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Unformatted text preview: Lecture 32 6.4 Tangent Spaces of Manifolds We generalize our earlier discussion of tangent spaces to tangent spaces of manifolds. First we review our earlier treatment of tangent spaces. Let p R n . We define T p R n = { ( p, v ) : v R n } . (6.31) Of course, we associate T p R n R n by the map ( p, v ) v . = If U is open in R n , V is open in R k , and f : ( U, p ) ( V, q ) (meaning that f maps U V and p p 1 ) is a C map, then we have the map df p : T p R n T q R k . Via the identifications T p R n = R n and T p R k R k , the map df p is just the map = Df ( p ) : R n R k . Because these two maps can be identified, we can use the chain rule for C maps. Specifically, if f : ( U, p ) ( V, q ) and g : ( V, q ) ( R , w ), then d ( g f ) p = ( dg ) q ( df ) p , (6.32) because ( Dg )( q )( Df ( p )) = ( Dg f )( p ). You might be wondering: Why did we make everything more complicated by using df instead of Df ? The answer is because we are going to generalize from Euclidean space to manifolds. Remember, a set X R N is an ndimensional manifold if for every p X , there exists a neighborhood V of p in R N , an open set U in R n , and a diffeomorphism : U V X . The map : U V X is called a parameterization of X at p . Let us think of as a map : U R N with Im X . Claim. Let 1 ( p ) = q . Then the map ( d ) q : T q R n T p R N is onetoone. Reminder of proof: The map 1 : V X U is a C map. So, shrinking V if necessary, we can assume that this map...
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 Fall '04
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