**Del**, or **nabla**, is an operator used in mathematics, in particular, 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 its standard derivative as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), del may denote the gradient (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, depending on the way it is applied.

## Contents

- Definition
- Notational uses
- Gradient
- Divergence
- Curl
- Directional derivative
- Laplacian
- Tensor derivative
- Product rules
- Second derivatives
- Precautions
- References

Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators, that makes many equations easier to write and remember. The del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, dot product, and cross product, respectively, of the del "operator" with the field. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:

## Definition

In the Cartesian coordinate system **R**^{n} with coordinates

In three-dimensional Cartesian coordinate system **R**^{3} with coordinates

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.

## Gradient

The vector derivative of a scalar field

It always points in the direction of greatest increase of

In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:

However, the rules for dot products do not turn out to be simple, as illustrated by:

## Divergence

The divergence of a vector field

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 repel from a point.

The power of the del notation is shown by the following product rule:

The formula for the vector product is slightly less intuitive, because this product is not commutative:

## Curl

The curl of a vector field

The curl at a point is proportional to the on-axis torque to which a tiny pinwheel would be subjected if it were centered at that point.

The vector product operation can be visualized as a pseudo-determinant:

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

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

## Directional derivative

The directional derivative of a scalar field

This gives the rate of change of a field

Note that

## Laplacian

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:

and the definition for more general coordinate systems is given in vector Laplacian.

The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation.

## Tensor derivative

Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field

For a small displacement

## Product rules

For vector calculus:

For matrix calculus (for which

## 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:

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved, two of them are always zero:

Two of them are always equal:

The 3 remaining vector derivatives are related by the equation:

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

## 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 relies on del's failure to commute:

A counterexample that relies on del's differential properties:

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.