Differentialform

Differentialform is a concept in differential geometry that generalizes functions and vector fields to higher dimensions. A differential p-form on a smooth manifold M is a smooth section of the p-th exterior power of the cotangent bundle, Λ^p(T^*M). It is alternating and multilinear when evaluated on p tangent vectors. The collection of all p-forms is denoted Ω^p(M).

Locally, with coordinates x^1,...,x^n, a p-form has the form ω = ∑ f_{i1...ip} dx^{i1} ∧ ... ∧ dx^{ip}, where the f's are smooth functions. The cases p=0 are smooth functions; p=1 are covector fields. In R^n the dx^i form a basis for Ω^1, and dx^i ∧ dx^j (i<j) form a basis for Ω^2.

Two central operations are the wedge product and the exterior derivative. The wedge product ∧ endows Ω^*(M) with a graded algebra. The exterior derivative d maps Ω^p(M) to Ω^{p+1}(M) with d^2=0 and the graded Leibniz rule d(ω ∧ η) = dω ∧ η + (-1)^p ω ∧ dη. For a smooth map φ: N → M, the pullback φ^* of a form is a form on N.

Integration and topology: top-degree forms on an oriented manifold can be integrated to yield numbers; Stokes' theorem relates the integral of dω over a manifold to the integral of ω over its boundary. The de Rham cohomology H^p(M) is the quotient of closed forms by exact forms and provides invariants of M.

Applications and related structures: differential forms underpin calculus on manifolds, appear in physics (electromagnetism, gravity), and are central to Hodge theory when combined with a metric via the Hodge star operator.