Del

This article is about the mathematical operator represented by the nabla symbol. For the symbol itself, seenabla symbol.For the operation associated with the symbol ∂, also sometimes referred to as "del", seePartial derivative.For other uses, seeDel (disambiguation).

Not to be confused withDell.

Del operator,
represented by
thenabla symbol

Del,ornabla,is anoperatorused in mathematics (particularly invector calculus) as avectordifferential operator,usually represented by thenabla symbol∇.When applied to afunctiondefined on aone-dimensionaldomain, it denotes the standardderivativeof the function as defined incalculus.When applied to afield(a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: thegradientor (locally) steepest slope of ascalar field(or sometimes of avector field,as in theNavier–Stokes equations); thedivergenceof a vector field; or thecurl(rotation) of a vector field.

Del is a very convenientmathematical notationfor those three operations (gradient, divergence, and curl) that makes manyequationseasier to write and remember. The del symbol (or nabla) can beformallydefined as a vector operator whose components are the correspondingpartial derivativeoperators. 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 multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formaldot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formalcross product—to give a vector field called the curl. These formal products do not necessarilycommutewith other operators or products. These three uses, detailed below, are summarized as:

• Gradient:$\operatorname {grad} f=\nabla f$
• Divergence:$\operatorname {div} \mathbf {v} =\nabla \cdot \mathbf {v}$
• Curl:$\operatorname {curl} \mathbf {v} =\nabla \times \mathbf {v}$

Definition

In theCartesian coordinate system$\mathbb {R} ^{n}$with coordinates$(x_{1},\dots,x_{n})$andstandard basis$\{\mathbf {e} _{1},\dots,\mathbf {e} _{n}\}$,del is a vector operator whose$x_{1},\dots,x_{n}$components are thepartial derivativeoperators${\partial \over \partial x_{1}},\dots,{\partial \over \partial x_{n}}$;that is,

$\nabla =\sum _{i=1}^{n}\mathbf {e} _{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots,{\partial \over \partial x_{n}}\right)$

Where the expression in parentheses is a row vector. Inthree-dimensionalCartesian coordinate system$\mathbb {R} ^{3}$with coordinates$(x,y,z)$and standard basis or unit vectors of axes$\{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}$,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 onedefines

$\left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)f:={\partial \over \partial x_{i}}(\mathbf {e} _{i}f)={\partial f \over \partial x_{i}}\mathbf {e} _{i}$

$\left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)\cdot \mathbf {F}:={\partial \over \partial x_{i}}(\mathbf {e} _{i}\cdot \mathbf {F} )={\partial F_{i} \over \partial x_{i}}$

$\left(\mathbf {e} _{x}{\partial \over \partial x}\right)\times \mathbf {F}:={\partial \over \partial x}(\mathbf {e} _{x}\times \mathbf {F} )={\partial \over \partial x}(0,-F_{z},F_{y})$

$\left(\mathbf {e} _{y}{\partial \over \partial y}\right)\times \mathbf {F}:={\partial \over \partial y}(\mathbf {e} _{y}\times \mathbf {F} )={\partial \over \partial y}(F_{z},0,-F_{x})$

$\left(\mathbf {e} _{z}{\partial \over \partial z}\right)\times \mathbf {F}:={\partial \over \partial z}(\mathbf {e} _{z}\times \mathbf {F} )={\partial \over \partial z}(-F_{y},F_{x},0),$

then using the above definition of$\nabla$,one may write

$\nabla f=\left(\mathbf {e} _{x}{\partial \over \partial x}\right)f+\left(\mathbf {e} _{y}{\partial \over \partial y}\right)f+\left(\mathbf {e} _{z}{\partial \over \partial z}\right)f={\partial f \over \partial x}\mathbf {e} _{x}+{\partial f \over \partial y}\mathbf {e} _{y}+{\partial f \over \partial z}\mathbf {e} _{z}$

