9964

9201

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1393

شش، 68ص.

ص.ع. به فارسي و انگليسي

انگليسي به فارسي

1396/09/27

كتابنامه

علوم رياضي

رياضي

ID9201

Another Approach to Stokes Theorem on Manifolds Gholamreza Allahyari g allahyari@math iut ac ir 2014 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Azam Etemad ae110mat@ cc iut ac ir Advisor Dr Sayed Ghahreman Taherian taherian@ cc iut ac ir 2010 MSC 26A39 Keywords manifold di erential forms Kurzweil Henstock integration AbstractThis thesis is an extension and generalization of the work s done by Varayu Boonpogkrong andJaroslav Kurzweil Stokes s theorem the generalization of the fundamental theorem of calculus isall about comparing integrals over manifolds and integrals over their boundaries Stokes theoremon manifolds says that the integral of a di erential k form over the boundary of a compact orienteddi erentiable manifold M is equal to the integral of the exterior derivative of that form over M Ourapproach to integration over a general manifold has several distinguishing features the manifold Mmust be oriented and the di erential form must have compact support Note that on a manifold ofdimension n one can integrate only n forms not functions Furthermore the boundary of M hasthe boundary orientation induced from M To orient a manifold M we orient the tangent space ateach point p M This can be done by simply assigning a nonzero n covector to each point of M in other words by giving a nowhere vanishing n form on M The assignment of an orientation ateach point must be done in a coherent way so that the orientation does not change abruptly ina neighborhood of a point The simplest way to guarantee this is to require that the n form on Mspecifying the orientation at each point be C It is proved that a manifold M of dimension n has aC nowhere vanishing n form if and only if it has an oriented atlas It should be recalled that anorientation on a manifold M with boundary induces in a natural way an orientation on the boundary M The prototype of a manifold with boundary is the closed upper half space Hn with the subspacetopology of Rn We use Hn to serve as a model for manifolds with boundary A topological n manifold

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