Differential forms

One-form (differential geometry)

In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space. Symbolically, where is linear. Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates: where the are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field. (Wikipedia).

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Related pages

Local coordinates | Tangent bundle | De Rham cohomology | Linear map | Local property | Winding number | Exterior derivative | Differential geometry | Covariance and contravariance of vectors | Tensor field | Derivative | Differentiable manifold | Section (fiber bundle) | Differentiable function | Atan2 | Open set | Total derivative | Cotangent bundle