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Unformatted text preview: x (Thm 5-2) with x = f ( a ). Then (denition) M x = f * ( R k a ). If v x M x then there is a (unique) vector u a = ( a, u ) R k a with v x = f * ( u a ). Consider the curve c ( ) = a + ( -t ) u in R k and then the curve c = f c on M x . What are the two tangent vectors c * (( e 1 ) t )? c * (( e 1 ) t )? #5-11 If M is an n-dimensional manifold-with-boundary in R n , dene x , as the usual orientation of M x = R n x (the orientation so dened is the usual orientation of M ). If x M , show that the two denitions of n ( x ) given above agree. hint: Read pages 119-121....
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- Spring '09