Unformatted text preview: ) ( 1 U C U U U ⊂ and such that G is onetoone 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 ) , ( θ r , we take to be an open set with content contained in U ) 2 , ( ) , ( π × +∞ . For spherical coordinates ) , , ( φ ρ , 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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 Fall '09
 RomauldStanczak
 Calculus, Topology, Empty set, Open set, measure, expression S1

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