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Unformatted text preview: 15.9: CHANGE OF VARIABLES IN MULTIPLE INTEGRALS KIAM HEONG KWA 1. C 1 Transformations and Their Jacobians By a transformation or change of variables T : S → R is meant a function whose domain S and range R are both subsets of R 2 . For ease of distinction, we denote the space in which R lies as the uv-plane, while the space in which S lies as the xy-plane. It is then customary to write T ( u,v ) = ( x ( u,v ) ,y ( u,v )) , ( u,v ) ∈ S, where x ( u,v ) and y ( u,v ) denote the first and the second components of T . The point ( x ( u,v ) ,y ( u,v )) is referred to as the image of the point ( u,v ), while the range R = T ( S ) is called the image of S un- der the transformation T . Such a transformation T is said to be continuously differentiable or C 1 provided x u , x v , y u , and y v are well- defined and continuous. We shall always assume that all transformations are C 1 . For our purpose in this section, we also require that a transformation T be injective or one-to-one in the sense that no two points in S has the same image. That is to say, if T ( u 1 ,v 1 ) = T ( u 2 ,v 2 ), then it is necessary that ( u 1 ,v 1 ) = ( u 2 ,v 2 ). In this case, there is an inverse transformation T- 1 : R → S from the xy-plane into the uv-plane such that T- 1 ◦ T ( u,v ) = ( u,v ) and T ◦ T- 1 ( x,y ) = ( x,y ) for all ( u,v ) ∈ S and all ( x,y ) ∈ R ....
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