A vector operator is a differential operator used in vector calculus. Vector operators include: Gradient is a vector operator that operates on a scalar field, producing a vector field. Divergence is a vector operator that operates on a vector field, producing a scalar field. Curl is a vector operator that operates on a vector field, producing a vector field.
Defined in terms of del:
The Laplacian operates on a scalar field, producing a scalar field:
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.
yields the gradient of f, but
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'Alembert operator
References
Further reading H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, .