pform

A p-form is an antisymmetric covariant tensor field of rank p on a smooth manifold. At each point, a p-form is a multilinear alternating map from p tangent vectors to the real numbers, equivalently an element of the pth exterior power Λ^p(T*_p M). The collection of smooth p-forms on a manifold M is denoted Ω^p(M). For an n-dimensional manifold, a p-form at a point has C(n,p) independent components in a local chart.

Locally, a p-form can be written as ω = ∑ ω_{i1...ip} dx^{i1} ∧ ... ∧ dx^{ip}, with i1 < ... < ip. 0-forms are smooth functions; 1-forms are linear combinations of dx^i; p-forms generalize differential objects such as area and volume elements.

The exterior derivative d maps Ω^p(M) to Ω^{p+1}(M), with d^2 = 0 and the Leibniz rule d(α ∧ β) = dα ∧ β + (-1)^p α ∧ dβ for α ∈ Ω^p(M). The pullback f^* allows transfer along smooth maps. One can integrate top-degree forms on oriented manifolds; Stokes' theorem ∫_M dω = ∫_{∂M} ω generalizes many fundamental theorems of calculus and geometry.

On a oriented Riemannian n-manifold, the Hodge star * gives a map Ω^p → Ω^{n-p}, enabling a codifferential δ and the Laplace–de Rham operator Δ = dδ + δd. The de Rham cohomology groups H^p(M) classify closed forms modulo exact forms. P-forms appear in physics (for example, the electromagnetic field strength F is a 2-form with dF = 0 and d*F = J).