13.8 Notes - If f is continuous on R, g and h have...

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Calculus III Section 13.8 – Jacobians Definition of the Jacobian If ) , ( v u g x = and ) , ( v u h y = , the Jacobian of x and y with respect to u and v , denoted by ) , ( ) , ( v u y x , is v y u y v x u x v u y x = ) , ( ) , ( Find the Jacobian for v u x cos 2 = and v u y sin 2 = . We are going to use the Jacobian to help us transform a “non-regular” region in the plane to one that is easier to integrate. Change of Variables for Double Integrals Let R and S be regions in the xy - and uv -planes that are related by the equations ) , ( v u g x = and ) , ( v u h y = such that each point in R is the image of a unique point in S.
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Unformatted text preview: If f is continuous on R, g and h have continuous partial derivatives on S, and ) , ( ) , ( v u y x is nonzero on S, then ( 29 = S R dudv v u y x v u h v u g f dxdy y x f ) , ( ) , ( ) , ( ), , ( ) , ( . Example: (#20) Use a change of variables to find the volume of the solid region lying below the surface 2 / 3 ) 2 )( 2 3 ( ) , ( x y y x y x f-+ = and above the plane region R that is bounded by the parallelogram with vertices (0,0), (-2,3), (2,5), (4,2)....
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This note was uploaded on 01/13/2012 for the course MATH 2574 taught by Professor Pamelasatterfield during the Spring '12 term at NorthWest Arkansas Community College.

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