Undergraduate Courses

M 301 College Algebra

Prerequisite and degree relevance

A passing score on the mathematics section of the Texas Higher Education Assessment (THEA) test (or an appropriate assessment test). May not be counted toward a degree in mathematics. Credit for Mathematics 301 may not be earned after a student has received credit for any calculus course with a grade of C- or better.

Course description

Topics include a brief review of elementary algebra; linear, quadratic, exponential, and logarithmic functions; polynomials; systems of linear equations; applications. Usually offered only in the summer session. 

Text

Durbin, College Algebra, preliminary third edition, McGraw-Hill College Custom Series, 1993

M 301 is the lowest-level "precalculus" course we offer. It should be an honest college algebra course, that is, not an intermediate algebra course (which is offered by community colleges and some four-year colleges and which is often equivalent to second-year high school algebra.) This syllabus is written for use in summer school (the only time we offer M 301). It assumes 26 lectures.

• Chapter 1 Five Fundamental Themes 5 sections 4 lectures
• Chapter 2 Algebraic Expressions 5 sections 4 lectures
• Chapter 3 Equations and Inequalities 5 sections 5 lectures
• Chapter 4 Graphs and Functions  4 sections 4 lectures
• Chapter 5 Polynomial and Rational Functions 4 sections 4 lectures
• Chapter 6 Exponential, Logarithmic Functions 4 sections 3 lectures
• Chapter 7 Systems of Equations, Inequalities 3 sections 2 lectures

M 302 Introduction to Mathematics

Prerequisite and degree relevance

Texas Success Initiative (TSI) exemption or a TSI Mathematics Assessment score of 350 or higher. The placement test is not required. It may be used to satisfy Area C requirements for the Bachelor of Arts degree under Plan I or the mathematics requirement for the Bachelor of Arts degree under Plan II.

M 302 is Intended primarily for general liberal arts students seeking knowledge of the nature of mathematics as well as training in mathematical thinking and problem-solving. Mathematics 302 and 303F may not both be counted. A student may not earn credit for Mathematics 302 after having received credit for any calculus course. May not be counted toward a degree in the College of Natural Sciences.

Course Description

Introduction to Mathematics is a terminal course satisfying the University's general education requirement in mathematics. Topics may may be chosen from: Fibonacci numbers, number theory (divisibility, prime numbers, the Fundamental Theorem of Arithmetic, gcd, Euclidean Algorithm, modular arithmetic, special divisibility tests), infinity, geometry (Pythagorean Theorem, Platonic Solids, the fourth dimension, rubber sheet geometry, the Moebius band), chaos and fractals, probability (definition, laws, permutations and combinations), network theory (Euler circuits, traveling salesman problem, bin packing), statistics, game theory, voting paradoxes. Some material is of the instructor's choosing. 

Texts

For all Practical Purpose or The Heart of Mathematics, Fourth Edition (preferred)

There is a broad spectrum of students who take M302. Some are quite good at math and may even have had some calculus in high school. These, however, are greatly outnumbered by the students who have weak math skills and poor backgrounds. It is not at all uncommon for the students to exhibit a fear of and dislike for math and most have very low self-confidence about their ability to succeed in a math class. In answer to this, the goal of the course should be to demonstrate that math is not about memorizing formulas, but is rather a process of thinking which is relevant to them on a daily basis. The two recommended books, both are geared toward this type of course. For All Practical Purpose emphasizes applications of math in today's world such as scheduling problems and consumer finance models, for example. The Heart of Mathematics, while dealing with more theoretical topics such as number theory and topology, emphasizes that the problem-solving strategies used to solve mathematical problems are universal and can be applied to solving day-to-day problems. Both texts have proven to be successful at engaging this population of students and giving them a new appreciation of math as well as boosting their self-confidence. 

The topics to be covered will depend on the choice of text. Both texts cover probability and statistics and at least 3 weeks of the course should be devoted to this topic. The coverage in For all Practical Purpose is more thorough, especially in the area of statistics. If this is the chosen text, then the syllabus should include chapter 5 and 7. Chapter 6 can be covered lightly, if at all, and chapter 8 should be considered optional. If The Heart of Mathematics is the chosen text, then all of chapter 7 should be covered.

Sample Syllabus for “For All Practical Purpose”

• Chapter 1 Street Networks All sections (3 days)
• Chapter 2Visiting Vertices(omit Minimum cost spanning trees) (4 days)
• Chapter 3 Planning and Scheduling(omit Bin Packing) (4 days)
• Chapter 5 Producing Data All sections(4 days)
• Chapter 6 Exploring Data Cover lightly (2 days)
• Chapter 7 Probability All sections (5 days)
• Chapter 10 Transmitting Information (supplement the modular arithmetic and cover cryptography only) (4 days)
• Chapter 15 Game Theory All sections (5 days)
• Chapter 20 Consumer Finance Models All section (time permitting) (6 days)

Notes: For all Practical Purposes 

Chapter 1 is an introduction to graph theory and is a good chapter for establishing the course as one which is not "formula-based."   Chapter 2 and 3 then follow up with some applications of graph theory.

As mentioned above, Chapters 5 and 7 should be covered thoroughly and Chapter 6 lightly.  Chapters 9 and 10 introduce the concept of modular arithmetic with applications to error detecting codes cryptography. Students tend to find the arithmetic challenging, but in general, they enjoy the ideas in these chapters.     

Chapter 13 on Fair Division is fun to do, however, it is difficult to get the ideas across. Students tend to get lost in the logic and may end up simply memorizing procedures. 

Chapter 15, Game Theory, also gives the students a work-out in the area of following a logical argument and again they tend to memorize algorithms for finding good strategies. This chapter does give a chance to revisit expected value and they also appreciate the real-world applications of the "Prisoner's Dilemma" problems. 

Chapter 20 deals with compound interest and annuities. The relevance of this material to their lives makes it one of the most widely-appreciated chapter on the part of the students. 

Sample syllabus for “The Heart of Mathematics”

• Chapter 1: Fun and Games All sections (2 days)
• Chapter 2: Number Contemplation 2.1-2.3; 2.6-2.7 (7 days)
• Chapter 3: Infinity Sections 3.1-3.3 (4 days)
• Chapter 4: Geometric Gems 4.1, 4.3, 4.5, 4.7 (6 days)
• Chapter 5: Contortions of Space 5.1-5.3 (4-5 days)
• Chapter 6: Modeling Our World through Graphs 6.1-6.2 (2 days)
• Chapter 7: Chaos and Fractals 7.1-7.2, 7.6 (3 days)
• Chapter 8: Taming Uncertainty 8.1-8.4 (5 days)

Notes: The Heart of Mathematics

Chapter 1 is excellent for setting the tone of the class and illustrating some problem-solving strategies. The puzzles also tie in with the material from later chapters. 

Chapter 2 covers some topics from number theory and gives an appreciation of number theory as an ancient area of mathematics.

Chapter 3, on infinity, is guaranteed to provoke lively discussions as well as controversy. 

Chapter 4 contains some nice sections on geometry. The section on the Pythagorean theorem gives the students several examples of geometric proofs. In the section on the Platonic solids, the students are encouraged to build the solids and explore the concept of duality. The section on the fourth dimension gives them the opportunity to experience an abstract idea through the process of generalization. The Moebius Band is a nice, concrete application.  

Chapter 5 deals with some ideas from topology. The section on rubber sheet geometry has some fun and surprising results, but the students will probably need a model to convince them that the results are indeed true.

Chapter 6 deals with an introduction to graph theory. The section on the Euler characteristic ties in with chapter 4's section on Platonic Solids.

Chapter 7 deals with chaos and fractals. The key point here is for students to understand iterative processes and how they relate to fractals.

Chapter 8 deals with basic probability.

Responsible Parties

Jennifer Austin and Amanda Hager, June 2019

M 305G Preparation for Calculus 

Texts

Abramson, Algebra and Trigonometry, ISBN 978-1-947172-10-4 (Units 1-3) and Abramson, Precalculus, ISBN 978-1-947172-06-7 (Unit 4)

Prerequisite and degree relevance

An appropriate score on the mathematics placement exam. Mathematics 305G and any college-level trigonometry course may not both be counted. A student may not earn credit for Mathematics 305G after having received credit for any calculus course with a grade of at least C-. Mathematics 301, 305G, and equivalent courses may not be counted toward a degree in mathematics.

Course description

The purpose of this course is to prepare students for calculus courses. Some students are taking this course as a review, many because they did not score high enough on the mathematics placement exam to enter calculus directly. The course emphasis is on techniques needed in calculus, with an emphasis on rigorous algebraic practice and on recognizing and interpreting graphs. It is assumed that the students have had at least three and a half years of high school mathematics.

Timing and optional sections

The following table contains suggestions for timing of topics and includes 36 class hours of content. Allowing for in-class exams, there remain 3-5 class hours for review or optional topics.

Unit 1: Algebra and function basics, 9 hours   

Unit 2: Exponential and logarithmic functions, 8 hours

Unit 3: Trigonometry, 12 hours   

Unit 4: Limits, 7 hours (uses Precalculus text)

Last update

Amanda Hager, December 2017

M 408C Differential and Integral Calculus

Text

Stewart, Calculus, Early Transcendentals, 9th Edition

Prerequisite and degree relevance

An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-.

Math majors are required to take both M 408C and M 408D (or either the equivalent sequence M 408K, M 408L, M 408M; or the equivalent sequence M 408N, M 408S, M 408M). Mathematics majors are required to make grades of C- or better in each of these courses.

408C may not be counted by students with credit for any of Mathematics 403K, 408K, 408N, or 408L. 

Course description

M 408C is the standard first-semester calculus course. It is directed at students in the natural sciences and engineering. The emphasis in this course is on problem-solving, not the theory of analysis. There should be some understanding of analysis, but the majority of the proofs in the text should not be covered in class. 

The syllabus for M408C includes most of the basic topics in the theory of functions of a real variable: algebraic, trigonometric, logarithmic and exponential functions and their limits, continuity, derivatives, maxima and minima, integration, area under a curve, and volumes of revolution.

Overview and Course Goals

The following pages comprise the syllabus for M408C, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes. You will do your students a disservice and leave them ill-equipped for subsequent courses.

For those instructors who have taught M408C previously, some changes should be noted.  Chapter 7 has been moved to M408D, allowing a slightly less hectic pace and more importantly the coverage of some topics which have been omitted or optional in the past.  The formal definition of a limit should be covered, although you still shouldn’t expect delta-epsilon proofs.  Sections 3.8, 3.9, 3.10, 4.7 are no longer optional.  Sections 6.3, 6.4, 6.5 have been included as optional sections and some, but not all, of these topics, should be covered.  These adjustments give more attention to applications of both differentiation and integration.

Remember that 408C/D is the fast sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take either the 408NSM or the 408KLM sequence.

