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.
Other articles related to "exterior derivative, exterior, derivative, exterior derivatives":
... 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 ...
... 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 ...
... 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 ...
Famous quotes containing the words derivative and/or exterior:
“Poor John Field!I trust he does not read this, unless he will improve by it,thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adams grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.”
—Henry David Thoreau (18171862)
“Had I been less resolved to work, I would perhaps had made an effort to begin immediately. But since my resolution was formal and before twenty four hours, in the empty slots of the next day where everything fit so nicely because I was not yet there, it was better not to choose a night at which I was not well-disposed for a debut to which the following days proved, alas, no more propitious.... Unfortunately, the following day was not the exterior and vast day which I had feverishly awaited.”
—Marcel Proust (18711922)