Vector operator

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:

:<math>\begin{align}

\operatorname{grad} &\equiv \nabla \\

\operatorname{div} &\equiv \nabla \cdot \\

\operatorname{curl} &\equiv \nabla \times

\end{align}</math>

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

:<math> \nabla^2 \equiv \operatorname{div}\ \operatorname{grad} \equiv \nabla \cdot \nabla </math>

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.

:<math> \nabla f </math>

yields the gradient of f, but

:<math> f \nabla </math>

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