Difference between revisions of "MAT3213"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
 
(32 intermediate revisions by 4 users not shown)
Line 1: Line 1:
−
The textbook for this course is
+
'''Catalog entry''':
−
Introduction to Real Analysis by Bartle and Sherbert
+
 
 +
''Prerequisites'': MAT 1224 and MAT 3013.
 +
 
 +
''Content'': Axiomatic definition of real numbers, including order properties and completeness; infinite sequences and their convergence; basic notions related to series and their convergence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving.
 +
 
 +
 
 +
 
 +
''Sample textbook'': Introduction to Real Analysis by Bartle and Sherbert
  
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
Line 23: Line 30:
 
* [[Sets:Operations]]
 
* [[Sets:Operations]]
 
* [[Functions:Definition]]  
 
* [[Functions:Definition]]  
−
* [[Functions:Domain]]  
+
* [[Domain of a Function|Functions:Domain]]  
−
* [[Functions:Range]]  
+
* [[Range of a Function|Functions:Range]]  
 
* [[Functions:Injective]]  
 
* [[Functions:Injective]]  
 
* [[Functions:Bijective]]  
 
* [[Functions:Bijective]]  
Line 30: Line 37:
 
* [[Functions:Inverses]]  
 
* [[Functions:Inverses]]  
 
* [[Functions:Forward Image]]  
 
* [[Functions:Forward Image]]  
−
* [[Functions:Inverse Image]]
+
* [[Functions:Forward Image|Functions:Inverse Image]]
 
* [[Functions:Restriction]]  
 
* [[Functions:Restriction]]  
  
Line 50: Line 57:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[NaturalNumbers:Well Ordering]]
+
* [[Natural Numbers:Well-Ordering]]
 
* [[Proofs:Induction]]
 
* [[Proofs:Induction]]
−
* [[Induction:Variants]]
+
* [[Proofs:Induction|Induction:Variants]]
 
* [[Sets:Cardinality]]
 
* [[Sets:Cardinality]]
 
* [[Sets:Finite]]
 
* [[Sets:Finite]]
Line 86: Line 93:
  
 
* [[Real Numbers]]
 
* [[Real Numbers]]
−
* [[RealNumbers:Algebraic Properties]]
+
* [[Real Numbers:Algebraic Properties]]
−
* [[RealNumbers:Order]]
+
* [[Real Numbers|Real Numbers:Order]]
−
* [[RealNumbers:Inequalities|Inequalities]]
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* [[RealNumbers:Absolute Value|Absolute Value]]
+
* [[Real Numbers:Absolute Value|Absolute Value]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
Line 110: Line 117:
 
|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
* [[RealNumbers:Completeness]]
+
* [[Real Numbers|Real Numbers:Completeness]]
−
* [[RealNumbers:Archimedean Property|Archimedean Property]]
+
* [[Real Numbers:Archimedean Property|Archimedean Property]]
−
* [[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
+
* [[Real Numbers:Suprema and Infima|Suprema of Subsets]]
−
* [[RealNumbers:Infima of Subsets|Infima of Subsets]]
+
* [[Real Numbers:Suprema and Infima|Infima of Subsets]]
−
* [[RealNumbers:Rational]]
+
* [[Real Numbers:Rational]]
−
* [[RealNumbers:Irrational]]
+
* [[Real Numbers:Irrational]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealNumbers:Algebraic Properties]]
+
* [[Real Numbers:Algebraic Properties]]
−
* [[RealNumbers:Inequalities|Inequalities]]
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* [[RealNumbers:Order]]
+
* [[Real Numbers|Real Numbers:Order]]
−
* [[RealNumbers:Absolute Value|Absolute Value]]
+
* [[Real Numbers:Absolute Value|Absolute Value]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 138: Line 145:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[RealNumbers:Intervals|Intervals]]
+
* [[Real Numbers:Intervals|Intervals]]
−
* [[RealNumbers:Cardinality]]
+
* [[Cardinality of important sets|Real Numbers:Cardinality]]
−
* [[RealNumbers:Sequences|Sequences]]
+
* [[Real Numbers:Sequences|Sequences]]
−
* [[Sequences:Convergence]]
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
 
* [[Sequences:Limits]]
 
* [[Sequences:Limits]]
  
Line 147: Line 154:
  
 
* [[Sets:Cardinality]]
 
* [[Sets:Cardinality]]
−
* [[RealNumbers:Inequalities|Inequalities]]
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* [[RealNumbers:Bounded Subsets|Bounded Sets]]
+
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 164: Line 171:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[Sequences:Tails]]
+
* [[Real Numbers:Sequences|Sequences:Tails]]
−
* [[Sequences:Limit Laws]]
+
* [[Sequences:Limits|Sequences:Limit Laws]]
−
* [[Sequences:Bounded when Convergent]]
+
* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]]
−
* [[Sequences:Squeeze Theorem]]
+
* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]]
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
−
* [[Sequences:Limits]]
+
* [[Sequences:Limits|Sequences:Limits]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 205: Line 212:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[Sequences:Monotone]]
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
 
