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Unformatted text preview: 5.4. DEGREE OF A MAP 137 Problem . Let M be a closed oriented n-dimensional manifold and : M S n be a smooth map. We identify ( x ) with a unit vector field v on S n in R n + 1 . Let vol be the volume ( n + 1)-form in R n + 1 . Let vol ( S n ) be the volume of the unit sphere S n . Let u , = 1 , . . . , n , be local coordinates on M . Show that deg( ) = 1 vol ( S n ) Z M vol v , v u 1 , , v u n ! du 1 du n 5.4.4 Index of a Vector Field Let M be a closed (compact without boundary) n-dimensional submanifold of R n + 1 that is a boundary of a compact region U R n + 1 , that is, M = U . Let N be an outward-pointing normal to M . Then N induces an orientation on M . Let v be a unit vector field on M . Let S n be the unit n-sphere embedded in R n centered at the origin. We identify the unit vectors in R n + 1 with points in S n ....
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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