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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 pchain in M , and be a pform 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 pform 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 pforms 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 pcycle 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 pform, p + 1 C p + 1 ( M ) be a ( p + 1)form, z p Z p ( M ) be a pcycle 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.
 Spring '10
 Wong
 Geometry, Vector Space

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