08 - calculus vs topology - HOMOTOPY FORMULA. COHOMOLOGY....

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Unformatted text preview: HOMOTOPY FORMULA. COHOMOLOGY. Analysis versus topology 1. Homotopy formula. Definition. The Lie derivative of a differential form d ( M ) of any degree d on a manifold M n along a vector field v is L v = lim t 1 t ( g t * - ) , where g t is the flow of the field v , and g t * is the associated pullback action: g t * ( v 1 ,...,v d ) = ( g t * v 1 ,...,g t * v d ) . Definition. If v is a vector field, then for any singular d-dimensional polyhedron its v-trace H v ( ) is the saturation of by pgase curves of the field v defined for t [0 , 1]: H v ( ) = [ x , t [0 , 1] g t ( x ) , where g t is the flow of v . Problem 1. Prove that H v ( ) is a ( d + 1)-dimensional singular polyhedron. Definition. We supply H v ( ) with the orientation in the following way: if e 1 ,...,e d is the declared-to-be-positive basis of vectors tangent to , then the ( d + 1)-tuple v,e 1 ,...,e d is the basis orienting H v ( )....
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This note was uploaded on 08/12/2008 for the course MATH 4540 taught by Professor Protsak during the Spring '08 term at Cornell University (Engineering School).

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08 - calculus vs topology - HOMOTOPY FORMULA. COHOMOLOGY....

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