Resources for Students

Many students find the study skills from high school are not sufficient for UT. Sanger Learning Center in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus. On request, (471-3614) they will come to your classroom and explain their services.

You can help your students by informing them of SLC services.

Timing and Optional Sections

A typical semester has 42-44 MWF days. The syllabus contains material for 37 days, allowing some time for testing and review.  Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh 75 min.  The purpose of Chapter 6 is to provide applications showing students what integration really means.  It does not matter which optional sections you cover, but it is crucial that you cover some of them or provide alternative examples.

37 Class Days As:

1 Functions and Models (Three Days )

• 1.4 Exponential Functions
• 1.5 Inverse Functions and Logarithms

2 Limits and Derivatives (Six Days)

• 2.1 The Tangent and Velocity Problems
• 2.2 The Limit of a Function
• 2.3 Calculating Limits Using the Limit Laws
• 2.4 The Precise Definition of a Limit
• 2.5 Continuity
• 2.6 Limits at Infinity; Horizontal Asymptotes
• 2.7 Derivatives and Rates of Change
• 2.8 The Derivative of a Function

3 Differentiation Rules (Eleven Days)

• 3.1 Derivatives of Polynomials and Exponential Functions
• 3.2 The Product and Quotient Rules
• 3.3 Derivatives of Trigonometric Functions
• 3.4 The Chain Rule
• 3.5 Implicit Differentiation
• 3.6 Derivatives of Logarithmic Functions
• 3.7 Rates of Change in the Natural and Social Sciences (optional)
• 3.8 Exponential Growth and Decay
• 3.9 Related Rates
• 3.10 Linear Approximations and Differentials
• 3.11 Hyperbolic Functions (quickly)

4 Applications of Differentiation (Eight Days)

• 4.1 Maximum and Minimum Values
• 4.2 The Mean Value Theorem
• 4.3 How Derivatives Affect the Shape of a Graph
• 4.4 Indeterminate Forms and L'Hospital's Rule
• 4.5 Summary of Curve Sketching
• 4.7 Optimization Problems
• 4.9 Antiderivatives

5 Integrals (Five Days)

• 5.1 Areas and Distances
• 5.2 The Definite Integral
• 5.3 The Fundamental Theorem of Calculus
• 5.4 Indefinite Integrals and the Net Change Theorem
• 5.5 The Substitution Rule

6 Applications of Integration (Four Days)

• 6.1 Areas between Curves
• 6.2 Volume
• 6.3 Volumes by Cylindrical shells (optional)
• 6.4 Work (optional)
• 6.5 Average Value of a Function (optional)

Last update

Eric Staron, July 2022

M 408D Sequences, Series, and Multivariable Calculus 

Text

Stewart, Calculus, Early Transcendentals, 9th Edition

Prerequisite and degree relevance

A grade of C- or better in M 408C, M 308L, M 408L, M 308S or M 408S. Only one of the following may be counted: Mathematics 403L, 408D, 408M (or 308M). Math majors are required to take both M 408C and M 408D (or either the equivalent sequence M 408K, M 408L, M 408M; or the equivalent sequence M 408N, M 408S, M 408M). Mathematics majors are required to make grades of C- or better in each of these courses.

Certain sections of this course are reserved as advanced placement or are honors sections; they are restricted to students who have scored well on the advanced placement AP exams or are honors students, or who have the approval of the faculty mathematics advisor. Such sections and their restrictions are listed in the Course Schedule each semester.

Course description

M 408C, M 408D is our standard first-year calculus sequence. It is designed for students in the natural and social sciences and engineering students. The emphasis in this course is on problem-solving, not on theory. While the course necessarily includes some discussion of theoretical notions, its primary objective is not the production of theorem-provers. M 408D contains a thorough treatment of integration techniques, a survey of first-order differential equations, and an introduction to multivariable calculus including parametric equations, partial derivatives, and multiple integrals.

Overview and Course Goals

The following pages comprise the syllabus for M 408D, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes (for example, skipping techniques of integration). You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors first found their passion for calculus; one ought not to bore them. In general it is fair to say that M 408D students will do better than M 408C students; on the other hand, M 408D is a more difficult course. Please keep in mind that students who pass this course meet the prerequisite for M 427K, where it assumed they have good calculus skills.  The M 408C/D sequence is the fast sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take either the M 408N/S/M or the M 408K/L/M sequence.

Resources for Students

Some of our students have weak study skills. The Sanger Learning Center in Jester has a wide variety of material (drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review), as well as tutoring options, all designed to help students through calculus. On request, (471-3614) they'll come to your classroom and explain their services.

You can help your students by informing them of SLC services.

Timing and Optional Sections

A typical semester has 42-44 MWF days. The syllabus contains material for 38 days; this allows some time for testing, reviews, and optional material.  In the spring semester, you will have more time to cover optional material.  Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh lasts 75 minutes.

38 Class Days As:

7 Techniques of Integration (eight days)

• Substitution Review
• 7.1 Integration by Parts
• 7.2 Trigonometric Integrals
• 7.3 Trigonometric Substitution
• 7.4 Integration of Rational Functions by Partial Fractions
• 7.5 Strategy for Integration (use as a reference with good problem set)
• 7.8 Improper Integrals

9 Differential Equations (six days)

• 9.1 Modeling with Differential Equations
• 9.2 Direction Fields and Euler’s Method
• 9.3 Separable Equations
• 9.4 Models for Population Growth
• 9.5 Linear Equations
• 9.6 Predator-prey Systems (optional)

10 Parametric Equations and Polar Coordinates (four days)

• 10.1 Curves Defined by Parametric Equations
• 10.2 Calculus with Parametric Curves
• 10.3 Polar Coordinates
• 10.4 Areas and Lengths in Polar Coordinates
• 10.5 Conic Sections (optional)
• 10.6 Conic Sections in Polar Coordinates (optional)

11 Infinite Sequences and Series (twelve days)

• 11.1 Sequences
• 11.2 Series
• 11.3 The Integral Test and Estimates of Sums
• 11.4 The Comparison Tests
• 11.5 Alternating Series
• 11.6 Absolute Convergence and the Ratio and Root Tests
• 11.7 Strategy for Testing Series
• 11.8 Power Series
• 11.9 Representations of Functions as Power Series
• 11.10 Taylor and Maclaurin Series
• 11.11 Applications of Taylor Polynomials

14 Partial Derivatives (three days)

• 14.1 Functions of Several Variables
• 14.2 Limits and Continuity
• 14.3 Partial Derivatives
• 14.5 The Chain Rule

15 Multiple Integrals (five days)

• 15.1 Double Integrals over Rectangles
• 15.2 Double Integrals over General Regions
• 15.3 Double Integrals in Polar Coordinates
• 15.4 Applications of Double Integrals (optional)
• 15.9 Change of Variables in Multiple Integrals (if time permits) 

Last Update

Keenan Kidwell and Bill Wolesensky, July 2022

M 408L Integral Calculus 

Text

Stewart, Calculus, Early Transcendentals, 9th Edition

Prerequisite and degree relevance

One of M 408C, M 408K, or M 408N, with a grade of at least C- or M 408R with a grade of at least B. Only one of the following may be counted: Mathematics 403L, 408L (or 308L), 408S.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence,  M 408C/D, and two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major.  For some degrees, the two-semesters  M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K.

Course description: M 408L is the second-semester calculus course of the three-course calculus sequence.  In comparison with M408D, it covers fewer chapters of the text. However, some material is covered in greater depth, and extra time is devoted to the development of skills in algebra and problem-solving. This is not a course in the theory of calculus.

Introduction to the theory and applications of integral calculus of functions of one variable.  The syllabus for M 408L includes most of the basic topics of integration on functions of a single real variable: the fundamental theorem of calculus, applications of integrations, techniques of integration, sequences, and infinite series.

The emphasis in this course is on problem-solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

Overview and Course Goals

The following pages comprise the syllabus for M 408L, and advice on teaching it.  Calculus is a service course, and the material in it was chosen after interdepartmental discussions.  Please do not make drastic changes (for example, skipping techniques of integration).  You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class.  At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them.  Remember that 408K/L/M is the sequence designed for students who may not have taken calculus previously.  Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

Resources for Students

Many students find their study skills from high school are not sufficient for UT.  The Sanger Learning Center (http://lifelearning.utexas.edu/) in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus.  On request, they will come to your classroom and explain their services.

You can help your students by informing them of these services.

Timing and Optional Sections

A typical fall semester has 42 hours of lecture, 42 MWF, and 28 TTh days, while a typical spring has 44 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases, there are three "hours" of lecture time per week).   The following syllabus contains suggestions as to timing and includes approximately 36 hours of required material.  Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.  

 Syllabus

• Ch. 5 Integrals (4 hours)
  • 5.3 The Fundamental Theorem of Calculus (review)
  • 5.4 Indefinite Integrals and the Net Change Theorem
  • 5.5 The Substitution Rule
• Ch. 6 Applications of Integration (2 hours)
  • 6.1 Areas between Curves
  • 6.2 Volumes
  • 6.3 Volumes by Cylindrical Shells (optional)
• Ch. 7 Techniques of Integration (9 hours)
  • 7.1 Integration by Parts
  • 7.2 Trigonometric Integrals (light)
  • 7.3 Trigonometric Substitution
  • 7.4 Integration of Rational Functions by Partial Fractions
  • 7.5 Strategy for Integration
  • 7.7 Approximate Integration (optional)
  • 7.8 Improper Integrals
• Ch. 9 Differential Equations (optional -- not in special UT version of the book)
  • 9.3 Separable Equations
  • 9.4 Models for Population Growth  
• Ch. 14 Partial Derivatives (1 hour)
  • 14.3 Partial Derivatives
• Ch. 15 Multiple Integrals (4 hours)
  • 15.1 Double Integrals over Rectangles
  • 15.2 Double Integrals over General Regions

Ch. 11 Infinite Sequences and Series (16 hours)

• 11.1 Sequences
• 11.2 Series
• 11.3 The Integral Test and Estimates of Sums
• 11.4 The Comparison Tests
• 11.5 Alternating Series
• 11.6 Absolute Convergence and the Ratio and Root Tests
• 11.7 Strategy for Testing Series
• 11.8 Power Series
• 11.9 Representations of Functions as Power Series
• 11.10 Taylor and Maclaurin Series
• 11.11 Applications of Taylor Polynomials (optional)

Last Update

Jane Arledge, Kathy Davis, Ray Heitmann, December  2011.

M 408M Multivariable Calculus

Text

Stewart, Calculus, Early Transcendentals, 9th Edition

Prerequisite and degree relevance

M 408L or M 408S with a grade of at least C-. Only one of the following may be counted: Mathematics 403L, 408D, 408M (or 308M).