* [[Euler's Number]]
 
* [[Euler's Number]]
  
Line 211: Line 218:
  
 
* [[Proofs:Induction]] <!--- 3213-1.2 --->
 
* [[Proofs:Induction]] <!--- 3213-1.2 --->
−
* [[RealNumbers:Bounded Subsets|Bounded Sets]]
+
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
−
* [[Sequences:Limit Laws]]
+
* [[Sequences:Limits|Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 228: Line 235:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[Sequences:Subsequences|Subsequences]]
+
* [[Real Numbers:Sequences|Subsequences]]
−
* [[Theorem:Monotone Sequence]]
+
* [[Real Numbers:Sequences|Theorem:Monotone Sequence]]
−
* [[Theorem:Bolzano-Weierstrass]]
+
* [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]]
−
* [[Sequences:Limit Superior]]
+
* [[Real Numbers:Sequences|Sequences:Limit Superior]]
−
* [[Sequences:Limit Inferior]]
+
* [[Real Numbers:Sequences|Sequences:Limit Inferior]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Monotone]]
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
−
* [[Sequences:Limit Laws]]
+
* [[Sequences:Limits|Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 253: Line 260:
 
|| <!--Topics-->  
 
|| <!--Topics-->  
 
    
 
    
−
* [[Sequences:Cauchy|The Cauchy Criterion]]
+
* [[Real Numbers:Sequences|The Cauchy Criterion]]
−
* [[Sequences:Contractive]]
+
* [[Real Numbers:Sequences|Sequences:Contractive]]
−
* [[Sequences:Divergent]]
+
* [[Properly Divergent Sequences]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]]
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
 
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
 
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
  
Line 277: Line 284:
  
 
* [[Series]]
 
* [[Series]]
−
* [[Series:Nth-Term Test]]
+
* [[Series|Series:Nth-Term Test]]
−
* [[Series:Cauchy Criterion]]
+
* [[Series|Series:Cauchy Criterion]]
−
* [[Series:Nonnegative Terms]]
+
* [[Series|Series:Nonnegative Terms]]
−
* [[Series:Comparison Tests]]
+
* [[Series|Series:Comparison Tests]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* [[Sequences:Cauchy|The Cauchy Criterion]] <!--- 3213-3.5 --->
+
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 302: Line 309:
 
|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
* [[Neighborhoods in 𝐑]]
+
* [[Neighborhoods in R]]
−
* [[Cluster Points in 𝐑]]
+
* [[Cluster Points]]
 
* [[The Limit of a Function|Limits of Real Functions]]
 
* [[The Limit of a Function|Limits of Real Functions]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* [[Sequences:Cauchy|The Cauchy Criterion]] <!--- 3213-3.5 --->
+
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 327: Line 334:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[RealFunctionLimits:Sequential Criterion|The Sequential Criterion for Limits]]
+
* [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]]
−
* [[RealFunctionLimits:Divergence Criteria|Divergence Criteria]]
+
* [[Divergence Criteria]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
  
Line 349: Line 356:
 
|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
* [[RealFunctionLimits:One-Sided|One-Sided Limits of Functions]]
+
* [[Real Function Limits:One-Sided|One-Sided Limits of Functions]]
−
* [[RealFunctionLimits:Infinite|Infinite Limits of Functions]]
+
* [[Real Function Limits:Infinite|Infinite Limits of Functions]]
−
* [[RealFunctionLimits:At Infinity|Limits of Functions at Infinity]]
+
* [[Real Function Limits:Infinite|Limits of Functions at Infinity]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealFunctions:Limits|Limits of Real Functions]]
+
* [[The Limit of a Function|Limits of Real Functions]]
−
* [[RealFunctionLimits:Divergence Criteria|Divergence Criteria]]
+
* [[Divergence Criteria]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 363: Line 370:
  
 
|-
 
|-
 +
 +
|}

Latest revision as of 20:56, 24 March 2023

Catalog entry:

Prerequisites: MAT 1224 and MAT 3013.

Content: Axiomatic definition of real numbers, including order properties and completeness; infinite sequences and their convergence; basic notions related to series and their convergence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving.


Sample textbook: Introduction to Real Analysis by Bartle and Sherbert

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Real Analysis.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

1.1


Week 2

1.2 & 1.3





Week 3

2.1-2.2


Week 4

2.3-2.4


Week 5

2.5 & 3.1


Week 6

3.2


Week 7
  • Review
  • Midterm exam


Week 8

3.3


Week 9

3.4


Week 10

3.5-3.6


Week 11

3.7


Week 12

4.1


Week 13

4.2


Week 14

4.3


Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT3213&oldid=4985"