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Change of Variable Theorem

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Date Posted: 7/23/2008 3:13:21 PM  Status: Live
Change of Variable Theorem
Course Textbook Chapter Problem
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Question Details:
Really appreciate the help!

Let D be a parallelogram with vertices (0,0), (2,0), (1,2), (3,2) and D* = [0,1] x [0,1].

a) Find a linear transformation such that T(D*)=D.
b) Find the jacobian determinant of T and apply the change of variables theorem in double integral to find the following integral
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Date Posted: 8/11/2008 11:49:02 AM  Status: Live
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Response:
a) A linear transformation is defined by (0,0)->(0,0) no affine part. (2,0)->(1,0) implies a21=0  and a11=1/2. (1,2)->(0,1) implies a11=1/2 and a12=-1/4. a22=1/2.
 det(T)=1/4. . x=2u+v, y=2v. The integrant transforms to 96u2vev. The double integration factors. u3/3 und ev(v-1) in each borders 0 to 1, gives 1/3 and 1. det(T)=1/4 is the Jacobian determinate. The integrant is 8.



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