Mach wave

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:

$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\equiv \mathrm{\nabla }$

$\mathrm{d}\mathrm{i}\mathrm{v}\equiv \mathrm{\nabla }\cdot$

$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\equiv \mathrm{\nabla }\times$

The Laplacian is

${\mathrm{\nabla }}^2\equiv \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }$

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.

$\mathrm{\nabla }f$

yields the gradient of f, but

$f\mathrm{\nabla }$

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

• del
• D'Alembertian operator
• Related articles on a diagram

Further reading

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