Basisforms

Basisforms refers to sets of differential forms that form a basis for a vector space of forms, commonly the spaces of k-forms on a smooth manifold. In differential geometry, let M be a smooth manifold and Ω^k(M) the space of smooth k-forms. At each point p ∈ M, the fiber Ω^k_p(M) is finite-dimensional, with dimension equal to the binomial coefficient C(n, k), where n = dim M. A basis of Ω^k_p(M) is a finite set of k-forms that spans the space.

In a local coordinate chart (U, x^i), the standard basis of Ω^k(U) is given by the wedge products dx^{i1} ∧ ... ∧ dx^{ik}, with i1 < ... < ik. These are often called basis forms or local basis forms for the chart. Any ω ∈ Ω^k(U) can be written uniquely as ω = Σ ω_{i1...ik} dx^{i1} ∧ ... ∧ dx^{ik}, where the ω_{i1...ik} are smooth functions on U.

Global considerations: A global basis of Ω^k(M) exists only if the k-form bundle Λ^k(T^*M) is trivial; otherwise one works with local bases and the appropriate transition relations on overlaps. The concept of a basis form is closely related to the notion of a coframe, which is a set of globally defined 1-forms that span the cotangent bundle.

Examples: In R^3 with standard coordinates, the basis 1-forms are dx, dy, dz; the basis 2-forms are dx∧dy, dx∧dz, dy∧dz; the basis 3-form is dx∧dy∧dz.

Applications: Basis forms enable coordinate expressions of differential forms, facilitate computations of exterior derivatives and wedge products, and support integration and cohomology procedures in a coordinate-independent framework. See also differential forms, exterior algebra, and frames.