del - wikipedia, the free encyclopedia

8/3/2019 Del - Wikipedia, The Free Encyclopedia

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Del operator,represented bythe nabla symbol

DelFrom Wikipedia, the free encyclopedia

In vector calculus, del is a vector differential operator, usually represented by the nabla

symbol . When applied to a function defined on a one-dimensional domain, it denotes

its standard derivative as defined in calculus. When applied to a field (a function defined

on a multi-dimensional domain), del may denote the gradient (locally steepest slope) of a

scalar field, the divergence of a vector field, or the curl (rotation) of a vector field,

depending on the way it is applied.

Strictly speaking, del is not a specific operator, but rather a convenient mathematical

notation for those three operators, that makes many equations easier to write and

remember. The del symbol can be interpreted as a vector of partial derivative operators,

and its three possible meaningsgradient, divergence, and curlcan be formally viewed

as the product of scalars, dot product, and cross product, respectively, of the del

"operator" with the field. These formal products do not necessarily commute with other operators or products.

Contents

1 Definition

2 Notational uses

2.1 Gradient

2.2 Divergence

2.3 Curl

2.4 Directional derivative2.5 Laplacian

2.6 Tensor derivative

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 in terms of

partial derivative operators as

where are the unit vectors in their respective directions. Though this page chiefly treats del in threedimensions, this definition can be generalized to the n-dimensional Euclidean space Rn. In the Cartesian

coordinate system with coordinates (x1,x2, ...,xn), del is:


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DCG chart: A simple chart depicting

all rules pertaining to second

derivatives. D, C, G, L and CC stand

for divergence, curl, gradient,

Laplacian and curl of curl,

respectively. Arrows indicate

existence of second derivatives. Blue

circle in the middle represents curl of

curl, whereas the other two red

circles(dashed) mean that DD and

GG do not exist.

equation, the heat equation, the wave equation, and the Schrdinger equationto name a few.

Tensor derivative

Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field

is a 9-term second-rank tensor, but can be denoted simply as , where represents the dyadic

product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to

space.

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

Second derivatives

When del operates on a scalar or vector, generally 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

fieldfor a vector field v; the use of the scalar Laplacian and vector

Laplacian gives two more:

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:

Two of them are always equal:

The 3 remaining vector derivatives are related by the equation:

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


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Precautions

Most of the above vector properties (except for those that rely explicitly on del's differential propertiesfor

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, thereverse is notnecessarily reliable, because del does not often commute.

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

A counterexample that relies on del's differential properties:

Central to these distinctions is the fact that del is not simply a vector; it is a vectoroperator. Whereas a vector

is an object with both a precise numerical magnitude and direction, del does not have a precise value for either

until it is allowed to operate on something.

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

Maxwell's equations

Navier-Stokes equations

Table of mathematical symbols


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Vector calculus identities

References

Div, Grad, Curl, and All That, H. M. Schey, ISBN 0-393-96997-5

Jeff Miller, Earliest Uses of Symbols of Calculus (http://jeff560.tripod.com/calculus.html)

"History of Nabla" (http://www.netlib.org/na-digest-html/98/v98n03.html#2) . netlib.org. January 26,

1998. http://www.netlib.org/na-digest-html/98/v98n03.html#2.

External links

A survey of the improper use of in vector analysis (http://hdl.handle.net/2027.42/7869) (1994)

Tai, Chen

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Categories: Calculus Vector calculus Mathematical notation Differential operators

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