k-form

A differential form of degree $k$ -- built to be integrated over a $k$ -dimensional oriented piece.

A $k$ -form in $\mathbb{R}^n$ is a differential form of degree $k$ , built from a linear combination of $k$ -fold wedge products $dx_{i_1}\wedge\cdots\wedge dx_{i_k}$ . In $\mathbb{R}^n$ there are $\binom{n}{k}$ independent basis $k$ -forms at each point, so the top degree is $n$ (an $n$ -form), beyond which every form is identically zero.

Examples

In $\mathbb{R}^3$ : 0-forms (functions), 1-forms, 2-forms, and 3-forms are the only nonzero degrees.