Calculus is offered in two equivalent sequences: a two-semester sequence, M 408C/D,  or either of two three-semester sequences, M 408N/S/M (for College of Natural Science Students) or M 408K/L/M.   Completion of one of these sequences is required for mathematics majors, with a C- or better in each course.

For some degrees, M 408N/S or M 408K/L satisfy the calculus requirement. This sequence is also a valid prerequisite for some upper-division mathematics courses, including M 325K, M 427K, M 340L, and M 362K.

Course description

M 408M is directed at students in the natural and social sciences and at engineering students. In comparison with M 408D, it covers fewer chapters of the text. However, some material is covered in greater depth. This is not a course in the theory of calculus.

The content includes an introduction to the theory and applications of differential and integral calculus of functions of several variables, including parametric equations, polar coordinates, vectors, vector calculus, functions of several variables, partial derivatives, gradients, and multiple integrals.

Forty Class Days As

10 Parametric Equations and Polar Coordinates (seven days)

• 10.1 Curves Defined by Parametric Equations
• 10.2 Calculus with Parametric Curves
• 10.3 Polar Coordinates
• 10.4 Areas and Lengths in Polar Coordinates
• 10.5 Conic Sections
• 10.6 Conic Sections in Polar Coordinates

12 Vectors and the Geometry of Space (eight days)

• 12.1 Three-Dimensional Coordinate Systems
• 12.2 Vectors
• 12.3 The Dot Product
• 12.4 The Cross Product
• 12.5 Equations of Lines and Planes
• 12.6 Cylinders and Quadric Surfaces

13 Vector Functions (five days)

• 13.1 Vector Functions and Space Curves
• 13.2 Derivatives and Integrals of Vector Functions
• 13.3 Arc Length and Curvature
• 13.4 Motion in Space: Velocity and Acceleration

14 Partial Derivatives (ten days)

• 14.1 Functions of Several Variables
• 14.2 Limits and Continuity
• 14.3 Partial Derivatives
• 14.4 Tangent Planes and Linear Approximations
• 14.5 The Chain Rule
• 14.6 Directional Derivatives and the Gradient Vector
• 14.7 Maximum and Minimum Values
• 14.8 Lagrange Multipliers

15 Multiple Integrals (ten days)(first three sections are review)

• 15.1 Double Integrals over Rectangles
• 15.2 Double Integrals over General Regions
• 15.3 Double Integrals in Polar Coordinates
• 15.4 Applications of Double Integrals (optional)
• 15.9 Change of Variables in Multiple Integrals (if time permits)

Last Update

Ray Heitmann and Jane Arledge, May 2012

M 408R Differential and Integral Calculus for Sciences

Prerequisite and degree relevance

An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. May not be counted by students with credit for Mathematics 403K, 408C, 408K, 408N, 408Q, or 408R.

Textbook (required)

The Six Pillars of Calculus - Biology Edition by Lorenzo Sadun.

Scope of course

M 408R is a 1-semester survey of calculus. As such, it covers more ground than the first semester of a 2-semester sequence, but with a very different emphasis. We will cover Chapters 1-6 of Callahan and part of Chapter 11.

Overview and Course Goals

1. Learning the key ideas of calculus, which I call the six pillars.
   1. Close is good enough (limits)
   2. Track the changes (derivatives)
   3. What goes up has to stop before it can come down (max/min)
   4. The whole is the sum of the parts (integrals)
   5. The whole change is the sum of the partial changes (fundamental theorem)
   6. One variable at a time.
2. Learning how to analyze a scientific situation and model it mathematically.
3. Learning to analyze a mathematical model using calculus.
4. Learning how to apply the results of the model back into the real world.
5. Learning enough formulas and calculational methods to make other goals possible. There
   are three questions associated with every mathematical idea in existence:
   1. What is it?
   2. How do you compute it?
   3. What is it good for?

Compared to most math classes, we're going to spend a lot more time on the first and third questions, but we still need to address the second. You can't spend all your time looking at the big picture! You need some practice sweating the details, too.

Timing and Sections

A context for Calculus: studying epidemics using the SIR model (7 hours)

• The idea behind a rate of change
• The equations in the SIR model
• Threshold values and peak infection
• Predictions using the SIR model: Euler's method
• Using MATLAB to get more accurate predictions

In the textbook, those topics can be found in sections 1.1, 1.2, 2.1, and 2.2.

Derivatives (10 hours)

• Local linearity and tangent lines
• The definition of the derivative of a function
• Some basic derivatives and properties
• The chain rule
• Product and quotient rules
• Critical points and extrema of a function
• Partial derivatives

In the textbook, those topics can be found in sections 3.1, 3.2, 3.3, 3.5, 3.6, 3.7, 5.1, and 5.4.

Differential equations (7 hours)

• Finding functions with prescribed rates of change: particular vs general solutions
• Solving dy/dx = f(x): antiderivatives
• Solving dy/dx = ky: exponential growth models
• Logistic growth models
• Numerical methods: Euler's method revisited

In the textbook, those topics can be found in chapter 4.

Integrals (13 hours)

• Accumulation functions
• Riemann sums and area estimate
• Definite integrals and some basic properties
• The fundamental theorem of Calculus
• Integration techniques: the substitution rule
• Integration techniques: integration by parts

In the textbook, those topics can be found in sections 6.1, 6.2, 6.3, 6.4, 11.1, 11.2, and 11.3.

UT Core Requirements

This course may be used to fulfill the mathematics component of the university core curriculum and addresses the following three core objectives established by the Texas Higher Education Coordinating Board: communication skills, critical thinking skills, and empirical and quantitative skills.

M 408S Integral Calculus for Science

Text

Stewart, Calculus, Early Transcendentals, 9th Edition

Prerequisite and degree relevance

Mathematics 408C, 408K, or 408N with a grade of at least C-, or Mathematics 408R with a grade of at least B. Only one of the following may be counted: Mathematics 403L, 408L (or 308L), 408S.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence,  M 408C/D, and two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major.  For some degrees, the two-semesters  M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K. 

Course description

M 408S is the second-semester calculus course of the three-course calculus sequence.  It is restricted to College of Natural Science Students.  It is an introduction to the theory and applications of integral calculus of functions of one variable. 

The syllabus for M 408S includes most of the basic topics of integration on functions of a single real variable: the fundamental theorem of calculus, applications of integrations, techniques of integration, sequences, and infinite series.  The emphasis in this course is on problem-solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

Overview and Course Goals

The following pages comprise the syllabus for M 408S, and advice on teaching it.  Calculus is a service course, and the material in it was chosen after interdepartmental discussions.  Please do not make drastic changes (for example, skipping techniques of integration).  You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class.  At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them.  Remember that 408N/S/M is the sequence designed for students who may not have taken calculus previously.  Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

Resources for Students

Many students find their study skills from high school are not sufficient for UT.  The Sanger Learning Center (http://lifelearning.utexas.edu/) in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus.  On request, they will come to your classroom and explain their services.

You can help your students by informing them of these services.

Timing and Optional Sections

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while a typical spring has 45 hours, 45 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week). The following syllabus contains suggestions as to timing and includes approximately 36 hours of required material. Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.  

Ch. 5 Integrals (4 hours)

• 5.3 The Fundamental Theorem of Calculus (review)
• 5.4 Indefinite Integrals and the Net Change Theorem
• 5.5 The Substitution Rule

Ch. 6 Applications of Integration (2 hours)

• 6.1 Areas between Curves
• 6.2 Volumes
• 6.3 Volumes by Cylindrical Shells (optional)

Ch. 7 Techniques of Integration (9 hours)

• 7.1 Integration by Parts
• 7.2 Trigonometric Integrals (light)
• 7.3 Trigonometric Substitution
• 7.4 Integration of Rational Functions by Partial Fractions
• 7.5 Strategy for Integration
• 7.7 Approximate Integration (optional)
• 7.8 Improper Integrals

Ch. 9 Differential Equations (optional -- not in special UT version of the book)

• 9.3 Separable Equations
• 9.4 Models for Population Growth  

Ch. 14 Partial Derivatives (1 hour)

• 14.3 Partial Derivatives

Ch. 15 Multiple Integrals (4 hours)

• 15.1 Double Integrals over Rectangles
• 15.2 Double Integrals over General Regions

Ch. 11 Infinite Sequences and Series (16 hours)

• 11.1 Sequences
• 11.2 Series
• 11.3 The Integral Test and Estimates of Sums
• 11.4 The Comparison Tests
• 11.5 Alternating Series
• 11.6 Absolute Convergence and the Ratio and Root Tests
• 11.7 Strategy for Testing Series
• 11.8 Power Series
• 11.9 Representations of Functions as Power Series
• 11.10 Taylor and Maclaurin Series
• 11.11 Applications of Taylor Polynomials (optional)

M 316 Elementary Statistical Methods 

Prerequisite and degree relevance

An appropriate score on the mathematics placement exam. M 316 is an elementary introduction to statistical methods for data analysis; knowledge of calculus is not assumed. Students with a background in calculus are advised to take M 362K plus either M 358K or M 378K instead.  This course may not be counted toward the major requirement for the Bachelor of Arts with a major in mathematics or toward the Bachelor of Science in Mathematics. Students taking the course should have good basic algebra skills. 

This course carries the Quantitative Reasoning flag.  QR courses are designed to equip you with skills that are necessary for understanding the types of quantitative arguments you will regularly encounter in your adult and professional life.  You should, therefore, expect a substantial portion of your grade to come from your use of quantitative skills to analyze real-world problems.

Text

StatsPortal; The Basic Practice of Statistics, 6th edition(2010) by David S. Moore

StatsPortal contains an interactive e-Book and numerous resources for students and instructors.  For students: Learning Curve, statistical videos, Stats Tutor, applets, software manuals, online quizzes, etc.  Resources for instructors include the e-Book, PowerPoint lecture slides, instructor's solution manual, printed test bank, i>clicker questions, grade book (which can be downloaded to Blackboard), extra exercises and solutions, etc.

To purchase StatsPortal and register your access code, go to http://courses.bfwpub.com/bps6e.php

Students can use the loose leaf version of the textbook packaged with StatsPortal for a nominal extra charge; the ISBN is 978-1-4641-2954-4.  You can ask the Coop to order copies for you.

You may go to www.whfreeman.com/bps6e to browse some of the resources mentioned above.

Topics

Part I: Exploring Data

•  Chapter 1 Picturing Distributions with Graphs
• Chapter 2 Describing Distributions with Numbers
• Chapter 3 The Normal Distributions
• Chapter 4 Scatterplots and Correlation
• Chapter 5 Regression
• Chapter 6 Two-Way Tables (optional)
• Chapter 7 Exploring Data: Part I Review (May be assigned as reading.)  

