation).

∇

Del operator,
represented by
the nabla symbol. Nabla is the Symbol \nabla The name comes from the Greek word for a Hebrew Harp, which had a similar shape

In vector calculus, del is a vector differential operator represented by the nabla symbol: $\nabla$ . Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator Nabla is the Symbol \nabla The name comes from the Greek word for a Hebrew Harp, which had a similar shape

Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. A convention is a set of agreed, stipulated or generally accepted Standards norms social norms or criteria, often taking the form of See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Depending on the way del is applied, it can describe the gradient (slope), divergence (degree to which something converges or diverges) or curl (rotational motion at points in a fluid). In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the cURL is a Command line tool for transferring files with URL syntax. More intuitive descriptions of each of the many operations del performs can be found below.

Mathematically, del can be viewed as the derivative in multi-dimensional space. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change When used in one dimension, it takes the form of the standard derivative of calculus. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives As an operator, it acts on vector fields and scalar fields with analogues of traditional multiplication. In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point As with all operators, these analogues should not be confused with traditional multiplication; in particular, del does not commute. In Mathematics, commutativity is the ability to change the order of something without changing the end result

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

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

where {i, j, k} is the standard basis in R³. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the standard basis (also called natural basis or canonical basis) of the n- dimensional Euclidean space

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:

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

where $\{ \vec e_i: 1\leq i\leq n\}$ is the standard basis in this space. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

More compactly, using the Einstein summation notation, del is written as

$\nabla = \vec e_i \partial_i$

Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational This is a list of some Vector calculus formulae of general use in working with various Coordinate systems Table

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. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the cURL is a Command line tool for transferring files with URL syntax. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after

Gradient

The vector derivative of a scalar field f is called the gradient, and it can be represented as:

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

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 Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar 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 In particular, if a hill is defined as a height function over a plane h(x,y), the 2d projection of the gradient at a given location will be a vector in the xy-plane (sort of like 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:

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

The rules for products do not always turn out so 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 of a vector field v(x,y,z) = v_x i + v_y j + v_z k is a scalar function that can be represented as:

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

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. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point

The power of the del notation is shown by the following product rule:

$\nabla \cdot (f \vec v) = f \nabla \cdot \vec v + \vec v \cdot \nabla f$

The formula for the vector product is slightly less intuitive, because this product is not commutative:

$\nabla \cdot (\vec u \times \vec v) = \vec v \cdot \nabla \times \vec u - \vec u \cdot \nabla \times \vec v$

Curl

The curl of a vector field $v(x, y, z) = v_x\mathbf{i} + v_y\mathbf{j} + v_z\mathbf{k}$ is a vector function that can be represented as:

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

The curl at a point is proportional to the on-axis torque a tiny pinwheel would feel if it were centered at that point. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which cURL is a Command line tool for transferring files with URL syntax. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

The vector product operation can be visualised as a pseudo-determinant:

$\nabla \times \vec v = \left|\begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ \\ v_x & v_y & v_z \end{matrix}\right|$

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 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 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:

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

This gives the change of a field f in the direction of a. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the 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 fluid flow: Fluids ( Liquids and Gases in motion

Laplacian

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; it is defined as:

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

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. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. In Mathematics, Poisson's equation is a Partial differential equation with broad utility in Electrostatics, Mechanical engineering and Theoretical The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system

Tensor derivative

Del can also be applied to a vector field with the result being a tensor. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually The tensor derivative of a vector field $\vec{v}$ is a 9-term second-rank tensor, but can be denoted simply as $\nabla \otimes \vec{v}$ , where $\otimes$ represents the dyadic product. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. In Mathematics, in particular Multilinear algebra, the dyadic product \mathbb{P} = \mathbf{u}\otimes\mathbf{v} of two This quantity is equivalent to the Jacobian matrix of the vector field with respect to space. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

For a small displacement $\delta \vec{r}$ , the change in the vector field is given by:

$\delta \vec{v} = (\nabla \otimes \vec{v}) \sdot \delta \vec{r}$

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:

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

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

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. Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space 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 simply 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. This is a listing of common symbols found within all branches of the science of Mathematics. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric This is a list of some Vector calculus formulae of general use in working with various Coordinate systems Table The following identities are important in Vector calculus: Single operators (summary This section explicitly lists what some symbols mean for clarity 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

Dictionary

-proper noun

(UK) A diminutive of the male given name Derek.

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