08wHW5 - ( u, v ) = ( uv, u 2 + v 2 ) at ( a, b ) = (3 ,...

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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 di±erentiable 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-de²ned 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) f
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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 nite subset of R : E 1 = { ( x k , y k ) : 0 x k , y k 1 , k = 1 , . . ., N } . Show that for any > 0 there exists a grid G 1 such that V ( E 1 ; G 1 ) < . ii) For E 2 = { ( x, y ) R : x = y } show that for any > 0 there exists a grid G 2 such that V ( E 2 ; G 2 ) < . iii) For E 3 = { ( x, y ) R : x 2 + y 2 = 1 } show that for any > 0 there exists a grid G 3 such that V ( E 3 ; G 3 ) < . Remark. I made a correction on 1-c). 1...
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This homework help was uploaded on 04/08/2008 for the course MATH 125B taught by Professor Fukuda during the Winter '07 term at UC Davis.

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