{{short description|Vector differential operator}}{{About|the mathematical operator represented by the nabla symbol|the symbol itself|nabla symbol|the operation associated with the symbol âˆ‚, also sometimes referred to as “del“|Partial derivative|other uses}}{{distinguish|Dell}}{{No footnotes|date=March 2010}}File:Del.svg|right|100px|thumb|Del operator,represented bythe nabla symbol nabla symbolDel, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol âˆ‡. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navierâ€“Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a “formal” scalar multiplicationâ€”to give a vector field called the gradient; second, it can act on vector fields by a “formal” dot productâ€”to give a scalar field called the divergence; and lastly, it can act on vector fields by a “formal” cross productâ€”to give a vector field called the curl. These “formal” products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:

Gradient: operatorname{grad}f = nabla f
Divergence: operatorname{div}vec v = nabla cdot vec v
Curl: operatorname{curl}vec v = nabla times vec v

Definition

In the Cartesian coordinate system mathbb{R}^n with coordinates (x_1, dots, x_n) and standard basis {vec e_1, dots, vec e_n }, del is a vector operator whose x_1, dots, x_n components are the partial derivative operators {partial over partial x_1}, dots, {partial over partial x_n}; that is,

nabla = sum_{i=1}^n vec e_i {partial over partial x_i} = left({partial over partial x_1}, ldots, {partial over partial x_n} right)

Where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system mathbb{R}^3 with coordinates (x, y, z) and standard basis or unit vectors of axes {vec e_x, vec e_y, vec e_z }, del is written as

nabla = mathbf{e}_x {partial over partial x} + mathbf{e}_y {partial over partial y} + mathbf{e}_z {partial over partial z}= left({partial over partial x}, {partial over partial y}, {partial over partial z} right)

As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.More specifically, for any scalar field f and any vector field mathbf{F}=(F_x, F_y, F_z), if one defines

left(mathbf{e}_i {partial over partial x_i}right) f := {partial over partial x_i}(mathbf{e}_i f) = {partial f over partial x_i}mathbf{e}_i left(mathbf{e}_i {partial over partial x_i}right) cdot mathbf{F} := {partial over partial x_i}(mathbf{e}_icdot mathbf{F}) = {partial F_i over partial x_i} left(mathbf{e}_x {partial over partial x}right) times mathbf{F} := {partial over partial x}(mathbf{e}_xtimes mathbf{F}) = {partial over partial x}(0, -F_z, F_y) left(mathbf{e}_y {partial over partial y}right) times mathbf{F} := {partial over partial y}(mathbf{e}_ytimes mathbf{F}) = {partial over partial y}(F_z,0,-F_x) left(mathbf{e}_z {partial over partial z}right) times mathbf{F} := {partial over partial z}(mathbf{e}_ztimes mathbf{F}) = {partial over partial z}(-F_y,F_x,0),

then using the above definition of nabla, one may write

nabla f =left(mathbf{e}_x {partial over partial x}right)f + left(mathbf{e}_y {partial over partial y}right)f + left(mathbf{e}_z {partial over partial z}right)f = {partial f over partial x}mathbf{e}_x + {partial f over partial y}mathbf{e}_y + {partial f over partial z}mathbf{e}_zand

nabla cdot mathbf{F} = left(mathbf{e}_x {partial over partial x}cdot mathbf{F}right) + left(mathbf{e}_y {partial over partial y}cdot mathbf{F}right) + left(mathbf{e}_z {partial over partial z}cdot mathbf{F}right)= {partial F_x over partial x} + {partial F_y over partial y} + {partial F_z over partial z}and

begin{align}

nabla times mathbf{F} &= left(mathbf{e}_x {partial over partial x}times mathbf{F}right) + left(mathbf{e}_y {partial over partial y}times mathbf{F}right) + left(mathbf{e}_z {partial over partial z}times mathbf{F}right)&= {partial over partial x}(0, -F_z, F_y) + {partial over partial y}(F_z,0,-F_x) + {partial over partial z}(-F_y,F_x,0)&= left({partial F_z over partial y}-{partial F_y over partial z}right)mathbf{e}_x + left({partial F_x over partial z}-{partial F_z over partial x}right)mathbf{e}_y + left({partial F_y over partial x}-{partial F_x over partial y}right)mathbf{e}_zend{align}

Example: f(x, y, z) = x + y + z nabla f = mathbf{e}_x {partial f over partial x} + mathbf{e}_y {partial f over partial y} + mathbf{e}_z {partial f over partial z} = left(1, 1, 1 right)

Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.

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, divergence, curl, directional derivative, and Laplacian.