Part II: From Exploration to Inference

• Chapter 8 Producing Data: Sampling
• Chapter 9 Producing Data: Experiments
  (Optional but strongly recommended: Commentary, Data Ethics. May be assigned as reading.)
• Chapter 10 Introducing Probability (Section on Personal Probability is optional.)
• Chapter 11 Sampling Distributions
• (Optional: Chapter 12 General Rules of Probability)
• (Optional: Chapter 13 Binomial Distributions)
• Chapter 14 Confidence Intervals: The Basics
• Chapter 15 Tests of Significance: The Basics
• Chapter 16 Inference in Practice (more focus on Power and less on Type II error)
• Chapter 17 From Exploration to Inference: Part II Review (May be assigned as reading.)

Part III: Inference about Variables

• Chapter 18 Inference about a Population Mean
• Chapter 19 Two-Sample Problems (The section on details of the t approximation is optional, and so are the sections on avoiding the pooled two-sample t procedures and avoiding inference about standard deviations.)
• Chapter 20 Inference about a Population Proportion
• Chapter 21 Comparing Two Proportions
• Chapter 22 Inference about Variables: Part III Review (May be assigned as reading)

Part IV: Inference about Variables

• Chapter 23 Two Categorical Variables: The Chi-Square Test (Section on the goodness of fit optional.)
• (Optional: Chapter 24 Inference for Regression)
• (Optional: Chapter 25 One-Way analysis of variance: comparing several means)
• (Optional: Chapter 26 Non-parametric Tests)
• (Optional: Chapter 27 Statistical Process Control)
• (Optional: Chapter 28 Multiple regression)

Comments for Instructors

If you choose to cover any of the optional chapters, save them (with the possible exception of the Commentary on Data Ethics) until the end of the semester. Don't try to do more than two of them.  The Commentary on Data Ethics is recommended, with chapter 24 second priority. Note that chapters 12 and 13 are not needed for the rest of the course, with the exception of conditional probability.

The book is readable enough that, especially in chapters 1 – 9, you may want to cover some topics as reading assignments, to be followed by class discussion, rather than lecturing.

The material on inference (beginning with chapter 14) is more challenging for most students than in the earlier chapters. To allow adequate time for the material on inference, chapter 14 should be started just before or at the midpoint of the semester.

Some instructors require students to do a (usually group) project involving designing an experiment or observational study, carrying it out, and analyzing the results.

Chapters 20 and 21: The sections on more accurate confidence intervals should be covered, reflecting currently recommended changes in statistical practice.

Statistical applets. These can be used for in-class demonstrations of concepts if your classroom is equipped for computer projection. They are also available as a resource on StatsPortal.

Access to the website StatsPortal is bundled with new copies of the textbook.

Last update

Evelyn Schultz, June 2012

M 316K Foundations of Arithmetic 

Prerequisite and degree relevance

Prerequisite is one of the following courses with a grade of C- or better:

• M 302: Introduction to Mathematics ("Math for Liberal Arts")
• M 303D: Applicable Mathematics
• M 305G or 505G: Elementary Functions and Coordinate Geometry ("Precalculus")
• M 316: Elementary Statistical Methods

This course is required for students preparing to teach elementary school.

Text

Beckmann

Course Description

An analysis, from an advanced perspective, of the concepts and algorithms of arithmetic, including sets; numbers; numeration systems; definitions, properties, and algorithms of arithmetic operations; and percents, ratios, and proportions. Problem-solving is stressed.

Topics and Format

The focus is on students working on Explorations supporting learning in the following sections of the textbook.

M 316L Foundations of Geometry, Statistics, and Probability 

Prerequisite and degree relevance

M 316K with a grade of at least C. Restricted to students in a teacher preparation program. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree. Credit for Mathematics 316L may not be earned after a student has received credit for any calculus course with a grade of C- or better unless the student is registered in the College of Education. This course is required for students preparing to teach elementary school, and for students in UTeach Liberal Arts planning to teach in the middle grades. It is also taken by some students preparing to teach middle grades mathematics. 

Text

Beckmann

Topics and Format

The focus is on students working on Explorations supporting learning in the following sections of the textbook.

ACF 329 Theory of Interest

Text

Vaaler & Daniel, Mathematical Interest Theory, Mathematical Association, Second Edition

Prerequisite and degree relevance

M 408D, M 308L, M 408L, or M 408S with a grade of at least C-, or consent of instructor.  This course covers the interest theory portion of the SOA/CAS Financial Mathematics exam (FM/2); this should be about 75-80% of the material on this professional exam, with the balance of the exam testing knowledge of elementary financial derivatives.   Topics include nominal and effective interest and discount rates, general accumulation functions and force of interest, yield rates, annuities including those with non-level payment patterns, amortization of loans, sinking funds, bonds, duration, and immunization.

Schedule of Topics

Chapter 0 An Introduction to the Texas Instruments BA II Plus (optional)

Chapter 1 The Growth of Money (7 days)

• 1.1 Introduction (optional)
• 1.2 What is interest?
• 1.3 Accumulation and Amount Functions
• 1.4 Simple Interest/Linear Accumulation Functions
• 1.5 Compound Interest (The usual case!)
• 1.6 Interest in Advance/The Effective Discount Rate
• 1.7 Discount Functions/The Time Value of Money
• 1.8 Simple Discount
• 1.9 Compound Discount
• 1.10 Nominal Rates of Interest and Discount
• 1.11 A Friendly Competition (Constant Force of Interest)
• 1.12 Force of Interest
• 1.14 Inflation

Chapter 2 Equations of Value and Yield Rates (4-5 days)

• 2.1 Introduction (optional)
• 2.2 Equations of Value for Investments Involving a Single Deposit made under Compound Interest
• 2.3 Equations of Value for Investments with Multiple Contributions
• 2.4 Investment Return
• 2.5 Reinvestment Considerations
• 2.6 Approximate Dollar Weighted Yield Rates (optional)
• 2.7 Fund Performance

Chapter 3 Annuities (Annuities Certain) (11-12 days)

• 3.1 Introduction (optional)
• 3.2 Annuities Immediate
• 3.3 Annuities Due
• 3.4 Perpetuities & 7.1 Common and Preferred Stock
• 3.5 Deferred Annuities and Values on any Date
• 3.6 Outstanding Loan Balances
• 3.7 Nonlevel Annuities
• 3.8 Annuities with Payments in Geometric Progression
• 3.9 Annuities with Payments in Arithmetic Progression
• 3.10 Yield Rate Examples Involving Annuities (optional)
• 3.11 Annuity Symbols for Nonintegral Terms (optional)
• 3.12 Annuities Governed by General Accumulation Functions (optional)
• 3.13 The Investment Year Method

Chapter 4 Annuities with Different Payment and Conversion Periods (1-2 days)

• 4.1 Introduction (optional)
• 4.2 Level Annuities with Payments Less Frequent Than Each Interest Period (optional)
• 4.3 Level Annuities with Payments More Frequent Than Each Interest Period (optional)
• 4.4 Annuities with Payments Less Frequent Than Each Interest Period and Payments in Arithmetic Progression (optional)
• 4.5 Annuities with Payments More Frequent Than Each Interest Period and Payments in Arithmetic Progression (optional)
• 4.6 Continuously Paying Annuities

Chapter 5 Loan Repayment (2-3 days)

• 5.1 Introduction (optional)
• 5.2 Amortized Loans and Amortization Schedules
• 5.3 The Sinking Fund Method
• 5.4 Loans with Other Repayment Patterns (optional)
• 5.5 Yield Rate Examples and Replacement of Capital (optional)

Chapter 6 Bonds (5-6 days)

• 6.1 Introduction (optional)    
• 6.2 Bond Alphabet Soup and the Basic Price Formula
• 6.3 The Premium-Discount Formula
• 6.4 Other Pricing Formulas for Bonds
• 6.5 Bond Amortization Schedules
• 6.6 Valuing a Bond After Its Date of Issue (optional)
• 6.9 Callable Bonds

Chapter 7 Stocks and Financial Markets (1 day)

• 7.1 Common and Preferred Stock (cover dividend discount model with §3.4)
• 7.4 Selling Short; Selling Borrowed Stocks (optional)

Chapter 8 Arbitrage, Term Structure of Interest Rates, and Derivatives (1-2 days)

• 8.1 Introduction (optional)
• 8.3 The Term Structure of Interest Rates

Chapter 9 Interest Rate Sensitivity (4-5 days)

• 9.1 Overview
• 9.2 Duration
• 9.3 Convexity
• 9.4 Immunization (optional)
• 9.5 Other Types of Duration  (optional)

Last Update

Shinko Harper, Milica Cudina, Alisa Havens, Jennifer Mann, January 2013

M 325K Discrete Mathematics (non-ECE)

Prerequisite

None. Faculty should not assume that students have taken calculus before taking M 325K.

Degree relevance

Our Introduction to Mathematical Proof Writing courses (M 325K, M 328K, and M 333L) provide an essential transition from the algorithmic approach of calculus to the entirely rigorous approach of more advanced proof writing courses such as Advanced Number Theory, M 361K/M 365C, M 343K/M 373K, or M 367K. The number of topics required for coverage in each course has been kept modest so as to allow adequate time for students to develop theorem-proving skills. Students are expected to become familiar with the language and techniques of proof; they should also see detailed, rigorous proofs presented in class. More importantly, they need to develop the ability to read and understand proofs on their own, and they must begin doing proofs; this cannot be slighted. Over the course, the generation of ideas in the class needs to transition from instructor-initiated to more student-initiated. At the beginning of the semester, it is necessary that the instructor heavily model this behavior. Then as the semester progresses the professor and the students need to meet the challenge of each student assuming responsibility/ownership. In teaching abstraction, it is critical to remember that almost no students can become truly comfortable with it in a single semester; it is self-defeating to establish this as a goal. All Introduction to Mathematical Proof Writing course professors are strongly encouraged to employ active learning strategies. Students will discuss, debate, and negotiate what counts as valid proof argumentation and why. Students will not merely watch the instructor present correct, completed mathematics and imitate with superficial understanding.

M 325K topics may include: fundamentals of logic and set theory; functions and relations; basic properties of integers, and elementary number theory; recursion and induction; counting techniques and combinatorics; introductory graph theory.

M 325K Student Experience Goals 

Students will: 

• approach mathematical problems with curiosity and creativity and persist in the face of difficulties,
• capture the essential elements of intuitive mathematical objects in precise language that can make them withstand rigorous mathematical analysis,
• participate in the creative process and social negotiation characteristic of mathematicians’ work,
• transition from algorithmic and computational mathematics to actually producing and creating mathematics,
• develop the skills to read, understand, critique, and construct logical arguments,
• develop effective thinking and communication skills applicable in mathematics and well beyond mathematics,
• develop mathematical independence and experience mathematical inquiry, and
• experience the beauty and power of mathematics.

