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