Unformatted text preview: x (Thm 5-2) with x = f ( a ). Then (deﬁnition) 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 ˆ c ( τ ) = a + ( τ-t ) u in R k and then the curve c = f ◦ ˆ c on M x . What are the two tangent vectors ˆ c * (( e 1 ) t )? c * (( e 1 ) t )? #5-11 If M is an n-dimensional manifold-with-boundary in R n , deﬁne μ x , as the usual orientation of M x = R n x (the orientation μ so deﬁned is the usual orientation of M ). If x ∈ ∂M , show that the two deﬁnitions of n ( x ) given above agree. hint: Read pages 119-121....
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- Spring '09
- Geometry, Manifold, Euclidean space