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:

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

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

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

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.

${\displaystyle \nabla f}$

yields the gradient of f, but

${\displaystyle f\nabla }$

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.