The Riemann Integral

In this chapter, you will see the full and mature development of analysis and how the paradigms would shift from the headlong pursuit of new results to the need for a solid theory. Indeed, only when the lack of a theory impeded progress did mathematicians invent one. Beginning with a simple observation about the solutions of the heat equation as trigonometric series, mathematics achieved notions of continuity, rigor in integration, set theory and transfinite analysis, measure theory, function spaces, functional analysis, and harmonic analysis. Each of these "notions" constitutes a graduate course or more in the modern mathematics curricula.

Just as the three problems of antiquity would require the development of new worlds of mathematics so to did the simple statement by Daniel Bernoulli about trigonometric series required the introduction of a massive mathematical structure that fueled much of 20th century analysis with no hint whatever of letting up in this one.

Goals

Among the many features you should regard are:

Philosophy and Mathematics.

• Note how infinite series continues to play an important role. What sort of theory was there in the beginning? Just what was a paradox then as opposed to now.
• What significant early approaches seem remarkable.
• Carefully note what was a function to these mathematicians. Is is in anyway a modern notion? How did it evolve?
• Why was continuity needed?
• What is area? You may think you know, but notice the role it played in the theory of the integral.
• Note the power and importance of symbolism at this juncture.
• Has the problem suggested by Bernoulli been fully resolved?

References