M 325K Student Learning Objectives 

Students will: 

• apply definitions effectively in proofs,
• use logical connectives and quantifiers correctly and with understanding,
• determine the validity of an argument,
• prove a given statement directly,
• prove a biconditional statement,
• prove a given statement indirectly (contradiction & contraposition),
• prove a given statement using mathematical induction,
• construct a proof by exhaustion,
• prove existence,
• prove existence and uniqueness,
• find a counterexample to disprove a given statement,
• write and speak about mathematics using precise mathematical language, and
• write well-organized, grammatically correct, and logically sound mathematical arguments.

Text

Faculty have a choice among the following recommended texts. The preferred texts are 

• Epp, Discrete Mathematics with Applications, 5th Edition 
• Scheinerman, Mathematics: A Discrete Introduction, 1st Edition. 

A text in use before these was Grimaldi, Discrete and Combinatorial Mathematics. Grimaldi is the most directed towards applications in Computer Science and Electrical Engineering. He also tends to integrate his applications directly into the flow of the text rather than discussing them separately.

Suggested Schedule of Topics

Discrete mathematics offers a variety of contexts in which the student can begin to understand mathematical techniques and appreciate the mathematical culture. Abstraction per se is not the goal; discrete mathematics offers very concrete computational contexts, and this can be exploited to develop a feeling for what it is that proofs, and proof techniques, say and do.

In Epp, Discrete Mathematics with Applications, one might include the topics below; this leaves time for the Instructor to cover additional topics of their choice. The instructor should focus on depth of understanding rather than breadth of coverage. However, subsequent courses will assume that students have seen induction and set theory in this course, so they must be covered.

Chapter 2 The Logic of Compound Statements (3-4 days)

• 2.1 Logical Form and Logical equivalence
• 2.2 Conditional Statements
• 2.3 Valid and Invalid Arguments

Chapter 3 The logic of Quantified Statements (4 days)

• 3.1 Introduction to Predicates and Quantified Statements I
• 3.2 Introduction to Predicates and Quantified Statements II
• 3.3 Statements Containing Multiple Quantifiers
• 3.4 Arguments with Quantified Statements

Chapter 4 Elementary Number Theory and Statements of Proof (8 days)

• 4.1 Direct Proof and Counterexample I: Introduction
• 4.2 Direct Proof and Counterexample II: Writing Advice
• 4.3 Direct Proof and Counterexample II: Rational Numbers
• 4.4 Direct Proof and Counterexample III: Divisibility
• 4.5 Direct Proof and Counterexample IV: Division Into Cases
• 4.7 Indirect Argument: Contradiction and Contraposition
• 4.8 Indirect Argument: Two Classical Theorems

Chapter 5 Sequences and Mathematical Induction (5-6 days)

• 5.1 Sequences
• 5.2 Mathematical Induction I
• 5.3 Mathematical Induction II
• 5.4 Strong Mathematical Induction and, optionally, Well-Ordering Principle for the Integers

Chapter 6 Set Theory (3-4 days)

• 6.1 Set Theory: Definitions & the Element Method of Proof
• 6.2 Properties of Sets
• 6.3 Disproofs & Algebraic Proofs

Chapter 7 Functions (3-4 days)

• 7.1 Functions Defined on General Sets
• 7.2 One-to-One and Onto; Inverse Functions
• 7.3 Composition of Functions

Chapter 8 Relations (3 days) (Optional)

• 8.1 Relations on Sets
• 8.2 Reflexive, Symmetric, and Transitive
• 8.3 Equivalence Relations

Chapter 9 Counting and Probability (1 day) (Optional)

• 9.4 The Pigeonhole Principle (usually covered after Section 7.2)

Last update

Jennifer Austin and Shinko Harper, March 2025 (Books updated)

 M 325K Discrete Mathematics ECE

Course Description

The course provides a transition from the problem-solving approach of Mathematics 408C and 408D to the rigorous approach of advanced courses. This is a course that emphasizes understanding and creating proofs of mathematical theorems. Successful students will leave this course with an understanding of introductory discrete techniques, as well as an ability to use the language and techniques of proof writing in a discrete context. Topics include logic, set theory, relations and functions, combinatorics, and graph theory, and graph algorithms.

Prerequisite

None. Faculty should not assume that students have taken calculus before taking M 325K.

Learning Goals

By the end of the semester, you should know how to read and critique a proof, and how to create your own. We will begin with proofs and logic, including several different patterns of proof (direct proof, proof by cases, proof by contrapositive, proof by contradiction, proof by induction). Later in the course, we will talk about some counting and combinatorics. This is partly because counting and combinatorics are very useful (for example, in probability) and partly because it is a great way to practice proving things. We will finish with some graph theory (basically for the same two reasons). You will learn by attending the lecture, reading proofs, writing proofs, and participating in every learning activity.

Students are expected to learn definitions and write proofs in class, homework, tests, and exam. You should be able to find the mistakes in a wrong proof and correct the mistakes. The whole course is about proving theorems using logical mathematical arguments and convincing the reader that the statement is true.

Course Outcomes

• Translate and construct arguments using the language of Mathematics, logical connectors, and quantifiers.
• Construct truth tables and verify whether a compound statement is true or false.
• Verify the correctness of an argument using truth tables.
• Read and explain correct proofs and find the errors in false proofs.
• Construct clear, correct, and organized proofs using a variety of patterns and techniques.
• Master fundamentals of Set Theory, Functions, Equivalence relations, and Equivalence classes and write proofs on these topics.
• Demonstrate essential concepts in Combinatorics.
• Solve problems and prove theorems using graphs, digraphs, and trees.

Textbook

Epp, Discrete Mathematics with Applications, 5th Edition (Not required to buy)

Other references

Mathematics for Computer Science, by Lehman, Leighton, and Meyer. Available for free at https://courses.csail.mit.edu/6.042/spring17/mcs.pdf.

Schedule of Topics

Last update

Kanthimathi Sathasivan, August 2021

M 326K Foundations of Number Systems 

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C-. Restricted to students in a teacher preparation program or who have the consent of the instructor.

Text

Optional: The Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, 2000. (Also known as NCTM Standards.)

Course Description

Number and operations, with emphasis on depth of understanding, mathematical communication, mathematical reasoning, mathematical representations, and pedagogical content knowledge in the context of number and operations. In this course, you may often need to re-examine things that have become “obvious” or automatic to you. For example,

• You know how to multiply. But can you help a student learn when to use multiplication in setting up an equation?
• You know how to divide fractions by inverting the divisor then multiplying. But can you explain to a questioning beginning algebra student why this procedure is legitimate?

There are many math problems you can solve. But most mathematics problems have several correct methods of solution and many more incorrect methods of solution. Will you as a math teacher be able to decide which is which?

Last update

Kathy Davis and Shinko Harper, Spring 2010

M 328K Introduction to Number Theory

Prerequisite

Mathematics 325K, 333L, or 341 with a grade of at least C-.

Degree relevance

Our Introduction to Mathematical Proof Writing courses (M 325K, M 328K, and M 333L) provide an essential transition from the algorithmic approach of calculus to the entirely rigorous approach of more advanced proof writing courses such as Advanced Number Theory, M 361K/M 365C, M 343K/M 373K, or M 367K. The number of topics required for coverage in each course has been kept modest so as to allow adequate time for students to develop theorem-proving skills. Students are expected to become familiar with the language and techniques of proof; they should also see detailed, rigorous proofs presented in class. More importantly, they need to develop the ability to read and understand proofs on their own, and they must begin doing proofs; this cannot be slighted. Over the course, the generation of ideas in the class needs to transition from instructor-initiated to more student-initiated. At the beginning of the semester, it is necessary that the instructor heavily model this behavior. Then as the semester progresses the professor and the students need to meet the challenge of each student assuming responsibility/ownership. In teaching abstraction, it is critical to remember that almost no students can become truly comfortable with it in a single semester; it is self-defeating to establish this as a goal. All Introduction to Mathematical Proof Writing course professors are strongly encouraged to employ active learning strategies. Students will discuss, debate, and negotiate what counts as valid proof argumentation and why. Students will not merely watch the instructor present correct, completed mathematics and imitate with superficial understanding.

M 328K Student Experience Goals 

Students will: 

• approach mathematical problems with curiosity and creativity and persist in the face of difficulties,
• capture the essential elements of intuitive mathematical objects in precise language that can make them withstand rigorous mathematical analysis,
• participate in the creative process and social negotiation characteristic of mathematicians’ work,
• transition from algorithmic and computational mathematics to actually producing and creating mathematics,
• develop the skills to read, understand, critique, and construct logical arguments,
• develop effective thinking and communication skills applicable in mathematics and well beyond mathematics,
• develop mathematical independence and experience mathematical inquiry, and
• experience the beauty and power of mathematics.

M 328K Student Learning Objectives 

Students will: 

• apply definitions effectively in proofs,
• use logical connectives and quantifiers correctly and with understanding,
• determine the validity of an argument,
• prove a given statement directly,
• prove a biconditional statement,
• prove a given statement indirectly (contradiction & contraposition),
• prove a given statement using mathematical induction,
• construct a proof by exhaustion,
• prove existence,
• prove existence and uniqueness,
• find a counterexample to disprove a given statement,
• write and speak about mathematics using precise mathematical language, and
• write well-organized, grammatically correct, and logically sound mathematical arguments.

Text

A list of texts from which the instructor may choose is maintained in the text office: Elementary Number Theory and its Applications by Rosen, 6th edition, Addison-Wesley -OR Number Theory Through Inquiry by Marshall/Odell/Starbird, 1st ed. MAA

The choice of text will determine the exact topics to be covered. The following subjects should definitely be included:

• Divisibility: divisibility of integers, prime numbers, and the fundamental theorem of arithmetic.
• Congruences: including linear congruences, the Chinese remainder theorem, Euler's -function, and polynomial congruences, primitive roots.

The following topics may also be covered, the exact choice will depend on the text and the taste of the instructor.

• Diophantine equations: (equations to be solved in integers), sums of squares, Pythagorean triples.
• Number theoretic functions: the Mobius Inversion formula, estimating and partial sums n(x) of other number-theoretic functions.
• Approximation of real numbers by rationals: Dirichlet's theorem, continued fractions, Pell's equation, Liousville's theorem, algebraic and transcendental numbers, the transcendence of e and/or n.

M 329F Theory of Interest

Text

Mathematical Interest Theory, Third Edition, Vaaler, Harper, & Daniel, AMS/MAA Press.

Prerequisite and degree relevance

Mathematics 408D, 308L, 408L, or 408S with a grade of at least C-. This course covers the content for the Society of Actuaries Financial Mathematics Exam (Exam FM) and the Casualty Actuarial Society Exam 2. Topics include nominal and effective interest and discount rates, general accumulation functions and force of interest, yield rates, annuities including those with non-level payment patterns, amortization of loans, bonds, spot and forward rates, interest rate swaps, duration, and immunization.

Course Goals

The goal of M 329F Theory of Interest is to provide an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows.

