Del

In the Cartesian coordinate system ${\displaystyle \mathbb {R} ^{n}}$ with coordinates ${\displaystyle (x_{1},\dots ,x_{n})}$ and standard basis ${\displaystyle \{{\vec {e}}_{1},\dots ,{\vec {e}}_{n}\}}$, 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,

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

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

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

Example:

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

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

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

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.

For vector calculus:

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

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

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

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

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

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:

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

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:

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

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:

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

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

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

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 (${\displaystyle \nabla \cdot {\vec {v}}}$) and the advection operator (${\displaystyle {\vec {v}}\cdot \nabla }$) are not commutative:

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

A counterexample that relies on del's differential properties:

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

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.

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

• 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.

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