# Changes of Variables and Integration of Forms

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## Abstract

Integration is one of the most a fundamental concepts in mathematics. In calculus you began by learning how to integrate one-variable functions on \(\mathbb {R}\). Then, you learned how to integrate two- and three-variable functions on \(\mathbb {R}^2\) and \(\mathbb {R}^3\). After this you learned how to integrate a function after a change-of-variables, and finally in vector calculus you learned how to integrate vector fields along curves and over surfaces. It turns out that differential forms are actually very nice things to integrate. Indeed, there is an intimate relationship between the integration of differential forms and the change-of-variables formulas you learned in calculus. In section one we define the integral of a two-form on \(\mathbb {R}^2\) in terms of Riemann sums. Integrals of *n*-forms on \(\mathbb {R}^n\) can be defined analogously. We then use the ideas from Chap. 6 along with the Riemann sum procedure to derive the change of coordinates formula from first principles. In section two we look carefully at a simple change of coordinates example. Section three continues by looking at changes from Cartesian coordinates to polar, cylindrical, and spherical coordinates. Finally in section four we consider a more general setting where we see how we can integrate arbitrary one- and two-forms on parameterized one- and two-dimensional surfaces.

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