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# asn3 - D f-1 f x(b Give an example of a diFerentiable...

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MAT237Y – Multivariable Calculus Summer 2009 Assignment # 3 Questions are worth 10 marks each. Due Tuesday, June 16, at 6:10pm sharp. 1. (a) Show that if A R n is convex, then the closure of A is convex. (b) Show that if A R n is convex, then the interior of A is convex. Hint: If r > 0 and x and y are such that B r ( x ) and B r ( y ) are subsets of A , show that B r ( z ) A for any z on the line between x and y . 2. A function f : U V is invertible if there is g : V U such that g ( f ( x )) = x for all x U , and f ( g ( y )) = y for all y V . Here, g is the inverse of f , denoted by f - 1 . (a) Suppose U, V R n are open and f : U V is differentiable and invertible, and that f - 1 is differentiable. Show that, for each x U , the matrix Df x is invertible, and that its inverse is
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Unformatted text preview: D ( f-1 ) f ( x ) . (b) Give an example of a diFerentiable, invertible function whose in-verse is not diFerentiable. 3. Consider the set S ⊂ R 5 de±ned by the equations x + 2 y + z = u + v xyz = uv 1 π sin ( ( x + 2 y + 3 z ) π ) + 1 = exp(5 u-3 v ) . Show in a neighbourhood of the point a = ( u, v, x, y, z ) = (6 , 10 , 3 , 4 , 5) that x, y, and z can be given as C 1 functions of u and v . Also, ±nd vectors w 1 , w 2 ∈ R 5 such that { a + sw 1 + tw 2 : s, t ∈ R } is the tangent plane to S at the point a . No late assignments. 1...
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