and

$\nabla \cdot \mathbf {F} =\left(\mathbf {e} _{x}{\partial \over \partial x}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\cdot \mathbf {F} \right)={\partial F_{x} \over \partial x}+{\partial F_{y} \over \partial y}+{\partial F_{z} \over \partial z}$

and

${\begin{aligned}\nabla \times \mathbf {F} &=\left(\mathbf {e} _{x}{\partial \over \partial x}\times \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\times \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\times \mathbf {F} \right)\\&={\partial \over \partial x}(0,-F_{z},F_{y})+{\partial \over \partial y}(F_{z},0,-F_{x})+{\partial \over \partial z}(-F_{y},F_{x},0)\\&=\left({\partial F_{z} \over \partial y}-{\partial F_{y} \over \partial z}\right)\mathbf {e} _{x}+\left({\partial F_{x} \over \partial z}-{\partial F_{z} \over \partial x}\right)\mathbf {e} _{y}+\left({\partial F_{y} \over \partial x}-{\partial F_{x} \over \partial y}\right)\mathbf {e} _{z}\end{aligned}}$

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 exampledel 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 thegradient,divergence,curl,directional derivative,andLaplacian.

Gradient

The vector derivative of ascalar field$f$is called thegradient,and it can be represented as:

$\operatorname {grad} f={\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}=\nabla f$

It always points in thedirectionof greatest increase of$f$,and it has amagnitudeequal 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:

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

However, the rules fordot productsdo not turn out to be simple, as illustrated by:

$\nabla (\mathbf {u} \cdot \mathbf {v} )=(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} +\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )$

Divergence

Thedivergenceof avector field $\mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}$is ascalar fieldthat can be represented as:

$\operatorname {div} \mathbf {v} ={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=\nabla \cdot \mathbf {v}$

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:

$\nabla \cdot (f\mathbf {v} )=(\nabla f)\cdot \mathbf {v} +f(\nabla \cdot \mathbf {v} )$

The formula for thevector productis slightly less intuitive, because this product is not commutative:

$\nabla \cdot (\mathbf {u} \times \mathbf {v} )=(\nabla \times \mathbf {u} )\cdot \mathbf {v} -\mathbf {u} \cdot (\nabla \times \mathbf {v} )$

Curl

Thecurlof a vector field$\mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}$is avectorfunction that can be represented as:

$\operatorname {curl} \mathbf {v} =\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right){\hat {\mathbf {x} }}+\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right){\hat {\mathbf {y} }}+\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right){\hat {\mathbf {z} }}=\nabla \times \mathbf {v}$

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:

$\nabla \times \mathbf {v} =\left|{\begin{matrix}{\hat {\mathbf {x} }}&{\hat {\mathbf {y} }}&{\hat {\mathbf {z} }}\\[2pt]{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\[2pt]v_{x}&v_{y}&v_{z}\end{matrix}}\right|$

Again the power of the notation is shown by the product rule:

$\nabla \times (f\mathbf {v} )=(\nabla f)\times \mathbf {v} +f(\nabla \times \mathbf {v} )$

The rule for the vector product does not turn out to be simple:

$\nabla \times (\mathbf {u} \times \mathbf {v} )=\mathbf {u} \,(\nabla \cdot \mathbf {v} )-\mathbf {v} \,(\nabla \cdot \mathbf {u} )+(\mathbf {v} \cdot \nabla )\,\mathbf {u} -(\mathbf {u} \cdot \nabla )\,\mathbf {v}$

Directional derivative

Thedirectional derivativeof a scalar field$f(x,y,z)$in the direction $\mathbf {a} (x,y,z)=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}$is defined as:

$(\mathbf {a} \cdot \nabla )f=\lim _{h\to 0}{\frac {f(x+a_{x}h,y+a_{y}h+z+a_{z}h)-f(x,y,z)}{h}}.$

Which is equal to the following when the gradient exists

$\mathbf {a} \cdot \operatorname {grad} f=a_{x}{\partial f \over \partial x}+a_{y}{\partial f \over \partial y}+a_{z}{\partial f \over \partial z}=\mathbf {a} \cdot (\nabla f)$

This gives the rate of change of a field$f$in the direction of$\mathbf {a}$,scaled by the magnitude of$\mathbf {a}$.In operator notation, the element in parentheses can be considered a single coherent unit;fluid dynamicsuses this convention extensively, terming it theconvective derivative—the "moving" derivative of the fluid.

