A vector operator is a

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

used in

vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

. Vector operators include: *

Gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

is a vector operator that operates on a

scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...

, producing a

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

. *

Divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...

is a vector operator that operates on a vector field, producing a

. *

Curl cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...

is a vector operator that operates on a vector field, producing a vector field. Defined in terms of

del 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 ...

: :

\begin
\operatorname &\equiv \nabla \\
\operatorname &\equiv \nabla \cdot \\
\operatorname &\equiv \nabla \times
\end

The

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

operates on a scalar field, producing a scalar field: :

\nabla^2 \equiv \operatorname\ \operatorname \equiv \nabla \cdot \nabla

Vector operators must always come right before the

on which they operate, in order to produce a result. E.g. :

\nabla f

yields the gradient of ''f'', but :

f \nabla

is just another vector operator, which is not operating on anything. A vector operator can operate on another vector operator, to produce a compound vector operator, as seen above in the case of the Laplacian.

See also

Further reading