Intrinsic Definitions
Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M. At any point p∈M, a k-form β defines an alternating multilinear map
(with k factors of TpM in the product), where TpM is the tangent space to M at p. Equivalently, β is a totally antisymmetric covariant tensor field of rank k.
The set of all differential k-forms on a manifold M is a vector space, often denoted Ωk(M).
For example, a differential 1-form α assigns to each point p∈M a linear functional αp on TpM. In the presence of an inner product on TpM (induced by a Riemannian metric on M), αp may be represented as the inner product with a tangent vector Xp. Differential 1-forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particularly within physics.
Read more about this topic: Differential Form
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