Ls5: /* Offered */

2019-11-14T16:46:31Z

‎Offered

Dac47: /* Textbooks */

2015-08-29T19:28:18Z

‎Textbooks

Ls5: moved Math 341 to Math 341: Theory of Analysis 1

2013-04-03T22:04:01Z

moved Math 341 to Math 341: Theory of Analysis 1

Cpg at 14:48, 19 July 2010

2010-07-19T14:48:24Z

Cpg: /* Minimal learning outcomes */

2009-12-31T21:03:01Z

‎Minimal learning outcomes

Cpg: New page: == Catalog Information == === Title === Theory of Analysis 1. === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Offered === F, W === Prerequisite === Math 113, [[Math 290|...

2009-06-04T19:36:23Z

New page: == Catalog Information == === Title === Theory of Analysis 1. === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Offered === F, W === Prerequisite === Math 113, [[Math 290|...

New page

== Catalog Information ==

=== Title ===
Theory of Analysis 1.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
F, W

=== Prerequisite ===
[[Math 113]], [[Math 290|290]].

=== 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 real-valued functions of a single real variable, with the focus being on the theoretical and logical foundations of single-variable 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 [[Math 113|113]] and [[Math 290|290]]. The first is to ensure that the student has had a complete course in single-variable 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.

<div style="-moz-column-count:2; column-count:2;">

# 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 Heine-Borel Theorem
#* The Bolzano-Weierstrass 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 ''M''-Test
#* The Weierstrass Approximation Theorem
#* Power series
#** Continuity
#** Absolute and uniform convergence
#** Termwise differentiability
#** Taylor's Theorem
#** Analyticity vs. smoothness

</div>

=== 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 [[Math 342|342]], [[Math 451|451]], [[Math 465|465]], [[Math 534|534]], [[Math 541|541]], and [[Math 634|634]].

[[Category:Courses|341]]

@@ Line 93: / Line 93: @@
 Possible textbooks for this course include (but are not limited to):
-* Stephen Abbott, ''Understanding Analysis'', Springer, 2001.
+* Stephen Abbott, ''Understanding Analysis, 2nd Edition'', Springer, 2015.
 === Additional topics ===

← Older revision		Revision as of 14:48, 19 July 2010
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	</div>		</div>
		+
		+	=== Textbooks ===
		+	Possible textbooks for this course include (but are not limited to):
		+
		+	* Stephen Abbott, ''Understanding Analysis'', Springer, 2001.

	=== Additional topics ===		=== Additional topics ===

@@ Line 8: / Line 8: @@
 === Offered ===
-F, W
+F, W, Sp
 === Prerequisite ===

← Older revision	Revision as of 22:04, 3 April 2013
(No difference)