18.100A Introduction to Analysis
MWF 1-2 4-163
Lectures:
Fall 2012
Prof. A. Mattuck 2-241 3-4345 mattuck@mit.edu
Text: Mattuck: Introduction to Analysis Prentice-Hall
(The 3rd or later printings require fewest corrections.)
Problem sets, two “hour” exams, final exam; each counts about 1/3
Grading:
Web page: http://math.mit.edu/∼apm/f12-18100A.html
Has reading and exercise assignments (as they are made), practice exams (when issued),
and links to corrections to sucessive textbook printings, plus general information about the
course.
The syllabus below is only approximate. In particular, topics after Thanksgiving may be
changed depending on the interests or needs of the students.
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Chap. 1, 2.1-2, App. A Monotone seqs.; completeness; inequalities
Chap. 2.3-6, 3.1 Estimations; limit of a sequence
Chap. 3.2-7 Examples of limits
Chap. 4.1-2, 5.1-.4 Error term; limit theorems
Chap. 5.4-5, 6.2 Subsequences, cluster points
Chap. 6.1-4 Nested intervals, B-W theorem, Cauchy seqs.
Chap. 6.5 Completeness property for sets (F 21 holiday)
Chap. 7.1-5 Infinite series
Chap. 7.6-7 Infinite series
Chap. 8 Power series
Chap. 9,10 Functions; local and global properties
Exam 1 (open book)
Chap. 11.1-3 Continuity (Mon., Tues holiday)
Chap. 11.4-5 Continuity (cont’d)
Chap. 12 Intermediate-value theorem
Chap. 13.1-3 Continuity theorems
Chap. 13.4-5 Uniform continuity
Chap. 14 Differentiation: local properties
Chap. 15 Differentiation: global properties
Chap. 16; 17 (lightly) Convexity; Taylor’s theorem (skip proofs)
Chap. 18 Integrability
Chap. 19 Riemann integral
Chap. 20.1-4 Fund. thms. of calculus
Chap. 21.1-3 Improper integrals, convergence, Gamma function
Chap. 20.5, 21.4 Stirling’s formula; conditional convergence
Exam 2 (open book)
(Fri. holiday)
Chap. 22.1-2 Uniform convergence of series
Chap. 22.3-4 Continuity of sum; integration term-by-term
Chap. 22.5-6 Differentiation term-by-term; analyticity
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30. Chap. 24.1-5 Continuous functions on the plane
31. App. B Quantifiers and Negation
(Thanksgiving Holiday Thur. and Fri.)
32 Chap. 24.6-7, 25.1 Plane point-set topology
33. Chap. 25.2-3 Compact sets and open sets
34. Chap. 26.1-2 Differentiating integrals w.r.t. a parameter
35. Chap. 26.2-3 Leibniz and Fubini theorems
36. Chap. 27.1-3 Improper integrals with a parameter
37. Chap. 27.4-5 Differentiating and integrating improper integrals
38. Chap. 23.1-.2 Countability; sets of measure zero
39. Chap. 23.3-4 Intro. to Lebesgue integral; review
Three-hour final exam during finals week (open book)