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Vector operator

Differential operator used in vector calculus

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Source updated October 21, 2025

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:

${\begin{aligned}\operatorname {grad} &\equiv \nabla \\\operatorname {div} &\equiv \nabla \cdot \\\operatorname {curl} &\equiv \nabla \times \end{aligned}}$

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

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

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.

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Vector operator

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Related topics

Del

D'Alembert operator