Differential forms

totally antisymmetric tensor field; section of exterior powers of the cotangent bundle

algebraic construction used in Euclidean geometry

In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to each fibre is a linear functional on the tangent space. Let U be an open subset of M and p \in U. Then \begin{align} \omega: U & \rightarrow \bigcup_{p \in U} T^*_p(M) \\ p & \mapsto \omega_p \in T_p^*(M) \end{align} defines a one-form \omega. \omega_p is a covector.

cohomology with real coefficients computed using differential forms

in differential geometry, a differential operation defined in differential forms that increases the form degree by 1

statement about integration of differential forms on manifolds

linear map from p-forms on an n-dimensional manifold to (n−p)-forms

top-dimensional differential form that can be defined on orientable manifolds

Mathematical condition

differential form on a manifold which is permitted to have complex coefficients

binary operation between a vector field and a differential form