# Vector operator

A vector operator is a differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl:

${\displaystyle \operatorname {grad} \equiv \nabla }$
${\displaystyle \operatorname {div} \ \equiv \nabla \cdot }$
${\displaystyle \operatorname {curl} \equiv \nabla \times }$

The Laplacian is

${\displaystyle \nabla ^{2}\equiv \operatorname {div} \ \operatorname {grad} \equiv \nabla \cdot \nabla }$

Vector operators must always come right before the scalar field or vector field on which they operate, in order to produce a result. E.g.

${\displaystyle \nabla f}$

yields the gradient of f, but

${\displaystyle f\nabla }$

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.