Vector operator

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:


 * $$\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.