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# TH - 1 U C U U U ⊂ and such that G is one-to-one on Let E...

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CHANGE OF VARIABLES THEOREM Here is the formulation of a strong version of Change of Variables Theorem which is more suitable in practice, and is more connected to topics that we have studied. Also the assumptions are easier to verify. .Note the meaning of particular terms and symbols: A set with “content” means Jordan measurable set (a set with defined Jordan measure), you should be familiar with the notion of the “zero content”. The expression 2 1 S S means (in your text notation) that is a subset of with possibility that . 1 S 2 S 2 1 S S = (certainly not always possible as for example a compact set cannot be equal to open set but we still write ), so in fact means the same as in other texts . 1 S 2 S 2 1 S S The expression denotes the set of all elements of that are not in . 2 1 \ S S 1 S 2 S You should be familiar with the notions of an open set and of the closure S of a set S . Here is the theorem: Let n R U be open and let n R U : G be of class . Let be an open set with content such that
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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 ) , ( 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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