Kforms

Kforms, usually written as k-forms, are a concept in differential geometry used to describe antisymmetric covariant tensors of rank k on a smooth manifold. They generalize functions (0-forms) and differential 1-forms, and they play a central role in integration on manifolds and in the formulation of physical laws. A k-form is a section of the k-th exterior power of the cotangent bundle, denoted Λ^k T* M. At each point, the space of k-forms has dimension equal to the binomial coefficient C(n,k), where n is the dimension of the manifold.

Locally, a k-form ω can be written as a sum of coefficient functions times wedge products of coordinate 1-forms: ω = sum f_{i1...ik} dx^{i1} ∧ ... ∧ dx^{ik}, where the indices are strictly increasing. The wedge product is associative and graded-commutative, and it enables the construction of higher-order forms from lower-order ones. The exterior derivative d maps k-forms to (k+1)-forms and satisfies d^2 = 0 and the Leibniz rule: d(α ∧ β) = dα ∧ β + (-1)^k α ∧ dβ for α a k-form. These operations allow for a rich calculus on manifolds, including the expression of many geometric and physical laws.

k-forms underpin de Rham cohomology, which classifies global properties of manifolds via closed forms (dω = 0) modulo exact forms (ω = dη). On oriented Riemannian manifolds, the Hodge star operator provides a duality between k-forms and (n−k)-forms, leading to the Laplacian and harmonic form theory. In physics, Maxwell’s equations can be written succinctly with differential forms, where the field strength is a 2-form and its exterior derivative encodes fundamental laws.