Difference between revisions of "MAT3213"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Fixed broken comments!)
(Removed Student Learning Outcomes (MAT3213).)
Line 28: Line 28:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Subsets
+
<!-- ------------------------------------------------------------ -->
−
* The definition of equality between two sets
 
−
* Commonly used sets
 
−
 
 
  
 
|-
 
|-
Line 54: Line 51:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Union, intersection and complements of sets
+
<!-- ------------------------------------------------------------ -->
−
* De Morgans Laws for sets
 
−
* Infinite Unions and intersections of sets
 
  
 
|-
 
|-
Line 79: Line 74:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The Cartesian Product
+
<!-- ------------------------------------------------------------ -->
−
* Definition of a function
 
−
* Domain and Range in terms of the Cartesian product
 
−
* Transformations and Machines
 
−
 
 
  
 
|-
 
|-
Line 106: Line 97:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of the Direct Image
+
<!-- ------------------------------------------------------------ -->
−
* Definition of the Inverse Image
 
  
 
|-
 
|-
Line 131: Line 121:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Injective functions
+
<!-- ------------------------------------------------------------ -->
−
* Surjective functions
 
−
* Bijective functions
 
  
 
|-
 
|-
Line 156: Line 144:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of Inverse functions
+
<!-- ------------------------------------------------------------ -->
−
* Criteria for an Inverse of a function to exist
 
  
 
|-
 
|-
Line 179: Line 166:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of a composition function
+
<!-- ------------------------------------------------------------ -->
−
* When function composition is defined
 
  
 
|-
 
|-
Line 202: Line 188:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Define the restriction of a function
+
<!-- ------------------------------------------------------------ -->
−
* Positive Square Root function
 
  
 
|-
 
|-
Line 225: Line 210:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Well-ordering principal
+
<!-- ------------------------------------------------------------ -->
−
* Principal of Mathematical induction
 
−
* The principal of Strong Induction
 
  
 
|-
 
|-
Line 250: Line 233:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of finite and infinite sets
+
<!-- ------------------------------------------------------------ -->
−
* Uniqueness Theorem
 
−
* If T is a subset of S and T is infinite, then S is also infinite.
 
  
 
|-
 
|-
Line 274: Line 255:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Countable and Uncountable sets
+
<!-- ------------------------------------------------------------ -->
−
* The set of rational numbers is countable
 
−
* Cantor's Theorem
 
−
 
 
  
 
|-
 
|-
Line 298: Line 276:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Algebraic properties of the Real Numbers
+
<!-- ------------------------------------------------------------ -->
−
 
 
  
 
|-
 
|-
Line 322: Line 299:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The Rational Numbers
+
<!-- ------------------------------------------------------------ -->
−
* Proof that the Square Root of 2 does not exist in the rational numbers
 
−
* The Irrational Numbers
 
−
 
 
  
 
|-
 
|-
Line 347: Line 321:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The ordering properties of the real numbers
+
<!-- ------------------------------------------------------------ -->
−
* Tricotomy property
 
−
* If 0 <= a < x for each x in the positive real numbers, then a = 0.
 
−
 
 
  
 
|-
 
|-
Line 372: Line 343:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Using the order properties to solve equations
+
<!-- ------------------------------------------------------------ -->
−
* Arithmetic-geometric mean
 
−
* Bernoulli's Inequality
 
−
 
 
  
 
|-
 
|-
Line 397: Line 365:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The absolute value function
+
<!-- ------------------------------------------------------------ -->
−
* The Triangle Inequality
 
−
* Distance between elements of the real numbers
 
−
* Definition of an epsilon neighborhood
 
−
 
 
  
 
|-
 
|-
Line 423: Line 387:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Upper and lower bounds of sets
+
<!-- ------------------------------------------------------------ -->
−
* Definition of the suprema and infima of a set
 
−
* Thed completeness property of the real numbers
 
−
 
 
−
 
 
  
 
|-
 
|-
Line 450: Line 410:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Bounded Functions
+
<!-- ------------------------------------------------------------ -->
−
* The Archimedean Property
 
−
* The existence of the square root of 2
 
−
* Density of the rational numbers in the real numbers
 
−
 
 
  
 
|-
 
|-
Line 476: Line 432:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Types of Intervals
+
<!-- ------------------------------------------------------------ -->
−
* Characterization of Intervals
 
−
* Nested intervals
 
−
* The Nested Intervals Property
 
−
* Demonstrate that the real numbers are not countable
 
−
 
 
  
 
|-
 
|-
Line 504: Line 455:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of the limit of a sequence
+
<!-- ------------------------------------------------------------ -->
−
* The uniqueness of limits in the real numbers
 
−
* Tails of sequences
 
−
* Examples of common sequences
 
−
 
 
−
 
 
  
 
|-
 
|-
Line 531: Line 477:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Bounded Sequences
+
<!-- ------------------------------------------------------------ -->
−
* Summation, difference, products, and quotients of sequences
 
−
* The squeeze theorem for sequences
 
−
* Divergent Sequences
 
−
 
 
−
 
 
  
 
|-
 
|-
Line 559: Line 500:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Increasing and Decreasing sequences
+
<!-- ------------------------------------------------------------ -->
−
* The Monotone Convergence theorem
 
