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15.9withScannedExamples

15.9withScannedExamples - 15.9 CHANGE OF VARIABLES IN...

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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 . Next, we define the Jacobian of T as (1.1) ( x, y ) ( u, v ) = x u x v y u y v = x u y v - x v y u .
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