Introduction to analysis

Lectures: 3 sessions / week, 1 hour / session

18.02 Multivariable Calculus; 18.03 Differential Equations; or 18.034 Honors Differential Equations

This course is an introduction to devising mathematical proofs and learning to write them up. It is primarily for students with no prior experience with this.

The subject matter for the first 2/3 of the syllabus (up to Exam 2) is the proofs of one-variable calculus theorems and arguments which use these theorems. The emphasis is on estimation and approximation, two basic tools of analysis. It is assumed that students know ordinary calculus fairly well, or once knew it and will review it when they need to. Calculus is used from the beginning as a source of examples.

The last third goes beyond calculus, getting into uniform convergence of series of functions, to justify differentiation and integration term-by-term; there is similar work involving integrals depending on a parameter, to justify differentiating under the integral sign with respect to the parameter. (Differentiating the Laplace transform F(s) = L(f(t)) with respect to the s-variable is an example.)

Toward the end, there is a brief introduction to point-set topology, which is used in upper-level courses having an analysis prerequisite, and if students are interested, at the very end an even briefer introduction to sets of measure zero and the Lebesgue integral.

Mattuck, Arthur. Introduction to Analysis. Pearson, 1998. ISBN: 9780130811325.

The textbook gives the clearest idea of the course, which follows the textbook closely. The Introduction to Analysis textbook website gives the preface, a detailed table of contents, fifteen pages of sample sections from the first three chapters illustrating the level and style of writing, and a brief account of why the book was written.

Homework is due in class twice weekly, on Monday and Friday, and returned graded at the following class meeting. There are usually from 3-6 problems, depending on their difficulty or length, or whether it's an assignment due Monday or Friday. Sometimes "Questions" are included (exercises having solutions at the end of the chapter), as an aid in learning how to write up solutions, and as a source of hints. Since the homework is really where the learning takes place, and timely feedback is essential to improving, handing in 3/4 of the assignments when they are due is a requirement for passing; any exceptions have to be for cause, and arranged in advance. Students who are accepted into the class late have to make up the missed assignments.

The textbook is by and large an adequate substitute for class attendance; students in the past have found it sufficiently clear. A few just read the book, get the assignments online, and slip the homework under my door before class, retrieving the returned homework from a box outside my door.

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