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Unformatted text preview: ) ( 1 U C U U U and such that G is one-to-one on . Let E be a compact set with zero content such that U ) ( det u G for . Suppose that E U \ u U T has content, U T , that R T f ) ( : G is bounded, and that f is continuous on ) \ ( E T G . Then u u G u G x x G n T T n d f d f = ) ( det )) ( ( ... ) ( ... ) ( . The advantage is that we may in many cases see immediately what is ! U For polar coordinates ) , ( r , we take to be an open set with content contained in U ) 2 , ( ) , ( + . For spherical coordinates ) , , ( , we take to be an open set with U content contained in ) 2 , ( ) , ( ) , ( + . Hence the applicability of the theorem follows immediately. You may try to define for any maps G you come to use U and verify easily the other assumptions if you use the map to evaluate the integral....
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.
- Fall '09