What gradients, fundamental forms, measures of curvature, and the Jacobian have in common is that they only produce local results. That is, they only provide information about a surface or a coordinate transformation near a given point. However, it is imperative in mathematics and science that we be able to work with global results, which are results that hold for an entire surface or for all possible values of a coordinate transformation.
In single variable calculus, we learned that global results such as area of a region, volume of a solid, and length of a curve are studied using definite integrals. Similarly, in order to determine global results in multivariable calculus, we need to develop a theory of integration of functions of two or more variables. Thus, this chapter is devoted to the development of the theory of multiple integrals.
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