Exterior Derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan.

The exterior derivative d has the property that d2 = 0 and is the differential (coboundary) used to define de Rham cohomology on forms. Integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology of a smooth manifold. The theorem of de Rham shows that this map is actually an isomorphism. In this sense, the exterior derivative is the "dual" of the boundary map on singular simplices.

Read more about Exterior DerivativeDefinition, Examples, The Exterior Derivative in Calculus

Other articles related to "exterior derivative, exterior, derivative, exterior derivatives":

Connection (vector Bundle) - Vector-valued Forms
... on E → M is a linear map A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms ... on E there is a unique way to extend ∇ to a covariant exterior derivative or exterior covariant derivative Unlike the ordinary exterior derivative one need ...
Differential Topology of Almost Complex Manifolds
... Just as we build differential forms out of exterior powers of the cotangent bundle, we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces ... We also have the exterior derivative d which maps Ωr(M)C to Ωr+1(M)C ... use the almost complex structure to refine the action of the exterior derivative to the forms of definite type so that is a map which increases the holomorphic part of the type by one (takes forms of ...
Geometric Calculus - Differentiation - Interior and Exterior Derivative
... Then we can define an additional pair of operators, the interior and exterior derivatives, In particular, if F is grade 1 (vector-valued function), then we ... Neither the interior derivative operator nor the exterior derivative operator is invertible ... Unlike the exterior product, the exterior derivative is not even associative ...

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