jacobian - ∂x 1 ∂y 1 = cos y 2 , ∂x 1 ∂y 2 =-y 1...

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Statistics 351 (Fall 2007) The Jacobian for Polar Coordinates Example. Determine the Jacobian for the change-of-variables from cartesian coordinates to polar coordinates. Solution. The traditional letters to use are x = r cos θ and y = r sin θ. However, to agree with the notation from class, we let x 1 = y 1 cos y 2 and x 2 = y 1 sin y 2 . In other words, our original variables are x = ( x 1 , x 2 ), and our new variables are y = ( y 1 , y 2 ). The variables x and y are related through the function g : R 2 R 2 defined implicitly by g ( x ) = g ( x 1 , x 2 ) = ( g 1 ( x 1 , x 2 ) , g 2 ( x 1 , x 2 )) = ( y 1 , y 2 ) . In other words, y 1 = q x 2 1 + x 2 2 and y 2 = arctan( x 2 /x 1 ) . We now compute the required partial derivatives:
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Unformatted text preview: ∂x 1 ∂y 1 = cos y 2 , ∂x 1 ∂y 2 =-y 1 sin y 2 , ∂x 2 ∂y 1 = sin y 2 , ∂x 2 ∂y 2 = y 1 cos y 2 . Therefore, the Jacobian is given by J = ± ± ± ± ± ± ± ∂x 1 ∂y 1 ∂x 1 ∂y 2 ∂x 2 ∂y 1 ∂x 2 ∂y 2 ± ± ± ± ± ± ± = ± ± ± ± cos y 2-y 1 sin y 2 sin y 2 y 1 cos y 2 ± ± ± ± = y 1 cos 2 y 2 + y 1 sin 2 y 2 = y 1 . Thus, using the traditional notation, J = r . i.e., d x d y = r d r d θ ....
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This note was uploaded on 03/26/2012 for the course STAT 351 taught by Professor Michaelkozdron during the Fall '08 term at University of Regina.

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