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A vector operator is a differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl:

\( \operatorname{grad} \equiv \nabla \)
\( \operatorname{div} \ \equiv \nabla \cdot \)
\( \operatorname{curl} \equiv \nabla \times \)

The Laplacian is

\( \nabla^2 \equiv \operatorname{div}\ \operatorname{grad} \equiv \nabla \cdot \nabla \)

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.

\( \nabla f \)

yields the gradient of f, but

\( f \nabla \)

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.
See also

D'Alembertian operator

Further reading

H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, ISBN 0-393-96997-5.

Mathematics Encyclopedia

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