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Change_of_variables_9-17

# Change_of_variables_9-17 - ∂ = 4 Not that the Jacobian of...

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1 ± It is sometimes convenient or necessary to change the variables x, y to u, v : ∫∫ ∫∫ = R R dxdy y x f dA y x f ) , ( ) , ( ± Double integrals in terms of two variables x and y may be written in the form Change of Variables in Double integrals ( ) () ∫∫ ∫∫ ∫∫ ∫∫ = = S R S R dudv v u J v u h v u g f dxdy y x f dA v u J v u h v u g f dA y x f ) , ( )) , ( ), , ( ( ) , ( ' ) , ( )) , ( ), , ( ( ) , ( ± The double integral after change of variables becomes ) , ( ) , ( v u h y v u g x = = x y u v

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2 1. The integrand function f(x,y) is expressed in terms of u and v 2. dA = dx dy is replaced by J(u,v) du dv , J(u,v) is the Jacobian of the transformation defined by: v y u y v x u x v u y x v u J = = ) , ( ) , ( ) , ( Steps of transformation (change of variables): 3. S is the region in the uv-plane corresponding to R in the xy -plane x y u v ) , ( ) , ( / 1 ) , ( / 1 ) , ( ) , ( ) , ( y x v u y x J v u y x v u J = =

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Unformatted text preview: ∂ = 4. Not that the Jacobian of transformation is related to the Jacobian to the inverse transformation through: 3 • Example Find the Jacobian of the transformation: v e y v e x u u cos , sin ) 8 . 17 . 9 3 3 = = /x y v y/x u 2 2 , ) 9 . 17 . 9 = = 4 • Example Calculate y x v y x u x y x y x y x y R dA y x (x-y) + = − = = + − = − = + − = + ∫∫ 3 , , 6 3 , , 3 3 : 3 cos ) 14 . 17 . 9 2 1 use R π ) , ( ) , ( y x J v u J or 5 y x v y x u-, x-y y x x y x R dA e y x y x − = + = = = + = = + + ∫∫ − use 1 y-R , 1 3 , , 1 : ) ( ) 26 . 17 . 9 4...
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Change_of_variables_9-17 - ∂ = 4 Not that the Jacobian of...

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