Use the transformation u = x − y, v = x + y to evaluate where R is the square with vertices (0, 2), (1, 1), (2, 2), and (1, 3).
To evaluate: The integral .
The region R is the square with vertices and .
The transformation is , .
Property used: Change of Variable
Change of Variable in double integral is given by, (1)
First find the value of x and y. Add the two transformations will yield, . Thus, . Similarly, subtract two transformations will give . Thus, .
Obtain the Jacobian,
Find the partial derivative of x and y with respect to u and v. then and and then and .
From the given conditions, rewrite the given integral as given below.
Find the boundary by using the given transformation.
For the point ,
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