math lessons - 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:

$\operatorname{grad} \equiv \nabla$

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

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

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

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.

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