In vector calculus, del is a vector differential operator represented by the nabla symbol, ∇.

In the three-dimensional Cartesian coordinate system R³ with coordinates (x, y, z), del can be defined as

$\backslash nabla\; =\; \backslash begin\{pmatrix\}$

{\partial \over \partial x}, {\partial \over \partial y}, {\partial \over \partial z} \end{pmatrix}

or alternatively,

$\backslash nabla\; =\; \backslash mathbf\{i\}\{\backslash partial\; \backslash over\; \backslash partial\; x\}\; +\; \backslash mathbf\{j\}\{\backslash partial\; \backslash over\; \backslash partial\; y\}\; +\; \backslash mathbf\{k\}\{\backslash partial\; \backslash over\; \backslash partial\; z\}$

where $(\backslash mathbf\{i\},\; \backslash mathbf\{j\},\; \backslash mathbf\{k\})$ is the standard basis in R³.

The del operator can be generalized to the n-dimensional Euclidean space Rⁿ. In the Cartesian coordinate system with coordinates (x₁, x₂, ..., x_n), del is:

$\backslash nabla\; =\; \backslash sum\_\{i=1\}^n\; \backslash vec\; e\_i\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_i\}$

where $\backslash \{\; e\_i:\; 1\backslash leq\; i\backslash leq\; n\backslash \}$ is the standard basis in this space.

More compactly, and using the Einstein summation notation del is written as

$\backslash nabla\; =\; \backslash hat\; e\_i\; \backslash partial\_i.$

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

Uses of del

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Intuitively, del can be described as the general form of the derivative in multiple dimensions. When used in one dimension, it takes the form of the standard derivative in calculus.

The vector derivative of a scalar field f is called the gradient, and it equals

$\backslash nabla\; f=\backslash left(\{\backslash partial\; f\; \backslash over\; \backslash partial\; x\},\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; y\},\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; z\}\backslash right)$

(also denoted $\backslash mbox\{grad\}\backslash ,f$).

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.

Another common use of del in vector calculus is in the directional derivative

$(\backslash vec\{a\}\backslash cdot\backslash nabla)f\; =\; a\_\{x\}\{\backslash partial\; f\; \backslash over\; \backslash partial\; x\}\; +\; a\_\{y\}\{\backslash partial\; f\; \backslash over\; \backslash partial\; y\}\; +\; a\_\{z\}\{\backslash partial\; f\; \backslash over\; \backslash partial\; z\}$

(also denoted $\backslash vec\{a\}\backslash cdot\backslash mbox\{grad\}\backslash ,f$) which gives the change of a field $f$ in the direction of $\backslash vec\{a\}$. This is used extensively in the dynamics of fluids where it describes the convective derivative, or 'moving' derivative of the fluid.

Del can also be applied to a vector field, with the result being a tensor. The tensor derivative of a vector field $v$ is $\backslash nabla\; \backslash otimes\; v$ where $\backslash otimes$ represents the dyadic product.

One may also take the dot product and cross product of del with a vector field, $\backslash vec\{v\}=(v\_1,\; v\_2,\; v\_3)$ when one obtains the divergence and curl, given respectively by the formulas

$\backslash nabla\; \backslash cdot\; \backslash vec\{v\}\; =\; \{\backslash partial\; v\_1\; \backslash over\; \backslash partial\; x\}\; +\; \{\backslash partial\; v\_2\; \backslash over\; \backslash partial\; y\}\; +\; \{\backslash partial\; v\_3\; \backslash over\; \backslash partial\; z\}$

(also denoted $\backslash mbox\{div\}\backslash ;\backslash vec\; v$) and

$\backslash nabla\; \backslash times\; \backslash vec\{v\}=\backslash left($

{\frac{\partial v_3}{\partial y}} - {\frac{\partial v_2}{\partial z}}, {\frac{\partial v_1}{\partial z}} - {\frac{\partial v_3}{\partial x}}, {\frac{\partial v_2}{\partial x}} - {\frac{\partial v_1}{\partial y}} \right) (also denoted $\backslash mbox\{curl\}\backslash ;\backslash vec\; v$).

Divergence is, briefly, a measure of "spreadiness" — it tells one how much of a field, if it represented a flow, is accumulated in a point. Curl is a measure of "spinniness" — it tells one how a field, if it were a force field, would spin a pinwheel.

Another useful operator involving del is the Laplacian, $\backslash Delta=\backslash nabla^2.$

Del and second derivatives

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For a scalar field f, the first derivative is, again, $\backslash nabla\; f$, which is a vector. There is really only one field to form.

