08wHW5 - u v = uv u 2 v 2 at a b =(3 10 2 Let R =[0 1 ×[0...

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MAT 125B Homework 5 M.Fukuda Please submit your answers at the discussion session on March 3 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. For each of the following functions, prove that f - 1 exists and is differentiable in some nonempty, open set containing the point ( a, b ), and compute D ( f - 1 )( a, b ). Note that several points may be mapped to ( a, b ) but f - 1 is a well-defined function with respect to each point. Consider all these points and these f - 1 ’s. a) f ( u, v ) = (2 u - v, 3 u + 5 v ) at ( a, b ). b) f ( u, v ) = ( u + 2 v, sin u + cos v ) at ( a, b ) = (0 , 1). c)
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Unformatted text preview: ( u, v ) = ( uv, u 2 + v 2 ) at ( a, b ) = (3 , 10). 2. Let R = [0 , 1] × [0 , 1] ⊂ R 2 . i) Suppose we have a ²nite subset of R : E 1 = { ( x k , y k ) : 0 ≤ x k , y k ≤ 1 , k = 1 , . . ., N } . Show that for any ǫ > 0 there exists a grid G 1 such that V ( E 1 ; G 1 ) < ǫ . ii) For E 2 = { ( x, y ) ∈ R : x = y } show that for any ǫ > 0 there exists a grid G 2 such that V ( E 2 ; G 2 ) < ǫ . iii) For E 3 = { ( x, y ) ∈ R : x 2 + y 2 = 1 } show that for any ǫ > 0 there exists a grid G 3 such that V ( E 3 ; G 3 ) < ǫ . Remark. I made a correction on 1-c). 1...
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