qform

qform, or a q-form, is a concept from differential geometry referring to an exterior differential form of degree q on a smooth manifold. For q = 0, a q-form is a smooth function; for q = 1, it is a covector field. More generally, q-forms are antisymmetric multilinear forms on the tangent spaces, and the collection of all q-forms on a manifold M is denoted Λ^q(T* M).

q-forms form a graded algebra under addition and the wedge product: the wedge of a p-form and a q-form is a (p+q)-form. The exterior derivative d maps Λ^q to Λ^{q+1} and satisfies d^2 = 0, allowing the definition of closed forms (dω = 0) and exact forms (ω = dη). These structures underpin de Rham cohomology, which classifies global topological properties of manifolds via the quotient of closed forms by exact forms.

In practice, 0-forms are functions, 1-forms are linear combinations of differentials such as dx, dy, and 2-forms can be written as combinations like f dx ∧ dy. The integral of a q-form over an oriented q-dimensional submanifold generalizes line and surface integrals, and Stokes’ theorem provides a unifying framework: the integral of dω over a region equals the integral of ω over the region’s boundary.

In physics and engineering, q-forms are used to model fields and fluxes; for example, the electromagnetic field is described by a 2-form on spacetime. The term qform can also appear as a stylized or project-specific name in software or literature, but its precise meaning is determined by context.