math21-2012-2up

math21-2012-2up - Intermediate Calculus and Linear Algebra...

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Unformatted text preview: Intermediate Calculus and Linear Algebra Jerry L. Kazdan Harvard University Lecture Notes, 1964–1965 i Preface These notes will contain most of the material covered in class, and be distributed before each lecture (hopefully). Since the course is an experimental one and the notes written before the lectures are delivered, there will inevitably be some sloppiness, disorganization, and even egregious blunders—not to mention the issue of clarity in exposition. But we will try. Part of your task is, in fact, to catch and point out these rough spots. In mathematics, proofs are not dogma given by authority; rather a proof is a way of convincing one of the validity of a statement. If, after a reasonable attempt, you are not convinced, complain loudly. Our subject matter is intermediate calculus and linear algebra. We shall develop the material of linear algebra and use it as setting for the relevant material of intermediate calculus. The first portion of our work—Chapter 1 on infinite series—more properly belongs in the first year, but is relegated to the second year by circumstance. Presumably this topic will eventually take its more proper place in the first year. Our course will have a tendency to swallow whole two other more advanced courses, and consequently, like the duck in Peter and the Wolf, remain undigested until regurgitated alive and kicking. To mitigate—if not avoid—this problem, we shall often take pains to state a theorem clearly and then either prove only some special case, or offer no proof at all. This will be true especially if the proof involves technical details which do not help illuminate the landscape. More often than not, when we only prove a special case, the proof in the general case is essentially identical—the equations only becoming larger. September 1964 ii Afterward I have now taught from these notes for two years. No attempt has been made to revise them, although a major revision would be needed to bring them even vaguely in line with what I now believe is the “right” way to do things. And too, the last several chapters remain unwritten. Because the notes were written as a first draft under panic pressure, they contain many incompletely thought-out ideas and expose the whimsy of my passing moods. It is with this—and the novelty of the material at the sophomore level—in mind, that the following suggestions and students’ reactions are listed. There are three categories, A), Material that turned out to be too difficult (they found rigor hard, but not many of the abstractions), B), changes in the order of covering the stuff, and C), material—mainly supplementary at this level—which is not too hard, but should be omitted if one ever hopes to complete the ”standard” topics within the confines of a year course....
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math21-2012-2up - Intermediate Calculus and Linear Algebra...

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