Learning Objectives

• Students will develop effective thinking and communication skills applicable in mathematics and well beyond mathematics.
• Students will be able to communicate mathematical ideas through symbolic expressions and graphs and be able to draw inferences from such presentations of data.
• Students will understand and be able to perform calculations relating to present value, current value, and accumulated value.
• Students will be able to calculate present value, current value, and accumulated value for sequences of non-contingent payments.
• Students will understand key concepts concerning loans and how to perform related calculations.
• Students will understand key concepts concerning bonds, and how to perform related calculations.
• Students will understand key concepts concerning yield curves, rates of return, and measures of duration and convexity, and how to perform related calculations.
• Students will understand key concepts concerning cash flow matching and immunization, and how to perform related calculations.
• Students will understand key concepts concerning interest rate swaps, and how to perform related calculations.
• Students will understand key concepts concerning the determinants of interest rates, the components of interest, and how to perform related calculations.

Timing and Optional Sections

A typical semester has 42 - 45 MWF days. The syllabus contains material for 35 – 39 days, allowing some time for testing and review.  Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh 75 min. It does not matter which optional sections you cover.

Chapter 0 An Introduction to the Texas Instruments BA II Plus (Optional)

Chapter 1 The Growth of Money (7 days)

• 1 Introduction
• 2 What is interest?
• 3 Accumulation and Amount Functions
• 4 Simple Interest/Linear Accumulation Functions
• 5 Compound Interest (The usual case!)
• 6 Interest in Advance/The Effective Discount Rate
• 7 Discount Functions/The Time Value of Money
• 8 Simple Discount
• 9 Compound Discount
• 10 Nominal Rates of Interest and Discount
• 11 A Friendly Competition (Constant Force of Interest)
• 12 Force of Interest
• 14 Quoted Rates for Treasury Bills (Optional: This section shows applications of simple discount and simple interest.)
• 15 Inflation (Optional)

Chapter 2 Equations of Value and Yield Rates (4 days)

• 2.1 Introduction
• 2.2 Equations of Value for Investments Involving a Single Deposit made under Compound Interest
• 2.3 Equations of Value for Investments with Multiple Contributions
• 2.4 Investment Return
• 2.5 Reinvestment Considerations
• 2.6 Dollar Weighted Yield Rates
• 2.7 Fund Performance

Chapter 3 Annuities (Annuities Certain) (8.5 days)

• 3.1 Introduction (cover lightly for vocabulary)
• 3.2 Annuities Immediate
• 3.3 Annuities Due
• 3.4 Perpetuities (Cover together with the dividend discount model in 7.1 Common and Preferred Stock.)
• 3.5 Deferred Annuities and Values on any Date
• 3.6 Outstanding Loan Balances
• 3.7 Nonlevel Annuities
• 3.8 Annuities with Payments in Geometric Progression
• 3.9 Annuities with Payments in Arithmetic Progression

Chapter 4 Annuities with Different Payment and Conversion Periods (1 day)

• Most of this chapter can be omitted, but §4.6 should be covered. Covering parts of §4.3 and §4.5 may be useful to aid in understanding §4.6 as listed below.
• 3 Level Annuities with Payments More Frequent Than Each Interest Period (Include content from pages 201-202 to aid in understanding §4.6.)
• 5 Annuities with Payments More Frequent Than Each Interest Period and Payments in Arithmetic Progression (Include content from pages 209-210 to aid in understanding §4.6.)
• 6 Continuously Paying Annuities

Chapter 5 Loan Repayment (2 days)

• 5.1 Introduction
• 5.2 Amortized Loans and Amortization Schedules
• 5.4 Loans with Other Repayment Patterns (Optional)
• 5.5 Yield Rate Examples and Replacement of Capital (Replacement of capital is optional.)

Chapter 6 Bonds (5 days)

• 6.1 Introduction (Cover lightly for vocabulary.)       
• 6.2 Bond Alphabet Soup and the Basic Price Formula
• 6.3 The Premium-Discount Formula
• 6.4 Other Pricing Formulas for Bonds (Optional)
• 6.5 Bond Amortization Schedules
• 6.6 Valuing a Bond After Its Date of Issue (Optional)
• 6.9 Callable Bonds

Chapter 7 Stocks and Financial Markets (0.5 days)

• 7.1 Common and Preferred Stock (Cover the dividend discount model with §3.4)

Chapter 8 Arbitrage, Term Structure of Interest Rates, and Derivatives (3 - 5 days)

• 8.1 Introduction
• 8.3 The Term Structure of Interest Rates
• 8.4 Loans with Floating Rate of Interest
• 8.5 Interest Rate Swaps: the Basics
• 8.6 Formulas for Interest Rate Swaps
• 8.7 Market value of an Interest Rate Swap

Chapter 9 Interest Rate Sensitivity (4.5 - 6 days)

• 9.1 Overview (cover asset-liability matching)
• 9.2 Duration
• 9.3 Convexity
• 9.4 Using Duration to Approximate Price (Optional: this section shows applications of Taylor polynomials.)
• 9.6 Immunization

Chapter 10 Determinants of Interest Rates

• The material in this chapter is included in the Exam FM syllabus, but appears to be a very minor part of Exam FM. Instructors may choose to use this chapter for giving reading assignments that relate to contemporary issues.

Last Update 

Jennifer Austin, Milica Cudina, Shinko Harper, Joel Nibert, Alisa Walch, Summer 2020

M 427J Differential Equations with Linear Algebra

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 427J and 427K may not both be counted.

Course description

This is an introduction to linear algebra and differential equations. Geared to the audience primarily consisting of engineering and science students, the course aims to teach the basic techniques for solving differential equations that arise in applications. The approach is problem-oriented and not particularly theoretical. Most of the time is devoted to first and second-order ordinary differential equations with an introduction to Fourier series and partial differential equations at the end.

Text

Differential Equations and Their Applications, by Martin Braun 4th edition. This text is required for most sections, and its chapter numbers are used for the outline below.

Topics Outline

First-order differential equations [6 hours]

• 1.1 Introduction

• 1.2 First-order linear differential equations

• 1.4 Separable equations

• 1.8 Modelling and applications

• 1.10 The existence-uniqueness theorem 

Second-order linear differential equations [5 hours]

• 2.1 Algebraic properties of solutions

• 2.2 Linear equations with constant coefficients

• 2.2.1 Complex roots

• 2.2.2 Equal roots; reduction of order

• 2.3 The nonhomogeneous equation

• 2.5 The method of judicious guessing

• 2.6 Mechanical vibrations (optional)

Linear Algebra [12 hours]

• 3.2 Vector spaces

• 3.3 Dimension of a vector space

• 3.5 The theory of determinants

• 3.6 Solutions of simultaneous linear equations

• 3.7 Linear transformations

Supplement

• 3.A Matrix multiplication as linear combination of columns

• 3.B Vectors as arrows in Rn and geometric meaning of operations (optional)

• 3.C Null and Column spaces

• 3.D Complete solution set of systems (RREF)

Systems of differential equations [6 hours]

• 3.1 Algebraic properties of solutions of linear systems

• 3.4 Applications of linear algebra to differential equations

• 3.8 The eigenvalue-eigenvector method of finding solutions

• 3.9 Complex roots

• 3.10 Equal roots

• 3.11 Fundamental matrix solutions; eAt

Qualitative theory of differential equations [3 hours]

• 4.1 Introduction

• 4.2 Stability of linear systems

• 4.4 The phase-plane

• 4.7 Phase portraits of linear systems

• 4.10 Predator-prey models

Chapter 5. Separation of variables and Fourier series [6 hours]

• 5.1 Two point boundary-value problems

• 5.2 Introduction to partial differential equations

• 5.3 The heat equation; separation of variables

• 5.4 Fourier series

• 5.5 Even and odd functions

• 5.6 Return to the heat equation

M 427K Advanced Calculus for Applications I

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 427J and 427K may not both be counted.

Course description

M 427K is a basic course in ordinary and partial differential equations, with Fourier series. It should be taken before most other upper-division, applied mathematics courses. The course meets three times a week for lecture and twice more for problem sessions. Geared to the audience primarily consisting of engineering and science students, the course aims to teach the basic techniques for solving differential equations that arise in applications. The approach is problem-oriented and not particularly theoretical. Most of the time is devoted to first and second-order ordinary differential equations with an introduction to Fourier series and partial differential equations at the end. Depending on the instructor, some time may be spent on applications, Laplace transformations, or numerical methods. Five sessions a week for one semester.

Note that some Engineering courses assume students have seen Laplace Transforms in M 427K.

Text

Boyce and DiPrima: Elementary Differential Equations and Boundary Value Problems. 10th Edition The text is required for most sections; honors classes, computer supplement sections, or innovative sections may use other texts.

Required Topics

It will be impossible to cover everything here adequately. The core material must be covered in selected sections from Chapters 1, 2, 3, 5, 10. Chapter 7 is so important that it ought to be covered, but be aware that most students have not already had matrix methods, and you will likely find yourself covering the 2 by 2 case. You might then do stability, etc. Numerical methods are becoming increasingly important, and covering this topic here is a good lead in to the department's new computational science degree. Again, some engineering courses need their students to have seen some Laplace transforms. This will leave time for other topics, and you may choose to equations, applications. Whichever approach you take, you will have to carefully plan your sections and time to be spent on them.

Resources

If you are new to this course, you might talk to the senior faculty who teach this course regularly: Profs. Arbogast, Beckner, Bichteler, Gamba, Koch,  Tsai, Uhlenbeck and others.

