differential geometry w notes from teacher_Part_94

differential geometry w notes from teacher_Part_94 - 7.4....

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Unformatted text preview: 7.4. DE RHAM COHOMOLOGY GROUPS 187 de Rham cohomology groups are vector spaces. Let c p = r X k = 1 a k k p be a real p-chain in M , and be a p-form on M . We define the integral of over c p by D , c p E = Z c p = r X k = 1 a k Z k p . Thus every p-form on M defines a linear functional on C p ( M ) : C p ( M ) R , by c p 7- D , c p E . The space of all p-forms can be naturally identified with the dual space C * p ( M ) C p ( M ) C * p ( M ) . Furthermore, by Stokes theorem we have for a ( p- 1)-form D d , c p E = D , c p E . Thus, for every p-cycle z p , that is, if z p = 0, D d , z p E = , and for every closed form , that is, if d = 0, D , c p E = . More generally, let p Z p ( M ) be a closed p-form, p + 1 C p + 1 ( M ) be a ( p + 1)-form, z p Z p ( M ) be a p-cycle and c p + 1 C p + 1 ( M ) be a ( p + 1)-chain....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_94 - 7.4....

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