Differential form

An object built to be integrated -- a 1-form eats a vector, a 2-form eats a pair, and so on.

A differential form of degree $k$ (a $k$ -form) is an object designed to be integrated over a $k$ -dimensional oriented region. A 1-form like $df$ eats a single tangent vector; a 2-form eats an oriented pair of them, returning the signed area they span, scaled by the form's coefficient.

Examples

$df = f_x\,dx + f_y\,dy$ is a 1-form; $dx\wedge dy$ is a 2-form.