Difference between revisions of "Math 341: Theory of Analysis 1"
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Revision as of 16:04, 3 April 2013
Contents
Catalog Information
Title
Theory of Analysis 1.
(Credit Hours:Lecture Hours:Lab Hours)
(3:3:0)
Offered
F, W
Prerequisite
Description
Rigorous treatment of calculus of a single real variable: topology, order, completeness of real numbers; continuity, differentiability, integrability, and convergence of functions.
Desired Learning Outcomes
The main purpose of this course is to provide students with an understanding of the real number system and of realvalued functions of a single real variable, with the focus being on the theoretical and logical foundations of singlevariable calculus. A secondary purpose of this course is to reinforce students' prior training in discovering and writing valid mathematical proofs.
Prerequisites
The prerequisites for this course are Math 113 and 290. The first is to ensure that the student has had a complete course in singlevariable calculus at the introductory level, and the second is to ensure that the student knows how to read and write proofs and is familiar with the fundamental objects of advanced mathematics.
Minimal learning outcomes
Outlined below are topics that all successful Math 341 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.
 Basic properties of R
 Characterization as a complete, ordered field
 Archimedean Property
 Density of Q and R \ Q
 Uncountability of each interval
 Convergence of real sequences and series
 Convergence of bounded monotonic sequences
 Algebraic and order rules for limits
 Cauchy criterion for sequences and series
 Common convergence tests for series
 Convergence of rearranged series
 Basic topology of R
 Open and closed sets
 Limit points and limits
 Characterizations of compactness
 Sequential
 Open coverings
 The HeineBorel Theorem
 The BolzanoWeierstrass Theorem
 Connectedness
 Continuity of f: D ⊆ R → R
 Functional limits
 Metric, sequential, and topological characterizations of continuity
 Combinations of continuous functions
 Continuity vs. uniform continuity
 Preservation of compactness
 The Extreme Value Theorem
 Uniform continuity for compact domains
 Preservation of connectedness
 The Intermediate Value Theorem
 Differentiability of f: D ⊆ R → R
 Algebraic differential rules
 Chain rule
 Characterizing extrema
 Rolle's Theorem
 The Mean Value Theorem
 The Generalized Mean Value Theorem
 L'Hôpital's Rule
 Integrability of f: [a,b] → R
 The Darboux integral
 The Riemann integral
 Integrability of continuous functions
 Integrability of monotonic functions
 Rules for combining and comparing integrals
 The Fundamental Theorem(s) of Calculus
 Convergence of sequences and series of functions
 Pointwise vs. uniform convergence
 Relation of uniform convergence to:
 Continuity
 Differentiation
 Integration
 The Weierstrass MTest
 The Weierstrass Approximation Theorem
 Power series
 Continuity
 Absolute and uniform convergence
 Termwise differentiability
 Taylor's Theorem
 Analyticity vs. smoothness
Textbooks
Possible textbooks for this course include (but are not limited to):
 Stephen Abbott, Understanding Analysis, Springer, 2001.
Additional topics
Among other options, instructors may want to discuss constructing the real numbers using Dedekind cuts or Cauchy sequences.
Courses for which this course is prerequisite
As the foundational course in real analysis, Math 341 is a prerequisite for many advanced undergraduate and graduate courses in pure and applied analysis: Math 342, 451, 465, 534, 541, and 634.