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hintsChap5

# hintsChap5 - x(Thm 5-2 with x = f a Then(deﬁnition M x =...

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Hints Chapter 5 December 12, 2004 Math 5010/6010 #5-5 Prove that a k -dimensional (vector) subspace of R n is a k -dimensional manifold. hint: Give the supspace a name and note that it’s equal to the span of a set A consisting of k linear independent vectors from R n . Use Thm 5-2 and the nice relationship between a point in that subspace and it’s coordinates with respect to the basis A . #5-6 If f : R n R m , the graph of f is { ( x, y ) | y = f ( x ) } . Show hat the graph of f is an n -dimensional manifold if and only if f is diﬀerentiable. hint: If f is diﬀerentiable then, for x R n and y R m , set G ( x, y ) = y - f ( x ). Then G : R n × R m R m and graph ( f ) = G - 1 (0). Identify R n × R m with R n + m . Is DG ( x, y ) of full rank m when ( x, y ) graph ( f )? If so, see Thm 5-1. #5-9 Show that M x consists of the tangent vectors at t of curves c in M with c ( t ) = x . hint: Let f be a coordinate system around
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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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