Note that$(\mathbf {a} \cdot \nabla )$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

TheLaplace operatoris 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 invector Laplacian.

The Laplacian is ubiquitous throughout modernmathematical physics,appearing for example inLaplace's equation,Poisson's equation,theheat equation,thewave equation,and theSchrödinger equation.

Hessian matrix

While$\nabla ^{2}$usually represents theLaplacian,sometimes$\nabla ^{2}$also represents theHessian matrix.The former refers to the inner product of$\nabla$,while the latter refers to thedyadic productof$\nabla$:

$\nabla ^{2}=\nabla \cdot \nabla ^{T}$.

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 atensor.Thetensor derivativeof a vector field$\mathbf {v}$(in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as$\nabla \otimes \mathbf {v}$,where$\otimes$represents thedyadic product.This quantity is equivalent to the transpose of theJacobian matrixof the vector field with respect to space. The divergence of the vector field can then be expressed as thetraceof this matrix.

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

$\delta \mathbf {v} =(\nabla \otimes \mathbf {v} )^{T}\cdot \delta \mathbf {r}$

Product rules

Forvector calculus:

${\begin{aligned}\nabla (fg)&=f\nabla g+g\nabla f\\\nabla (\mathbf {u} \cdot \mathbf {v} )&=\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )+(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} \\\nabla \cdot (f\mathbf {v} )&=f(\nabla \cdot \mathbf {v} )+\mathbf {v} \cdot (\nabla f)\\\nabla \cdot (\mathbf {u} \times \mathbf {v} )&=\mathbf {v} \cdot (\nabla \times \mathbf {u} )-\mathbf {u} \cdot (\nabla \times \mathbf {v} )\\\nabla \times (f\mathbf {v} )&=(\nabla f)\times \mathbf {v} +f(\nabla \times \mathbf {v} )\\\nabla \times (\mathbf {u} \times \mathbf {v} )&=\mathbf {u} \,(\nabla \cdot \mathbf {v} )-\mathbf {v} \,(\nabla \cdot \mathbf {u} )+(\mathbf {v} \cdot \nabla )\,\mathbf {u} -(\mathbf {u} \cdot \nabla )\,\mathbf {v} \end{aligned}}$

Formatrix calculus(for which$\mathbf {u} \cdot \mathbf {v}$can be written$\mathbf {u} ^{\text{T}}\mathbf {v}$):

${\begin{aligned}\left(\mathbf {A} \nabla \right)^{\text{T}}\mathbf {u} &=\nabla ^{\text{T}}\left(\mathbf {A} ^{\text{T}}\mathbf {u} \right)-\left(\nabla ^{\text{T}}\mathbf {A} ^{\text{T}}\right)\mathbf {u} \end{aligned}}$

Another relation of interest (see e.g.Euler equations) is the following, where$\mathbf {u} \otimes \mathbf {v}$is theouter producttensor:

${\begin{aligned}\nabla \cdot (\mathbf {u} \otimes \mathbf {v} )=(\nabla \cdot \mathbf {u} )\mathbf {v} +(\mathbf {u} \cdot \nabla )\mathbf {v} \end{aligned}}$

Second derivatives

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 fieldfor a vector fieldv;the use of the scalarLaplacianandvector Laplaciangives two more:

