vector operator

{{short description|Differential operator used in vector calculus}}

A vector operator is a differential operator used in vector calculus.{{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}} 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:

:$\backslash begin\{align\}$

\operatorname{grad} &\equiv \nabla \\

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

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

\end{align}

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

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

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.

:$\backslash nabla\; f$

yields the gradient of f, but

:$f\; \backslash nabla$

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.