Chapter 6 Integration in Several Variables
There are many aspects of integration in several variables. In this chapter we first discuss the definition and evaluation of the integral of well behaved scalar functions on simple domains in the Euclidean space \(\bbR^{n}\) which are generalizations of rectangular boxes, triangles and tetrahedrons in two and three dimensions, called hypercubes and simplexes respectively. We then extend the discussion to domains with piecewise \(C^{1}\) boundary. In the next chapter we will extend the discussion, where the domain of integration is a lower dimensional surface in the Euclidean space \(\bbR^{n}\text{,}\) and, first, the integrand is a scalar function as usual, then, the integrand is given in terms of coupling of a differential form of degree \(k\) and a \(k\)-dimensional surface, namely, the integration of a differential form over a surface or a singular chain. This is a generalization of the notion of the circulation of a vector field along a curve or of the flux of a vector field over a surface, where the coupling is done involving the tangent of the curve or the normal of the tangent plane of the surface,
Below are some main issues to be addressed.
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It is not too different from integration in one dimension if one only discusses integrals of a scalar function defined in a multi-dimensional rectangle. One can establish a similar integrability criterion and prove a version of the Fubini Theorem, which reduces the evaluation of the integral to iterated one variable integrals.
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Complications come in when one tries to define integrals in more general domains in multi-dimensions. One main issue is how to properly define partitions. Partitions in the Riemann sense require that there is a well defined notion of area (volume) for the cells used in the partition, which is why traditionally we only confine to rectangular cells in partitions. Another aspect is that we want the cells to be non-overlapping, and this is relatively easy to achieve using rectangular cells.One way to resolve the issue of decomposing a general domain as a non-overlapping union of rectangular cells is to enclose the domain (required to be bounded in Riemann integration) by a rectangular box and extend the integrand to be \(0\) in the complement. This would require accounting for the contributions to the upper and lower integrals from those rectangular cells in the partition of the enclosing rectangular box that "straddle" between the domain and its complement. This is not too difficult to address, but one does need to control the measure of the boundary of the domain of integration.
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In any integration theory one expects the integral \(\int_{A}f\) to be linear in the integrand \(f\text{,}\) namely, for any integrable \(f, g\text{,}\) and constants \(a, b\text{,}\)\begin{equation*} \int_{A} \left( a f+ b g\right)\, = a \int_{A}f + b \int_{A}g. \end{equation*}It is also natural to expect that when \(A\) can be decomposed as a non-overlapping union of two subsets \(A_{1}\cup A_{2}\text{,}\)\begin{equation*} \int_{A}f =\int_{A_{1}}f + \int_{A_{2}}f\text{.} \end{equation*}One also hopes that the converse holds: if the terms on the right hand side are well defined, then the left hand side should be well defined and equals the right hand side.In one dimension this is certainly true when \(A_{1}, A_{2}\) are intervals. But even in that context, Riemann integration theory places restrictions on these sets: if we take \(A=[0,1], A_{1}={\mathbb Q}\cap [0, 1], A_{2}=[0,1]\setminus A_{1}\text{,}\) then it is reasonable to argue that Dirichlet’s non-Riemann integrable function on \([0,1]\) could be regarded as integrable on both \(A_{1}\) and \(A_{2}\) as it is a constant on either set, but its Riemann integral on \([0, 1]\) is not defined. One key issue is that Riemann’s integration theory does not allow general sets such as \(A_{2}\) here as cells in the partition, and each partition has to be a finite partition. As a result one can’t directly define Riemann integral on a domain with infinite length (area or volume) or when the integrand is unbounded.
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A more serious issue is that reasonable limits of Riemann integrable functions may not be Riemann integrable. The rectification of this issue requires by Lebesgue’s integration theory.
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The simplistic idea of using "nice cells" to partition a domain of integration becomes more challenging when one tries to carry it out to surfaces or hypersurfaces. The issue of defining the area (volume) of cells on such surfaces has to be tackled first.
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A basic difficulty in discussing the integration on a surface is how to properly define a surface. Unlike in the case of a curve, which can always be thought of as defined through a map on an interval, there is no simple or canonical domain which can be used to define a surface, except for a patch of a surface.
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For the purpose of integration theory on a surface, it suffices that one can partition the surface into some non-overlapping union of patches such that each patch has a parametric representation through a map defined on a nice domain such as a rectangular box or simplex, and one defines the integration on each such patch on this nice domain through this parametrization.
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The implementation of the above approach still requires substantial preparation work. First, the existence and construction of such a partition would require careful proof. Second, there is usually no canonical parametrization for a patch of a surface, so one has to verify that different parametrizations do not lead to different values of the integrals.It is here that the change of variables in multiple integrals becomes crucial.
