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A vector operator is a differential operator used in vector calculus.cite-ref-1[1] 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:

grad ≡ ≡ ∇ ∇ div ≡ ≡ ∇ ∇ ⋅ ⋅ curl ≡ ≡ ∇ ∇ × × {\displaystyle {\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:

∇ ∇ 2 ≡ ≡ div ⁡ ⁡ grad ≡ ≡ ∇ ∇ ⋅ ⋅ ∇ ∇ {\displaystyle \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.

∇ ∇ f {\displaystyle \nabla f}

yields the gradient of f, but

f ∇ ∇ {\displaystyle 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.

• See also

• Further reading

• del

• d'Alembert operator

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

cite-note-11. ↑ "12.2: Vector Operators". Physics LibreTexts. 2020-05-09. Retrieved 2025-05-14.