Gradient

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

operatorname{grad}f = {partial f over partial x} vec e_x + {partial f over partial y} vec e_y + {partial f over partial z} vec e_z=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 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 products 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 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{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, 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 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_xvec e_x + v_yvec e_y + v_zvec e_z is a vector function that can be represented as:

operatorname{curl}vec v = left({partial v_z over partial y} - {partial v_y over partial z} right) vec e_x + left({partial v_x over partial z} - {partial v_z over partial x} right) vec e_y + left({partial v_y over partial x} - {partial v_x over partial y} 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 centered at that point.The vector product operation can be visualized as a pseudo-determinant:

nabla times vec v = left|begin{matrix} vec e_x & vec e_y & vec e_z [2pt] {frac{partial}{partial x}} & {frac{partial}{partial y}} & {frac{partial}{partial z}} [2pt] 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)

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 of a scalar field f(x,y,z) in the directionvec a(x,y,z) = a_x vec e_x + a_y vec e_y + a_z vec e_z is defined as:

vec acdotoperatorname{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 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—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 = {partial^2 over partial x^2} + {partial^2 over partial y^2} + {partial^2 over partial z^2} = nabla cdot nabla = nabla^2

and the definition for more general coordinate systems is given in vector Laplacian.The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace’s equation, Poisson’s equation, the heat equation, the wave equation, and the SchrÃ¶dinger equation.

Hessian matrix

While nabla^2 usually represents the Laplacian, sometimes nabla^2 also represents the Hessian matrix. 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. The tensor derivative of a vector field vec{v} (in three dimensions) is a 9-term second-rank tensor â€“ that is, a 3Ã—3 matrix â€“ but can be denoted simply as nabla otimes vec{v}, where otimes represents the dyadic product. 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{r}, the change in the vector field is given by:

delta vec{v} = (nabla otimes vec{v})^T sdot delta vec{r}

Product rules

For vector calculus:

begin{align}

nabla (fg) &= fnabla g + gnabla 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{align}For matrix calculus (for which vec u cdot vec v can be written vec u^text{T} vec v):

begin{align}

left(mathbf{A}nablaright)^text{T} vec u &= nabla^text{T} left(mathbf{A}^text{T}vec uright) - left(nabla^text{T} mathbf{A}^text{T}right) vec u

end{align}Another relation of interest (see e.g. Euler equations) is the following, where vec u otimes vec v is the outer product tensor:

begin{align}

nabla cdot (vec u otimes vec v) = (nabla cdot vec u) vec v + (vec u cdot nabla) vec v

end{align}

Second derivatives

(File:DCG chart.svg|thumb|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. )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 gives two more:

begin{align}

operatorname{div}(operatorname{grad}f) &= nabla cdot (nabla f) = nabla^2 f
operatorname{curl}(operatorname{grad}f) &= nabla times (nabla f)
operatorname{grad}(operatorname{div}vec v) &= nabla (nabla cdot vec v)
operatorname{div}(operatorname{curl}vec v) &= nabla cdot (nabla times vec v)
operatorname{curl}(operatorname{curl}vec v) &= nabla times (nabla times vec v)
Delta f &= nabla^2 f
Delta vec v &= nabla^2 vec v

end{align}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{align}

operatorname{curl}(operatorname{grad}f) &= nabla times (nabla f) = 0
operatorname{div}(operatorname{curl}vec v) &= nabla cdot (nabla times vec v) = 0

end{align}Two of them are always equal:

operatorname{div}(operatorname{grad}f) = 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 vright) = 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 (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 demonstrates the divergence (nabla cdot vec v ) and the advection operator (vec v cdot nabla ) are not commutative:

begin{align}

(vec u cdot vec v) f &equiv (vec v cdot vec u) f
(nabla cdot vec v) f &= left (frac{partial v_x}{partial x} + frac{partial v_y}{partial y} + frac{partial v_z}{partial z} right)f
= frac{partial v_x}{partial x}f + frac{partial v_y}{partial y}f + frac{partial v_z}{partial z}f
(vec v cdot nabla) f &= left (v_x frac{partial}{partial x} + v_y frac{partial}{partial y} + v_z frac{partial}{partial z} right)f
= v_x frac{partial f}{partial x} + v_y frac{partial f}{partial y} + v_z frac{partial f}{partial z}
Rightarrow (nabla cdot vec v) f &ne (vec v cdot nabla) f

end{align}A counterexample that relies on del’s differential properties:

begin{align}

(nabla x) times (nabla y) &= left (vec e_x frac{partial x}{partial x}+vec e_y frac{partial x}{partial y}+vec e_z frac{partial x}{partial z} right) times left (vec e_x frac{partial y}{partial x}+vec e_y frac{partial y}{partial y}+vec e_z frac{partial y}{partial z} 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{align}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 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.

References

Willard Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, Yale University Press, 1960: Dover Publications.
BOOK, Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, H. M., Schey, 1997, New York, Norton, 0-393-96997-5,
WEB, Jeff, Miller,jeff560.tripod.com/calculus.html, Earliest Uses of Symbols of Calculus,
WEB, Arnold Neumaier, Cleve Moler,www.netlib.org/na-digest-html/98/v98n03.html2, History of Nabla, NA Digest, Volume 98, Issue 03, netlib.org, January 26, 1998,

External links

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

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del

Definition

Notational uses

Gradient

Divergence

Curl

Directional derivative

Laplacian

Hessian matrix

Tensor derivative

Product rules

Second derivatives

Precautions

See also

References

External links