−
* Inductively defined sequences
 
−
* The existence of Euler's Number
 
−
 
 
  
 
|-
 
|-
Line 585: Line 522:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of a Subsequence
+
<!-- ------------------------------------------------------------ -->
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
  
 
|-
 
|-
Line 611: Line 543:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The Bolzano Weierstrass Theorem
+
<!-- ------------------------------------------------------------ -->
−
* Examples using the Bolzano Weierstrass Theorem
 
−
 
 
  
 
|-
 
|-
Line 635: Line 565:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of the limit superior and limit inferior
+
<!-- ------------------------------------------------------------ -->
−
* Equivalent statements defining the limit superior and limit inferior
 
−
* A bounded sequence converges if and only if its limit superior equals its limit inferior
 
−
 
 
  
 
|-
 
|-
Line 660: Line 587:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of a Cauchy sequence
+
<!-- ------------------------------------------------------------ -->
−
* A sequence converges if and only if it is a Cauchy sequence
 
−
* Contractive sequences
 
−
 
 
  
 
|-
 
|-
Line 685: Line 609:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Limits that tend to infinity
+
<!-- ------------------------------------------------------------ -->
−
* Properly divergent sequences
 
−
 
 
  
 
|-
 
|-
Line 709: Line 631:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Sequences of partial sums
+
<!-- ------------------------------------------------------------ -->
−
* If a series converges, then the sequence of coefficients for that series  must converge to zero.
 
−
* Examples of common series
 
−
* Comparison tests for series
 
  
 
|-
 
|-
Line 734: Line 653:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Definition of a cluster point
+
<!-- ------------------------------------------------------------ -->
−
* The cluster point as the limit of a sequence
 
−
 
 
  
 
|-
 
|-
Line 758: Line 675:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The definition of the limit of a function at a point
+
<!-- ------------------------------------------------------------ -->
−
* The uniqueness of limits at cluster points
 
−
* Examples of limits of functions
 
  
 
|-
 
|-
Line 782: Line 697:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The sequential criterion for limits of functions at a point
+
<!-- ------------------------------------------------------------ -->
−
* Divergence criteria for limits
 
−
* The signum function
 
−
 
 
  
 
|-
 
|-
Line 807: Line 719:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* Functions bounded on a neighborhood of a cluster point
+
<!-- ------------------------------------------------------------ -->
−
* Sums, differences, products, and quotients of limits
 
−
* The squeeze theorem for limits of functions
 
−
* Examples of Limits using the limit theorems
 
−
 
 
  
 
|-
 
|-
Line 833: Line 741:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The definition of the right and left hand limits of a function at a point
+
<!-- ------------------------------------------------------------ -->
−
* The sequential criterion for the left and right hand limits
 
−
* The limit of a function at a point exists if and only if its left and right hand limits are equal
 
−
 
 
  
 
|-
 
|-
Line 858: Line 763:
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
* The definition of an infinite limit
+
<!-- ------------------------------------------------------------ -->
−
* If the function f is less than the function g on a specified domain and f tends to infinity, then g tends to infinity on this domain as well.
 
−
* The definition of a limit has its independent variable approaches infinity
 
−
* The sequential criterion for limits at infinity
 
−
 
 
  
 
|-
 
|-

Revision as of 18:08, 13 August 2020

The textbook for this course is 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

Basic Terminology


Week 1

1.1


Set Operations


Week 1

1.1

Functions (The Cartesian product definition)


Week 1/2

1.1

Direct and Inverse Images


Week 1/2

1.1


Injective and Surjective Functions


Week 1/2

1.1


Inverse Functions


Week 1/2

1.1

Composition of Functions


Week 1/2

1.1


Restrictions on Functions


Week 2

1.2

Mathematical Induction


Week 2

1.3


Finite and Infinite Sets


Week 2

1.3

Countable Sets


Week 3

2.1

Algebraic Properties of the Real Numbers


Week 3

2.1

Rational and Irrational Numbers


Week 2

2.1

The Ordering Properties of the Real Numbers


Week 2

2.1

Inequalities


Week 2/3

2.2

Absolute Value and the Real Line


Week 3

2.3

Suprema, Infima, and the Completeness Property


Week 3

2.4

Applications of the Supremum Property


Week 3/4

2.5

Intervals


Week 4

3.1

Sequences and Their Limits


Week 4

3.2

The Limit Laws for Sequences


Week 4/5

3.3

Monotone Sequences


Week 5

3.4

Subsequences


Week 5

3.4

The Bolzano Weierstrass Theorem


Week 5/6

3.4

The Limit Superior and Limit Inferior


Week 6

3.5

The Cauchy Criterion for Convergence


Week 6

3.6

Properly Divergent Sequences


Week 6/7

3.7

Introduction to Infinite Series


Week 12

4.1

Cluster Points


Week 12

4.1

The Definition of the Limit of a Function


Week 12/13

4.1

The Sequential Criterion and Divergence Criteria


Week 13

4.2

The Limit Theorems for Functions


Week 14

4.3

One Sided Limits


Week 14/15

4.3

Infinite Limits and Limits at Infinity


Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT3213&oldid=514"