${\begin{aligned}\operatorname {div} (\operatorname {grad} f)&=\nabla \cdot (\nabla f)=\nabla ^{2}f\\\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)\\\operatorname {grad} (\operatorname {div} \mathbf {v} )&=\nabla (\nabla \cdot \mathbf {v} )\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=\nabla \cdot (\nabla \times \mathbf {v} )\\\operatorname {curl} (\operatorname {curl} \mathbf {v} )&=\nabla \times (\nabla \times \mathbf {v} )\\\Delta f&=\nabla ^{2}f\\\Delta \mathbf {v} &=\nabla ^{2}\mathbf {v} \end{aligned}}$

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:

${\begin{aligned}\operatorname {curl} (\operatorname {grad} f)&=\nabla \times (\nabla f)=0\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=\nabla \cdot (\nabla \times \mathbf {v} )=0\end{aligned}}$

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:

$\nabla \times \left(\nabla \times \mathbf {v} \right)=\nabla (\nabla \cdot \mathbf {v} )-\nabla ^{2}\mathbf {v}$

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

$\nabla (\nabla \cdot \mathbf {v} )=\nabla \cdot (\mathbf {v} \otimes \nabla )$

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 isnotnecessarily reliable, because del does not commute in general.

A counterexample that demonstrates the divergence ($\nabla \cdot \mathbf {v}$) and theadvection operator($\mathbf {v} \cdot \nabla$) are not commutative:

${\begin{aligned}(\mathbf {u} \cdot \mathbf {v} )f&\equiv (\mathbf {v} \cdot \mathbf {u} )f\\(\nabla \cdot \mathbf {v} )f&=\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}\right)f={\frac {\partial v_{x}}{\partial x}}f+{\frac {\partial v_{y}}{\partial y}}f+{\frac {\partial v_{z}}{\partial z}}f\\(\mathbf {v} \cdot \nabla )f&=\left(v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}\right)f=v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}\\\Rightarrow (\nabla \cdot \mathbf {v} )f&\neq (\mathbf {v} \cdot \nabla )f\\\end{aligned}}$

A counterexample that relies on del's differential properties:

${\begin{aligned}(\nabla x)\times (\nabla y)&=\left(\mathbf {e} _{x}{\frac {\partial x}{\partial x}}+\mathbf {e} _{y}{\frac {\partial x}{\partial y}}+\mathbf {e} _{z}{\frac {\partial x}{\partial z}}\right)\times \left(\mathbf {e} _{x}{\frac {\partial y}{\partial x}}+\mathbf {e} _{y}{\frac {\partial y}{\partial y}}+\mathbf {e} _{z}{\frac {\partial y}{\partial z}}\right)\\&=(\mathbf {e} _{x}\cdot 1+\mathbf {e} _{y}\cdot 0+\mathbf {e} _{z}\cdot 0)\times (\mathbf {e} _{x}\cdot 0+\mathbf {e} _{y}\cdot 1+\mathbf {e} _{z}\cdot 0)\\&=\mathbf {e} _{x}\times \mathbf {e} _{y}\\&=\mathbf {e} _{z}\\(\mathbf {u} x)\times (\mathbf {u} y)&=xy(\mathbf {u} \times \mathbf {u} )\\&=xy\mathbf {0} \\&=\mathbf {0} \end{aligned}}$

Central to these distinctions is the fact that del is not simply a vector; it is avector 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 anddifferentiationidentities such as the product rule.

See also

• Del in cylindrical and spherical coordinates
• Notation for differentiation
• Vector calculus identities
• Maxwell's equations
• Navier–Stokes equations
• Table of mathematical symbols
• Quabla operator

References

• Willard Gibbs&Edwin Bidwell Wilson(1901)Vector Analysis,Yale University Press,1960:Dover Publications.
• Schey, H. M. (1997).Div, Grad, Curl, and All That: An Informal Text on Vector Calculus.New York: Norton.ISBN0-393-96997-5.
• Miller, Jeff."Earliest Uses of Symbols of Calculus".
• Arnold Neumaier (January 26, 1998). Cleve Moler (ed.)."History of Nabla".NA Digest, Volume 98, Issue 03. netlib.org.

External links

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