Del

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Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol). 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 Navier–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 multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl. These formal products do not necessarily commute with other operators or products. These three uses are summarized as:

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

Definition

In the Cartesian coordinate system ${\displaystyle \mathbb {R} ^{n}}$ with coordinates ${\displaystyle (x_{1},\dots ,x_{n})}$ and standard basis ${\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}}$, del is a vector operator whose ${\displaystyle x_{1},\dots ,x_{n}}$ components are the partial derivative operators ${\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \partial x_{n}}}$; that is,

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

where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system ${\displaystyle \mathbb {R} ^{3}}$ with coordinates ${\displaystyle (x,y,z)}$ and standard basis or unit vectors of axes ${\displaystyle \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}}$, del is written as:

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

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, in three dimensions, for any scalar field ${\displaystyle f}$ and any vector field ${\displaystyle \mathbf {F} =(F_{x},F_{y},F_{z})}$, if one defines

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

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

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

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

${\displaystyle \left(\mathbf {e} _{z}{\partial \over \partial z}\right)\times \mathbf {F}$ :={\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 ${\displaystyle \nabla }$, one may write

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

and

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

and

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

Example:

${\displaystyle f(x,y,z)=x+y+z}$

${\displaystyle \nabla f=\mathbf {e} _{x}{\partial f \over \partial x}+\mathbf {e} _{y}{\partial f \over \partial y}+\mathbf {e} _{z}{\partial f \over \partial z}=\left(1,1,1\right)}$

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.

Product rules

For vector calculus:

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

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

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

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

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

Second derivatives

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:

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

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved (${\displaystyle C^{\infty }}$ in most cases), two of them are always zero:

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

Two of them are always equal:

${\displaystyle \operatorname {div} (\operatorname {grad} f)=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}$

The 3 remaining vector derivatives are related by the equation:

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

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

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

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 is not necessarily reliable, because del does not commute in general.

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

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

A counterexample that relies on del's differential properties:

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

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

See also

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

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. ISBN 0-393-96997-5.
• 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

• Tai, Chen-To (1994). A survey of the improper use of ∇ in vector analysis (Report). Radiation Laboratory, University of Michigan. hdl:2027.42/7869.