# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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More Integrals… *whine*. Topic: Calculus. June 4th, 2007

Definition: (Jacobian) If is the transformation from the -plane to the -plane defined by the equations and , then the Jacobian of is denoted by or by and is defined by

,

i.e. the determinant of the matrix of the partial derivatives (also known as the Jacobian matrix). Naturally, this can be generalized to more variables.

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Theorem: If the transformation , maps the region in the -plane into the region in the -plane, and if the Jacobian is nonzero and does not change sign on , then (with appropriate restrictions on the transformation and the regions) it follows that

.

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Problem: Evaluate , where is the region in the first quadrant enclosed by the trapezoid with vertices .

Solution: The bounding lines can be written as , , , and . Now consider the transformation and . In the -plane, the bounding lines of the new region can now be written as , , , and .

We can write and as functions of and : simply and . So the Jacobian .

Then our original integral becomes . And this is equivalent to

.

QED.

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Comment: Note that the above theorem is probably very important in multivariable calculus, as it is the equivalent to -substitution in one variable, which we all know is the ultimate integration technique. It functions in the same way, giving you a lot more flexibility on the function you are integrating and the region you are integrating on.

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Practice Problem: Evaluate , where is the rectangular region enclosed by the lines , , , .