Vector Operator

A vector operator is a type of differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl:
\operatorname{grad} \equiv \nabla
\operatorname{div} \ \equiv \nabla \cdot
\operatorname{curl} \equiv \nabla \times
The Laplacian is
\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.

Further Reading

  • div, grad, curl, and all that (an informal text on vector calculus), by h. m. schey
See also: del, D'Alembertian operator.

 

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