Del

For other uses, see Del (disambiguation).

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

This symbol is also sometimes called the nabla operator, after the Greek word for a kind of harp with a similar shape. Related words also exist in Aramaic and Hebrew. Another, less-common name is atled, because it is a reversed delta.

The symbol ∇ was introduced by William Rowan Hamilton.

The symbol is available in standard HTML as ∇ and in LaTeX as \nabla. In Unicode, it is the character at decimal number 8711, or the hexadecimal number 0x2207.

In differential geometry, the nabla symbol is also used to refer to a connection.

Del in different conventions

Del is most familiar in three dimensions. For any three-dimensional Cartesian coordinate system {x,y,z}, del can be written as

${\vec {\nabla }}={\hat {x}}{\partial \over \partial x}+{\hat {y}}{\partial \over \partial y}+{\hat {z}}{\partial \over \partial z}$

(Depending on the person, {$\mathbf {i} ,\mathbf {j} ,\mathbf {k}$} may be used instead of {${\hat {x}},{\hat {y}},{\hat {z}}$}.)

Del can be generalized outside of three dimensions. For any n-dimensional Cartesian coordinate system {x₁, x₂, ... x_n}, del can be written as:

${\vec {\nabla }}=\sum _{i=1}^{n}{\hat {e}}_{i}{\partial \over \partial x_{i}}$

With a couple of conventions, this can be written in a more compact form involving Einstein summation notation:

${\vec {\nabla }}={\hat {e}}_{i}\partial _{i}$

In the matrix/vector forms used for linear algebra, it can be written as:

${\vec {\nabla }}={\begin{pmatrix}{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\end{pmatrix}}$

While del is defined using a Cartesian coordinate system, that does not mean that this is the only way to use it. For more on this subject, see nabla in cylindrical and spherical coordinates.

Finally, one last term often comes up. It is the scalar operator del-squared. In three dimensions, it is:

$\nabla ^{2}={\vec {\nabla }}\cdot {\vec {\nabla }}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}$

It is important to notice that this is a scalar operator, not a vector.

The Meaning of Del

Del pops up in vector calculus enough to be given its own name and symbol. It can roughly be described as the general form of the derivative in multiple dimensions. When del is used in only one dimension, it takes the form of the standard derivative found in non-vector calculus.

The vector derivative of a scalar field f is called the gradient, and is equal to:

$\operatorname {grad} \;f={\vec {\nabla }}f$

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. It's interesting to note that direction of greatest decrease of f at the point is the opposite of the Del and the maximum rate of decrease at the point is the negative value of the magnitude of f.

Del can also be applied to a vector field, to obtain an analogous idea. Where $\otimes$ represents the dyadic product, the tensor derivative of a vector field ${\vec {v}}$ is:

$\operatorname {grad} \;{\vec {v}}={\vec {\nabla }}\otimes {\vec {v}}$

The two play analogous roles. When dotted into a displacement vector, both derivatives give an approximation of the change in the field. In the former, this change is a scalar quantity. In the latter, this change is a vector quantity.

Three other operations that occur frequently in vector calculus can be written with del:

Divergence:	$\operatorname {div} \;{\vec {u}}={\vec {\nabla }}\cdot {\vec {u}}$
Curl:	$\operatorname {curl} \;{\vec {u}}={\vec {\nabla }}\times {\vec {u}}$
Laplacian:	$\operatorname {Laplacian} \;f=\nabla ^{2}f$

Divergence is, briefly, a measure of "spreadiness" -- it tells you how much vectors increase in whatever direction they travel. Curl is a measure of "spinniness" -- it tells you how a field, if it were a force field, would spin a pinwheel. And the Laplacian is a general second derivative.

Del and Second Derivatives

For a scalar field f, the first derivative is, again, ${\vec {\nabla }}f$, 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 $f$
${\vec {\nabla }}\cdot {\vec {\nabla }}f$	${\vec {\nabla }}\times {\vec {\nabla }}f$	${\vec {\nabla }}\otimes {\vec {\nabla }}f$
For a vector field ${\vec {v}}$
${\vec {\nabla }}\cdot {\vec {\nabla }}\times {\vec {v}}$	${\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {v}}$	${\vec {\nabla }}\otimes {\vec {\nabla }}\times {\vec {v}}$
${\vec {\nabla }}({\vec {\nabla }}\cdot {\vec {v}})$	${\vec {\nabla }}\cdot {\vec {\nabla }}\otimes {\vec {v}}$	${\vec {\nabla }}\otimes {\vec {\nabla }}\otimes {\vec {v}}$

Now, in what might be described as a remarkable coincidence, del works like any other vector in two of these identities. Since del doesn't really have a direction, this is hardly expectable. However, so long as the functions are well-behaved,

${\vec {\nabla }}\times {\vec {\nabla }}f={\vec {0}}$

${\vec {\nabla }}\cdot {\vec {\nabla }}\times {\vec {v}}=0$

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

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

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

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

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

Finally, two more identities hold:

${\vec {\nabla }}\cdot {\vec {\nabla }}f=\nabla ^{2}f$

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

See also

• Table of mathematical symbols
• Maxwell's equations

Further reading

• 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).