• Chapter I Introduction (2 - 3 weeks for Chapters 1 and 2)
  • 1.1Some Basic Mathematical Models; Direction Fields
  • 1.2 Solutions of Some Differential Equations 
  • 1.3 Classification of Differential Equations
  • 1.4 Historical Remarks  
• Chapter 2 First-Order Differential Equations(2 - 3 weeks for Chapters 1 and 2)
  • 2.1 Linear Equations with Variable Coefficients  
  • 2.2 Separable Equations  
  • 2.3 Modeling with First Order Equations  (optional)
  • 2.4 Differences Between Linear and Nonlinear Equations
  • 2.5 Autonomous Equations and Population Dynamics  (optional)
  • 2.6 Exact Equations and Integrating Factors  
  • 2.7 Numerical Approximations: Euler's Method   (optional unless you do Ch 8)
  • 2.8 The Existence and Uniqueness Theorem  
  • 2.9 First Order Difference Equations 115 (optional unless you do stability)
• Chapter 3 Second Order Linear Equations (2 - 3 weeks)
  • 3.1 Homogeneous Equations with Constant Coefficients  
  • 3.2 Fundamental Solutions of Linear Homogeneous Equations  
  • 3.3 Complex Roots of the Characteristic Equation
  • 3.4   Repeated Roots; Reduction of Order
  • 3.5   Nonhomogeneous Equations; Method of Undetermined Coefficients
  • 3.6   Variation of Parameters   (optional)
  • 3.7 Mechanical and Electrical Vibrations   (optional)
  • 3.8 Forced Vibrations   (optional)
• Chapter 4 Higher Order Linear Equations (cover quickly)
  • 4.1 General Theory of nth Order Linear Equations  
  • 4.2 Homogeneous Equations with Constant Coefficients  
  • 4.3 The Method of Undetermined Coefficients  (optional)
  • 4.4 The Method of Variation of Parameters  
• Chapter 5 Series Solutions of Second Order Linear Equations (2 weeks)
  • 5.1 Review of Power Series  (optional)
  • 5.2 Series Solutions Near an Ordinary Point, Part I  
  • 5.3 Series Solutions Near an Ordinary Point, Part II  
  • 5.4 Euler Equations, Regular Singular Points  
  • 5.5 Series Solutions Near a Regular Singular Point, Part I
  • 5.6 Series Solutions Near a Regular Singular Point, Part II
• Chapter 6 The Laplace Transform (1 week: Important for some Engineers))
  • 6.1 Definition of the Laplace Transform  
  • 6.2 Solution of Initial Value Problems  
  • 6.3 Step Functions  
  • 6.4 Differential Equations with Discontinuous Forcing Functions
  • 6.5 Impulse Functions  
  • 6.6 The Convolution Integral 
• Chapter 7 Systems of First Order Linear Equations (1 – 2 weeks)
• Chapter 8 Numerical Methods (1 week if covered) (optional)
• Chapter 9 Nonlinear Differential Equations and Stability
• Chapter 10 Partial Differential Equations and Fourier Series (3 weeks)
  • 10.1 Two-Point Boundary Value Problems
  • 10.2 Fourier Series
  • 10.3 The Fourier Convergence Theorem
  • 10.4 Even and Odd Functions
  • 10.5 Separation of Variables; Heat Conduction in a Rod
  • 10.6 Other Heat Conduction Problems (optional)
  • 10.7 The Wave Equation; Vibrations of an Elastic String
  • 10.8 Laplace's Equation (optional)
  • Appendix A Derivation of the Heat Equation (optional)
  • Appendix B Derivation of the Wave Equation (optional)

M 427L Advanced Calculus for Applications II

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C-.

Course description

Topics include matrices, elements of vector analysis, and calculus functions of several variables, including gradient, divergence, and curl of a vector field, multiple integrals, and chain rules, length and area, line and surface integrals, Greens theorem in the plane and space. If time permits, topics in complex analysis may be included. This course has three lectures and two problem sessions each week. It is anticipated that most students will be engineering majors. Five sessions a week for one semester.

Text

Marsden & Tromba, Vector Calculus 6th edition

Outline

• I THE GEOMETRY OF EUCLIDEAN SPACE (6 days)
  • 1. I Vectors in two- and three-dimensional space
  • 1.2 The inner product, length, and distance
  • 1.3 Matrices, determinants, and the cross product
  • 1.4 Cylindrical and spherical coordinates
  • 1.5 n-dimensional Euclidean space
• 2 DIFFERENTIATION (5-6 days)
  • (add discussion of linear maps, matrices) 2.1 The geometry of real-valued functions
  • 2.2 Limits and continuity (assign to read)
  • 2.3 Differentiation
  • 2.4 Introduction to paths
  • 2.5 Properties of the derivative
  • 2.6 Gradients and directional derivatives
• 3 HIGHER-ORDER DERIVATIVES (3 days)
  • 3.1 Iterated partial derivatives (briefly)
  • 3.2 Taylor's theorem
  • 3.3 Extrema of real-valued functions
  • 3.4 Constrained extrema and Lagrange multipliers
  • 3.5 The implicit function theorem(if time permits) 3,5
• 4 VECTOR-VALUED FUNCTIONS (5 days)
  • 4.1 Acceleration and Newton's Second Law
  • 4.2 Arc length
  • 4.3 Vector fields
  • 4.4 Divergence and curl
• 5 DOUBLE AND TRIPLE INTEGRALS (3 days)(cover first three sections in one lecture)
  • 5.1 Introduction
  • 5.2 The double integral over a rectangle
  • 5.3 The double integral over more general regions
  • 5.4 Changing the order of integration
  • 5.6 The triple integral
• 6 THE CHANGE OF VARIABLES FORMULA (3 days)
  • 6.1 The geometry of maps (not crucial)
  • 6.2 The change of variables theorem (lightly)
  • 6.3 Applications of double, triple integrals(if time permits)
• 7 INTEGRALS OVER PATHS AND SURFACES (7 days) (next chapter
  • depends heavily on this) 7.1 The path integral
  • 7.2 Line integrals
  • 7.3 Parametrized Surfaces
  • 7.4 Area of a surface
  • 7.5 Integrals of scalar functions over surfaces
  • 7.6 Surface integrals of vector functions
• 8 THEOREMS OF VECTOR ANALYSIS (5-6 days) (may reorder as (8.1, 8.4, 8.2, 8.3)
  • 8.1 Green's theorem
  • 8.2 Stokes' theorem
  • 8.3 Conservative fields
  • 8.4 Gauss' theorem

Last Updated

Kathy Davis 1998

M 333L Structure of Modern Geometry

Degree relevance

Our Introduction to Mathematical Proof Writing courses (M 325K, M 328K, and M 333L) provide an essential transition from the algorithmic approach of calculus to the entirely rigorous approach of more advanced proof writing courses such as Advanced Number Theory, M 361K/M 365C, M 343K/M 373K, or M 367K. The number of topics required for coverage in each course has been kept modest so as to allow adequate time for students to develop theorem-proving skills. Students are expected to become familiar with the language and techniques of proof; they should also see detailed, rigorous proofs presented in class. More importantly, they need to develop the ability to read and understand proofs on their own, and they must begin doing proofs; this cannot be slighted. Over the course, the generation of ideas in the class needs to transition from instructor-initiated to more student-initiated. At the beginning of the semester, it is necessary that the instructor heavily model this behavior. Then as the semester progresses the professor and the students need to meet the challenge of each student assuming responsibility/ownership. In teaching abstraction, it is critical to remember that almost no students can become truly comfortable with it in a single semester; it is self-defeating to establish this as a goal. All Introduction to Mathematical Proof Writing course professors are strongly encouraged to employ active learning strategies. Students will discuss, debate, and negotiate what counts as valid proof argumentation and why. Students will not merely watch the instructor present correct, completed mathematics and imitate with superficial understanding.

M 333L Student Experience Goals 

Students will: 

• approach mathematical problems with curiosity and creativity and persist in the face of difficulties,
• capture the essential elements of intuitive mathematical objects in precise language that can make them withstand rigorous mathematical analysis,
• participate in the creative process and social negotiation characteristic of mathematicians’ work,
• transition from algorithmic and computational mathematics to actually producing and creating mathematics,
• develop the skills to read, understand, critique, and construct logical arguments,
• develop effective thinking and communication skills applicable in mathematics and well beyond mathematics,
• develop mathematical independence and experience mathematical inquiry, and
• experience the beauty and power of mathematics.

M 333L Student Learning Objectives 

Students will: 

• apply definitions effectively in proofs,
• use logical connectives and quantifiers correctly and with understanding,
• determine the validity of an argument,
• prove a given statement directly,
• prove a biconditional statement,
• prove a given statement indirectly (contradiction & contraposition),
• prove a given statement using mathematical induction,
• construct a proof by exhaustion,
• prove existence,
• prove existence and uniqueness,
• find a counterexample to disprove a given statement,
• write and speak about mathematics using precise mathematical language, and
• write well-organized, grammatically correct, and logically sound mathematical arguments.

Course Description

In this course, we will study 2-dimensional geometry from an axiomatic perspective. The emphasis of the course is on conceptual understanding and the development of proof-writing skills. Topics include an introduction to axiomatic systems, Euclidean plane geometry, and a glimpse of non-Euclidean geometries.

Texts

Edward Wallace and Stephen West's "Roads to Geometry" (3rd Ed, with corrections) Waveland Press, 2015 or STRUCTURE OF MODERN GEOMETRYSTRUCTURE OF MODERN GEOMETRY: A DISCOVERY APPROACH by Mark Daniels.

Major Topics Covered

1. Euclidean Geometry including some famous solved and unsolved proofs and problems.
2. Constructions in Euclidean Geometry and Constructibility of “Numbers”
3. Proofs in Analytic Geometry
4. Transformations & Iterated Function Systems
5. Inversion and Non-Euclidean Geometries Including Some Constructions and Proofs.

Last Updated

Jennifer Austin and Mark Daniels, August 2021.

M 339D Introduction to Financial Mathematics for Actuaries

Prerequisite and degree relevance

Actuarial Foundations 329 or Mathematics 329F; and Mathematics 362K with a grade of at least C-. Moreover, the instructor advises that students will need a thorough understanding and operational knowledge of (at least) calculus, finite-stage-space probability, and the term structure of interest rates.

Text

Robert L. McDonald, Derivative Markets, 2nd Edition (2015) Prentice Hall, ISBN 9780321280305

Description of the Course

This course is intended to provide the mathematical foundations necessary to prepare for a portion of

1. the joint SOA/CAS exam FM/2, as well as
2. the SoA exam MFE and the financial economics portion of the CAS Exam 3.

Additionally, the course is aimed at building up the vocabulary and the techniques indispensable in the workplace at current financial and insurance institutions.  This is not an exam-prep seminar.  There is intellectual merit to the course beyond the ability to prepare for a professional exam.

The material exhibited includes elementary risk management, forward contracts, options, futures, swaps, the simple random walk, the binomial asset pricing model, and its application to option pricing.  The remainder of the Exam MFE/3F curriculum is exhibited in course M339W (also offered by the Department of Mathematics).