For vectors, there are three primary modes of multiplication — cross products, dot products, and dyadic products. Hence, there are three possible second derivatives for a scalar field. While there are three possible first derivatives for a vector field, one of these is a scalar, one a vector, and one a tensor. Since the cross product is not well defined on tensors, we get 1 + 3 + 2 = 6 second derivatives for a vector field:

For a scalar field $f$
$\backslash nabla\; \backslash cdot\; \backslash nabla\; f$	$\backslash nabla\; \backslash times\; \backslash nabla\; f$	$\backslash nabla\; \backslash otimes\; \backslash nabla\; f$
For a vector field $v$
$\backslash nabla\; \backslash cdot\; \backslash nabla\; \backslash times\; \backslash vec\{v\}$	$\backslash nabla\; \backslash times\; \backslash nabla\; \backslash times\; \backslash vec\{v\}$	$\backslash nabla\; \backslash otimes\; \backslash nabla\; \backslash times\; \backslash vec\{v\}$
$\backslash nabla\; (\; \backslash nabla\; \backslash cdot\; \backslash vec\{v\}\; )$	$\backslash nabla\; \backslash cdot\; \backslash nabla\; \backslash otimes\; \backslash vec\{v\}$	$\backslash nabla\; \backslash otimes\; \backslash nabla\; \backslash otimes\; \backslash vec\{v\}$

As long as the functions are well-behaved,

$\backslash nabla\; \backslash times\; \backslash nabla\; f\; =\; 0$

$\backslash nabla\; \backslash cdot\; \backslash nabla\; \backslash times\; \backslash vec\{v\}\; =\; 0.$

Also, so long as the functions are well-behaved, two of these second derivatives are always equal:

$\backslash nabla\; \backslash cdot\; \backslash nabla\; \backslash otimes\; \backslash vec\{v\}\; =\; \backslash nabla\; (\; \backslash nabla\; \backslash cdot\; \backslash vec\{v\}\; ).$

So, there are only really 6 nontrivial unique second derivatives for well-behaved functions:

$\backslash nabla\; \backslash cdot\; \backslash nabla\; f$	$\backslash nabla\; \backslash otimes\; \backslash nabla\; f$	$\backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash vec\{v\})$
$\backslash nabla\; \backslash times\; \backslash nabla\; \backslash times\; \backslash vec\{v\}$	$\backslash nabla\; \backslash otimes\; \backslash nabla\; \backslash times\; \backslash vec\{v\}$	$\backslash nabla\; \backslash otimes\; \backslash nabla\; \backslash otimes\; \backslash vec\{v\}$

The Laplacian $\backslash nabla^2\; f$ is easily the most important of these second derivatives; however, for well-behaved functions the matrix $\backslash nabla\; \backslash otimes\; \backslash nabla\; f$ is a symmetric matrix, and, consequently, it is usually also a Hermitian matrix. Hermitian matrices have real eigenvalues and orthogonal eigenvectors.

Finally, four more identities hold:

(1) $\backslash nabla\; \backslash cdot\; \backslash nabla\; f\; =\; \backslash nabla^2\; f$
(2) $\backslash nabla\; \backslash times\; \backslash nabla\; \backslash times\; \backslash vec\{v\}\; =\; \backslash nabla(\backslash nabla\; \backslash cdot\; \backslash vec\{v\})\; -\; \backslash nabla^2\; \backslash vec\{v\}$
(3) $\backslash nabla(\backslash vec\{v\}\backslash cdot\backslash vec\{g\})=\backslash vec\{v\}\backslash cdot\backslash nabla\backslash vec\{g\}+\backslash vec\{g\}\backslash cdot\backslash nabla\backslash vec\{v\}\; +\backslash vec\{g\}\backslash times(\backslash nabla\backslash times\backslash vec\{v\})+\backslash vec\{v\}\backslash times(\backslash nabla\backslash times\backslash vec\{g\})$
If $\backslash vec\{v\}\; =\; \backslash vec\{g\}$ then (3) becomes:
$\backslash frac\{1\}\{2\}\backslash nabla(\backslash vec\{v\}\backslash cdot\backslash vec\{v\})=\backslash frac\{1\}\{2\}\backslash nabla(v^2)=\backslash vec\{v\}\backslash cdot\backslash nabla\backslash vec\{v\}+\backslash vec\{v\}\backslash times(\backslash nabla\backslash times\backslash vec\{v\})$
(4)$\backslash nabla\backslash times(f\backslash vec\{v\})+\backslash nabla\; f\; \backslash times\backslash vec\{v\}$

See also

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• Table of mathematical symbols
• Maxwell's equations
• Nabla in cylindrical and spherical coordinates

References

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• Div, Grad, Curl, and All That, H. M. Schey, ISBN 0-393-96997-5
• Jeff Miller, Earliest Uses of Symbols of Calculus (Aug. 30, 2004).
• Cleve Moler, ed., "History of Nabla", NA Digest 98 (Jan. 26, 1998).

External links

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

Vector calculus | Mathematical notation

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