Editing 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 (mathematics)|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]]

==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]]