Vector operator

{{short description|Differential operator used in vector calculus}} A '''vector operator''' is a differential operator used in vector calculus.<ref>{{Cite web |date=2020-05-09 |title=12.2: Vector Operators |url=https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/12:_Mathematical_Formulas/12.02:_Vector_Operators |access-date=2025-05-14 |website=Physics LibreTexts |language=en}}</ref> 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:

:<math>\begin{align} \operatorname{grad} &\equiv \nabla \\ \operatorname{div} &\equiv \nabla \cdot \\ \operatorname{curl} &\equiv \nabla \times \end{align}</math>

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

:<math> \nabla^2 \equiv \operatorname{div}\ \operatorname{grad} \equiv \nabla \cdot \nabla </math>

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. :<math> \nabla f </math> yields the gradient of ''f'', but :<math> f \nabla </math> 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== {{Reflist}}

==Further reading== * H. M. Schey (1996) ''Div, Grad, Curl, and All That: An Informal Text on Vector Calculus'', {{ISBN|0-393-96997-5}}.

Category:Vector calculus