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:

$\operatorname {grad} \equiv \nabla$

$\operatorname {div} \ \equiv \nabla \cdot$

$\operatorname {curl} \equiv \nabla \times$

The Laplacian is

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

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.

$\nabla f$

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

• del
• D'Alembertian operator
• Related articles on a diagram

Further reading

• H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, ISBN 0-393-96997-5.