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If you are stuck in a calculus problem and don’t know what else to do, try integrating by parts or changing variables.
—Jerry Kazdan
God does not care about our mathematical difficulties. He integrates empirically.
—Albert Einstein
The change of variables formula is one of the most powerful integration methods in single-variable calculus; it enables us to evaluate integrals such as \[ \int^{1}_{0} \boldsymbol{\mathit{xe}}^{{x}^{2}} \boldsymbol{\mathit{dx}} \] by using the substitution, or change of variables \(u = x^2\), which reduces the problem to the easy task of integrating \(e^{u}\) with respect to \(u\). In this chapter, we develop the multidimensional change of variables formula, which is especially important and useful in evaluating multiple integrals in polar, cylindrical, and spherical coordinates.
One of the key ingredients in the change of variables formula is how to change variables in several dimensions. This involves the notion of mapping, which occurs in various interesting situations. For example, consider a deforming object, such as a swimming fish. As it changes its shape, one can imagine the instantaneous correspondence between points on the fish in its rest state and in its current shape. This type of correspondence is, in fact, the main idea behind a change of variables, in this case, of one three-dimensional region (the fish in its rest state) to another (the fish in its current shape).
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The first section in this chapter describes the key concepts for mappings between regions of the plane. It goes on to develop the change of variables technique for double and then triple integrals. The chapter also includes some of the important physical applications of the integral.