Vector operator

Short description: Differential operator used in vector calculus

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:

[math]\displaystyle{ \begin{align} \operatorname{grad} &\equiv \nabla \\ \operatorname{div} &\equiv \nabla \cdot \\ \operatorname{curl} &\equiv \nabla \times \end{align} }[/math]

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

[math]\displaystyle{ \nabla^2 \equiv \operatorname{div}\ \operatorname{grad} \equiv \nabla \cdot \nabla }[/math]

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.

[math]\displaystyle{ \nabla f }[/math]

yields the gradient of f, but

[math]\displaystyle{ f \nabla }[/math]

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.