Preface Preface
These notes have been constructed to cover the topics in Math412. While there are many excellent textbooks covering the analysis of differential and integral calculus in a single variable---the bulk of the topics for Math411, there are fewer choices covering the integration of functions and differential forms in several variables that provide more geometric discussions and motivation for some of the more abstract notions related to differential forms. This was the initial motivation for constructing these notes. But the notes have been expanded to cover aspects of analysis of a single variable which often form a bridging part with the first semester course mostly surrounding the notions of pointwise convergence and uniform convergence.
Among the most basic questions of this part are
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If a sequence of continuous functions \(f_{k}\) on \([a, b]\) converges pointwise to a function \(f\text{,}\) what can be said about \(f\text{?}\) Is \(f\) necessarily continuous, Riemann integrable, or bounded? The answers to these questions are no in general.
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If a sequence of continuous functions \(f_{k}\) on \([a, b]\) converges uniformly on \([a, b]\) to a function \(f\text{,}\) it is known that \(f\) is continuous, therefore Riemann integrable, and \(\int_{a}^{b}|f_{k}(x)-f(x)|\, dx\to 0\) as \(k\to \infty\text{.}\) Furthermore, the space \(C[a, b]\) of continuous functions with \(\Vert f \Vert_{C[a, b]} :=\max_{[a, b]}|f(x)|\) as a norm is a complete normed space.Is there an analogue in \(C[a, b]\) of the Bolzano-Weierstrass Theorem on \(\bbR\) that any bounded sequence has a convergent subsequence? What if we know that a sequence in \(C[a, b]\) already converges pointwise on a countable and dense subset of \([a, b]\text{?}\) Is this enough to imply that the sequence would converge pointwise or uniformly? The answer turns out to be no in general, and a positive conclusion would require equi-continuity on the sequence.
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\(\Vert f \Vert_{L[a, b]} :=\int_{a}^{b} |f(x)|\, dx\) also defines a norm on \(C[a, b]\text{.}\) Does this norm also make \(C[a, b]\) a complete normed space? The answer is no, and we will discuss in some detail characterization of elements in the completion of \(C[a, b]\) under this and other integral norms.
We will also discuss relations between convergence in integral norms and pointwise or uniform convergence.
The issues above also have their analogues in the context of sequences and series: We know that if \(a_{k, l}\to b_{l}\) as \(k\to \infty\) for each \(l\) and that if \(\sum_{l=1}^{\infty}a_{k, l}\) converges for each \(k\text{,}\) then \(\sum_{l=1}^{\infty}b_{l}\) does not necessarily converge, and even if it does, it may not equal \(\lim_{k\to \infty} \sum_{l=1}^{\infty}a_{k, l}\) when the latter exists, even if \(a_{k, l}\to b_{l}\) converges uniformly in \(l\text{.}\) The relevant condition needed here is equi-summability of the sequence of series. In the general theory of integration, these issues are usually addressed by the so-called dominated convergence theorem. The concrete contexts here reveal the central issues in addressing such convergence problems.
To prepare for our discussion, we will first briefly discuss the Riemann-Stieltjes integral at the beginning of chapter 1. We then discuss properties of elements in the completion of the space \(C[a, b]\) of continuous functions under integral norms adapted from an approach taken by P. Lax. Here we discuss the essential features of Lebesgue’s integral on \(\bbR\) without relying on the measure theory on the so-called \(\sigma\)-algebra of measurable sets. Finally we discuss the notions of absolutely continuous functions and of bounded variation, and their relations to the extension of the Fundamental Theorem of Calculus.
Chapter 2 records the most useful properties related to the convergence of sequences and series of functions. Chapter 3 records the basic properties of power series. A large part of these two chapters is usually done in the first semester.
Chapter 4 discusses the basics of Fourier series. Because there may not be enough time to have a thorough discussion on the properties of elements in the completion of the space \(C[a, b]\) of continuous functions under integral norms, Chapter 4 is written in a way that can be studied without a detailed knowledge of the functions in the completion references above.
Chapters 5--7 focus on the differential and integral calculus of functions of several variables. Here my approach is very close to that of Spivak’s, but I also try to provide more motivated discussions on the origin of the basic notions involved: curl and divergence of vector fields and differential of forms.
The construction of the course notes was sponsored by an ’Open and Affordable Textbooks Program’ (OAT Program) award from the Rutgers University Libraries in spring 2023.
The OAT Program supports textbook affordability at Rutgers by encouraging courses to adopt educational materials that are freely available, available at a low cost (compared to similar courses), or part of the Rutgers University Libraries’ electronic collections, and thereby free of charge to Rutgers University students.
