Difference between revisions of "MAT3213"
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| − | + | '''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 112: | Line 119: | ||
* [[Real Numbers|Real Numbers:Completeness]] | * [[Real Numbers|Real Numbers:Completeness]] | ||
* [[Real Numbers:Archimedean Property|Archimedean Property]] | * [[Real Numbers:Archimedean Property|Archimedean Property]] | ||
| − | * [[Real Numbers:Suprema | + | * [[Real Numbers:Suprema and Infima|Suprema of Subsets]] |
| − | * [[Real Numbers:Infima | + | * [[Real Numbers:Suprema and Infima|Infima of Subsets]] |
* [[Real Numbers:Rational]] | * [[Real Numbers:Rational]] | ||
* [[Real Numbers:Irrational]] | * [[Real Numbers:Irrational]] | ||
| Line 121: | Line 128: | ||
* [[Real Numbers:Algebraic Properties]] | * [[Real Numbers:Algebraic Properties]] | ||
* [[Real Numbers:Inequalities|Inequalities]] | * [[Real Numbers:Inequalities|Inequalities]] | ||
| − | * [[Real Numbers:Order]] | + | * [[Real Numbers|Real Numbers:Order]] |
* [[Real Numbers:Absolute Value|Absolute Value]] | * [[Real Numbers:Absolute Value|Absolute Value]] | ||
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* [[Real Numbers:Sequences|Sequences]] | * [[Real Numbers:Sequences|Sequences]] | ||
* [[Real Numbers:Sequences|Sequences:Convergence]] | * [[Real Numbers:Sequences|Sequences:Convergence]] | ||
| − | * [[ | + | * [[Sequences:Limits]] |
|| <!--Prerequisite Skills--> | || <!--Prerequisite Skills--> | ||
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* [[Real Numbers:Sequences|Sequences:Tails]] | * [[Real Numbers:Sequences|Sequences:Tails]] | ||
| − | * [[ | + | * [[Sequences:Limits|Sequences:Limit Laws]] |
* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]] | * [[Real Numbers:Sequences|Sequences:Bounded when Convergent]] | ||
* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]] | * [[Real Numbers:Sequences|Sequences:Squeeze Theorem]] | ||
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* [[Real Numbers:Sequences|Sequences:Convergence]] | * [[Real Numbers:Sequences|Sequences:Convergence]] | ||
| − | * [[ | + | * [[Sequences:Limits|Sequences:Limits]] |
|| <!--Student Learning Outcomes--> | || <!--Student Learning Outcomes--> | ||
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* [[Proofs:Induction]] <!--- 3213-1.2 ---> | * [[Proofs:Induction]] <!--- 3213-1.2 ---> | ||
* [[Real Numbers:Bounded Subsets|Bounded Sets]] | * [[Real Numbers:Bounded Subsets|Bounded Sets]] | ||
| − | * [[ | + | * [[Sequences:Limits|Sequences:Limit Laws]] |
|| <!--Student Learning Outcomes--> | || <!--Student Learning Outcomes--> | ||
| Line 228: | Line 235: | ||
|| <!--Topics--> | || <!--Topics--> | ||
| − | * [[Sequences | + | * [[Real Numbers:Sequences|Subsequences]] |
* [[Real Numbers:Sequences|Theorem:Monotone Sequence]] | * [[Real Numbers:Sequences|Theorem:Monotone Sequence]] | ||
| − | * [[Theorem:Bolzano-Weierstrass]] | + | * [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]] |
* [[Real Numbers:Sequences|Sequences:Limit Superior]] | * [[Real Numbers:Sequences|Sequences:Limit Superior]] | ||
* [[Real Numbers:Sequences|Sequences:Limit Inferior]] | * [[Real Numbers:Sequences|Sequences:Limit Inferior]] | ||
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* [[Real Numbers:Sequences|Sequences:Monotone]] | * [[Real Numbers:Sequences|Sequences:Monotone]] | ||
| − | * [[ | + | * [[Sequences:Limits|Sequences:Limit Laws]] |
|| <!--Student Learning Outcomes--> | || <!--Student Learning Outcomes--> | ||
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|| <!--Topics--> | || <!--Topics--> | ||
| − | * [[Sequences | + | * [[Real Numbers:Sequences|The Cauchy Criterion]] |
| − | * [[Sequences:Contractive]] | + | * [[Real Numbers:Sequences|Sequences:Contractive]] |
| − | * [[Sequences | + | * [[Properly Divergent Sequences]] |
|| <!--Prerequisite Skills--> | || <!--Prerequisite Skills--> | ||
| − | * [[Sequences:Convergence]] | + | * [[Real Numbers:Sequences|Sequences:Convergence]] |
* [[Sequences:Subsequences]] <!--- 3213-3.4 ---> | * [[Sequences:Subsequences]] <!--- 3213-3.4 ---> | ||
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* [[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 | + | * [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 ---> |
|| <!--Student Learning Outcomes--> | || <!--Student Learning Outcomes--> | ||
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|| <!--Topics--> | || <!--Topics--> | ||
| − | * [[Neighborhoods in | + | * [[Neighborhoods in R]] |
| − | * [[Cluster Points | + | * [[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 | + | * [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 ---> |
|| <!--Student Learning Outcomes--> | || <!--Student Learning Outcomes--> | ||
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* [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]] | * [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]] | ||
| − | * [[ | + | * [[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 ---> | ||
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* [[Real Function Limits:One-Sided|One-Sided Limits of Functions]] | * [[Real Function Limits:One-Sided|One-Sided Limits of Functions]] | ||
* [[Real Function Limits:Infinite|Infinite Limits of Functions]] | * [[Real Function Limits:Infinite|Infinite Limits of Functions]] | ||
| − | * [[Real Function Limits: | + | * [[Real Function Limits:Infinite|Limits of Functions at Infinity]] |
|| <!--Prerequisite Skills--> | || <!--Prerequisite Skills--> | ||
| − | * [[ | + | * [[The Limit of a Function|Limits of Real Functions]] |
| − | * [[ | + | * [[Divergence Criteria]] |
|| <!--Student Learning Outcomes--> | || <!--Student Learning Outcomes--> | ||
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|- | |- | ||
| + | |||
| + | |} | ||
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 |
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| Week 2 |
1.2 & 1.3 |
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| Week 3 |
2.1-2.2 |
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| Week 4 |
2.3-2.4 |
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| Week 5 |
2.5 & 3.1 |
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| Week 6 |
3.2 |
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| Week 7 |
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| Week 8 |
3.3 |
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| Week 9 |
3.4 |
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| Week 10 |
3.5-3.6 |
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| Week 11 |
3.7 |
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| Week 12 |
4.1 |
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| Week 13 |
4.2 |
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| Week 14 |
4.3 |
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