dualform

Dualform is a mathematical term used to denote objects that are dual to another structure, typically in linear algebra or differential geometry. In its common form, a dualform refers to an element of the dual space V*, the set of all linear functionals on a vector space V over a field F. Elements of V* are called covectors or dual forms. For finite-dimensional V, the dual space V* has the same dimension as V, and a basis {e_i} in V induces a dual basis {e^i} in V* with e^i(e_j) = δ^i_j.

In differential geometry, a 1-form is a smooth section of the cotangent bundle, i.e., a dual form that assigns to each tangent vector a real number. More generally, k-forms are alternating multilinear maps on k tangent vectors, forming the exterior algebra Λ^k(T^*M) of a manifold M. These dual forms enable coordinate-free expressions for integration, differentiation, and other geometric operations.

Dual maps are another aspect: for a linear map T: V -> W, the dual map T*: W* -> V* is defined by T*(φ) = φ ∘ T. When a inner product on V is present, the Riesz representation theorem provides a canonical identification between V and V*, allowing one to convert vectors to dual forms via the inner product.

Examples and notation: in R^n with the standard basis, the dual basis consists of coordinate functionals dx^i, forming canonical 1-forms. In calculus on manifolds, gradients, differential forms, and pullbacks are naturally described using dual forms. In physics, differential forms offer a coordinate-free language for fields and conservation laws.

See also: Covector, Differential form, Exterior derivative, Hodge star.