Problem Set 9 Solution

12 let u y 2x and v 2y x then d is the region 0

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Unformatted text preview: he map is linear, it follows that the map is injective, and so by the Inverse function theorem it defines a diffeomor­ phism. Therefore � � � 2 �� (x/2)+1 � �� 2 12 2 5 (2y −x) 5 v2 x (2y − x)e dx dy = u v e dv du 20 0 x/2 0 � 1 2 5 � v2 �2 = u e du 40 0 � e4 − 1 2 5 = u du 4 0 e4 − 1 � 6 �2 = u0 24 8(e4 − 1) = . 3 11. Let u = 2x + y and v = x − y . Then � � �2 1 � ∂ (u, x) � � = −3. (x, y ) = � � 1 −1 ∂ (x, y ) So ∂ (x, y ) 1 (u, v ) = − . ∂ (u, v ) 3 This is nowhere zero. As the map is linear, it follows that the map is injective, and so by the Inverse function theorem it defines a diffeomor­ phism. Therefore, � ��...
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