Vector operator

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

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

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.

References

^ "12.2: Vector Operators". Physics LibreTexts. 2020-05-09. Retrieved 2025-05-14.