# Vector operator

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{{short description|Differential operator used in vector calculus}}
A '''vector operator''' is a [differential operator](/source/differential_operator) used in [vector calculus](/source/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](/source/Gradient) is a vector operator that operates on a [scalar field](/source/scalar_field), producing a [vector field](/source/vector_field).
* [Divergence](/source/Divergence) is a vector operator that operates on a vector field, producing a [scalar field](/source/scalar_field).
* [Curl](/source/Curl_(mathematics)) is a vector operator that operates on a vector field, producing a vector field.

Defined in terms of [del](/source/del):

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

The [Laplacian](/source/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](/source/scalar_field) or [vector field](/source/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](/source/del)
* [d'Alembert operator](/source/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

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Adapted from the Wikipedia article [Vector operator](https://en.wikipedia.org/wiki/Vector_operator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Vector_operator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.