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Vector operator

From Wikipedia, the free encyclopedia

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

  • 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:

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

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.

yields the gradient of f, but

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

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Further reading

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