nforms

In mathematics, an n-form (often called a differential n-form) is a smooth section of the nth exterior power of the cotangent bundle. At each point on a smooth manifold, an n-form assigns a completely antisymmetric multilinear form to n tangent vectors. Collectively, these objects form the space of differential n-forms, denoted Ω^n(M), and they can be integrated over oriented n-dimensional submanifolds.

Locally, in a coordinate chart with coordinates x^1, ..., x^m, an n-form has the form ω = sum over i1<...<in f_{i1...in}(x) dx^{i1} ∧ ... ∧ dx^{in}, where the dx^i are basis 1-forms and the f_{i1...in} are smooth functions. The wedge product ∧ provides a graded-commutative algebra structure, so higher-degree forms are built from lower-degree ones.

A fundamental operation on forms is the exterior derivative d, which maps n-forms to (n+1)-forms: d: Ω^n(M) → Ω^{n+1}(M). It satisfies d^2 = 0 and the Leibniz rule. This leads to concepts of closed forms (dω = 0) and exact forms (ω = dη).

Integration of n-forms is defined on oriented, compact n-manifolds: ∫_M ω. Stokes' theorem relates the integral of an exterior derivative to a boundary integral: ∫_M dη = ∫_{∂M} η.

Examples and applications include volume forms in R^3 or R^n, where dx^1 ∧ ... ∧ dx^n is the canonical volume form, and physical theories such as electromagnetism, where field strength is represented by a 2-form F and Maxwell’s equations can be expressed as dF = 0 and d*F = J.

In topology, n-forms are central to de Rham cohomology, which classifies closed forms modulo exact forms. The concept generalizes to manifolds and to other geometric structures, providing a coordinate-free framework for integration and differentiation.