Vector operator

Short description: Differential operator used in vector calculus

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

$\begin{array}{rl}\mathrm{grad}& \equiv \mathrm{\nabla }\\ \mathrm{div}& \equiv \mathrm{\nabla }\cdot \\ \mathrm{curl}& \equiv \mathrm{\nabla }\times \end{array}$

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

${\mathrm{\nabla }}^2\equiv \mathrm{div}\mathrm{grad}\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }$

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'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.