Topics Covered

• Orientation. Role of financial markets.  Bid-ask spread.  Commissions.
• Standing assumptions.  Conventions.
• Outright purchase of an asset.  Discrete dividends.  Simple return.
• Continuous-dividend-paying assets.  Market  Indices.
• Short sales.
• Static financial portfolios.  Initial cost.  Payoff.
• Profit.  Definition of long/short positions.  Basic risk management. Forward contracts.
• European call options (rationale,  definition, implementation).
• European call options (payoff/profit).
• Hedging using European call options.
• Caps, i.e., short intrinsic position hedged with a call.
• Covered/naked option writing.  Covered calls. European put options (definition).
• Hedging using put options.  Floors.  Covered puts.
• Parallels between classical property-insurance policies and put options.
• Derivative securities.
• Examples of “simplest” derivative securities: All-or-nothing options.
• Review of finite probability spaces. Dynamic portfolios.  Profit.
• Arbitrage portfolio.  Arbitrage.
• Law of the unique price.
• Prepaid forward contracts. Forward and prepaid forward pricing (stocks).
• Annualized forward premium.  Arbitrage and forwards’ pricing.
• Commodity Swaps.
• Futures.
• Put-call parity.
• Replicating portfolios.  “Synthetic forward contracts”. Chooser options.  Straddles.
• Gap calls and puts.  Gap-option parity.
• American options.  Options on futures contracts.
• Options on currencies.
• Exchange options.
• Maximum option.  Generalized put-call parity.
• Option price bounds and monotonicity.  Bull spreads.
• Option price “slope” bounds.  Bear Spreads.
• Option price convexity.  Butterfly  Spreads.  Speculating on volatility.
• Strangles.  Collars.  Ratio  Spreads.  Equity-linked  CDs.
• Binomial asset-pricing model.
• Derivative-pricing by replication.  Risk-neutral probability.
• The forward tree.  Cox-Ross-Rubinstein binomial tree.  Jarrow-Rudd binomial tree.
• Two-period binomial pricing.  Multiple binomial periods.
• Early exercise. Bermudan options.
• Pricing American options.
• Properties of American-option prices.
• Asian options and their binomial pricing.
• Barrier options and their binomial pricing.
• Compound options and their binomial pricing.
• Binomial pricing of options on currencies.
• Binomial pricing of options on futures contracts.
• Interest-rate swaps.

Last Updated

Milica Cudina March 2015

M 339J Probability Models with Actuarial Applications

Prerequisite and degree relevance

Mathematics 358K or 378K with a grade of at least C-. Please note that a thorough knowledge of calculus, probability, and statistics will be assumed.

Course description

Introductory actuarial models for short-term insurance. 

Textbook

Klugman, S.A., Panjer, H.H. and Willmot, G.E., Loss Models: From Data to Decisions, (Fifth Edition), 2019

Topics Covered

PART I INTRODUCTION

• 2 Random variables
  • 2.1 Introduction
  • 2.2 Key functions (cumulative distribution function, survival function, probability density function, probability mass function, and hazard rate (= force of mortality or failure rate)
• 3 Basic distributional quantities 
  • 3.1 Moments 
  • 3.2 Percentiles 
  • 3.3 Generating functions and sums of random variables 
  • 3.4 Tails of distributions 
  • 3.5 Measures of Risk 

PART II ACTUARIAL MODELS

• 4 Characteristics of Actuarial Models 
  • 4.1 Introduction 
  • 4.2 The role of parameters 
• 5 Continuous models 
  • 5.3 Selected distributions and their relationships       
• 6 Discrete Distributions 
  • 6.1 Introduction 
  • 6.2 The Poisson distribution 
  • 6.3 The negative binomial distribution 
  • 6.4 The binomial distribution 
  • 6.5 The (a, b, 0) class 
  • 6.6 Truncation and modification at zero 
• 8 Frequency and severity with coverage modifications 
  • 8.1 Introduction 
  • 8.2 Deductibles 
  • 8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles 
  • 8.4 Policy limits 
  • 8.5 Coinsurance, deductibles, and limits 
  • 8.6 The impact of deductibles on claim frequency
• 9 Aggregate loss models 
  • 9.1 Introduction 
  • 9.2 Model choices 
  • 9.3 The compound model for aggregate claims 
  • 9.7 The impact of individual policy modifications on aggregate payments 
  • 9.8 The individual risk model 

PART III MATHEMATICAL STATISTICS

• 11 Maximum Likelihood Estimation 
  • 11.1 Introduction 
  • 11.2 Individual Data 
  • 11.3 Grouped Data
  • 11.4 Truncated or Censored Data
• 12 Frequentist Estimation for Discrete Distributions 
  • 12.1 The Poisson Distribution 
  • 12.2 The Negative Binomial Distribution 
  • 12.3 The Binomial Distribution 

Calculators

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.  

Actuarial Examinations

M 339J covers much of the short-term insurance content for SOA Exam FAM.  Students are expected to be familiar with survival, severity, frequency, and aggregate models, and use statistical methods to estimate parameters of such models given sample data. Students are further expected to identify steps in the modeling process, understand the underlying assumptions implicit in each family of models, recognize which assumptions are applicable in a given business application, and appropriately adjust the models for the impact of insurance coverage modifications.     

Last Updated

Alisa Havens Walch and Mark Maxwell, August 2014

M 339U Actuarial Contingent Payments I

Prerequisite and degree relevance: Mathematics 362K with a grade of at least C-; credit with a grade of at least C- or registration for Actuarial Foundations 329 or Mathematics 329F; and credit with a grade of at least C- or registration for Mathematics 340L or 341. Please note that a thorough knowledge of calculus, probability, and interest theory will be assumed.

Course description: Intermediate actuarial models for life insurance, property insurance, and annuities.

Textbook: Actuarial Mathematics for Life Contingent Risks, 2nd Edition (2013) David C. Dickson, Mary R. Hardy, and Howard R. Waters, Cambridge University Press, ISBN 9781107044074

Topics:

Chapter 1 Introduction to life insurance (1 hour)

    1.1 Summary

    1.2 Background

    1.3 Life insurance and annuity contracts

    1.4 Other insurance contracts

    1.5 Pension benefits

    1.6 Mutual and proprietary insurers

Chapter 2 Survival models (4 hours)

    2.1 Summary

    2.2 The future lifetime random variable

    2.3 The force of mortality

    2.4 Actuarial notation

    2.5 Mean and standard deviation of Tx

    2.6 Curtate future lifetime

Chapter 3 Life tables and selection (5 hours)

    3.1 Summary

    3.2 Life tables 

    3.3 Fractional age assumptions

    3.4 National life tables

    3.5 Survival models for life insurance policyholders

    3.6 Life insurance underwriting

    3.7 Select and ultimate survival models

    3.8 Notation and formulae for select survival models

    3.9 Select life tables

    3.10 Some comments on heterogeneity in mortality

    3.11 Mortality trends

Chapter 4 Insurance benefits (8 hours)

    4.1 Summary

    4.2 Introduction

    4.3 Assumptions

    4.4 Variation of insurance benefits

    4.5 Relatingand 

    4.6 Variable insurance benefits

    4.7 Functions for select lives

Chapter 5 Annuities (9 hours)

    5.1 Summary

    5.2 Introduction

    5.3 Review of annuities certain

    5.4 Annual life annuities

    5.5 Annuities payable continuously

    5.6 Annuities payable 1/mthly

    5.7 Comparison of annuities by payment frequency

    5.8 Deferred annuities

    5.9 Guaranteed annuities

    5.10 Increasing annuities

    5.11 Numerical illustrations

    5.12 Functions for select lives

Chapter 6 Premium calculation (7 hours)

    6.1 Summary

    6.2 Preliminaries

    6.3 Assumptions

    6.4 The present value of future loss random variable

    6.5 The equivalence principle

    6.6 Gross premiums

    6.7 Profit

    6.8 The portfolio percentile premium principle

    6.9 Extra risks

Chapter 7 Policy values (3 hours)

    7.1 Summary 

    7.2 Assumptions

    7.3 Policies with annual cash flows

For the suggested time devoted to each chapter, 1 hour corresponds to 50 minutes of actual class time. The total number of hours listed do not constitute an entire semester. They allow for review and examinations.

Calculators

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.  

Actuarial Examinations

In conjunction with M 339V, M 339U covers the content of SOA Exam MLC and CAS Exam LC. Topics covered: life insurance, survival models, life tables, insurance benefits, annuities, and premium calculation. See https://www.soa.org/education/exam-req/edu-asa-req.aspx for further details regarding these exams.

M 339V Actuarial Contingent Payments II

Prerequisite and degree relevance

Actuarial Foundations 329 or Mathematics 329F, and Mathematics 339U with a grade of at least C- in each. Please note that thorough knowledge of calculus, probability, interest theory, and Actuarial Contingent Payments I will be assumed.

Text

David C. Dickson, Mary R. Hardy, and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks, 2nd Edition (2013) Cambridge University Press, ISBN 9781107044074

Description of the Course

M 339V = M 389V Actuarial Contingent Payments II. Topics covered include Policy Values, Multiple State Models, Pensions, Interest Rate Risk, and Emerging Costs for Traditional Life Insurance.  

This is an actuarial capstone course and students are expected to do some independent learning and improve verbal and written acumen.  Three graded components of the course are 1) communication, 2) content, and 3) contribution to the class.  This class carries the Independent Inquiry Flag.  This class carries the Quantitative Reasoning flag.  

Meets with M389V, the corresponding graduate-course number.  Offered every spring semester only. This is a 3-credit course.

Topics

Chapter 7 Policy values 

• 7.4 Policy values for policies with cash flows at 1/mthly intervals 
• 7.4.1 Recursions 
• 7.4.2 Valuation between premium dates 
• 7.5 Policy values with continuous cash flows
• 7.5.1 Thiele’s differential equation 
• 7.5.2 Numerical solution of Thiele’s differential equation 
• 7.6 Policy alterations 
• 7.7 Retrospective policy values 
• 7.7.1 Prospective and retrospective valuation 
• 7.7.2 Defining the retrospective net premium policy value 
• 7.8 Negative policy values 
• 7.9 Deferred acquisition expenses and modified premium reserves 
• 7.10 Notes and further reading 
• 7.11 Exercises 

Chapter 8 Multiple state models 

• 8.1 Summary 
• 8.2 Examples of multiple state models 
• 8.2.1 The alive–dead model 
• 8.2.2 Term insurance with increased benefit on accidental death 
• 8.2.3 The permanent disability model 
• 8.2.4 The disability income insurance model 
• 8.3 Assumptions and notation 
• 8.4 Formulae for probabilities 
• 8.4.1 Kolmogorov’s forward equations 
• 8.5 Numerical evaluation of probabilities 
• 8.6 Premiums 
• 8.7 Policy values and Thiele’s differential equation 
• 8.7.1 The disability income insurance model 
• 8.7.2 Thiele’s differential equation – the general case 
• 8.8 Multiple decrement models 
• 8.9 Multiple decrement tables 
• 8.9.1 Fractional age assumptions for decrements 
• 8.10 Constructing a multiple decrement table 
• 8.10.1 Deriving independent rates from dependent rates 
• 8.10.2 Deriving dependent rates from independent rates 
• 8.11 Comments on multiple decrement notation 
• 8.12 Transitions at exact ages 
• 8.13 Markov multiple state models in discrete time 
• 8.13.1 The Chapman–Kolmogorov equations 
• 8.13.2 Transition matrices 
• 8.14 Notes and further reading 
• 8.15 Exercises 

Chapter 9 Joint life and last survivor benefits 

• 9.1 Summary 
• 9.2 Joint life and last survivor benefits 
• 9.3 Joint life notation 
• 9.4 Independent future lifetimes 
• 9.5 A multiple state model for independent future lifetimes 
• 9.6 A model with dependent future lifetimes 
• 9.7 The common shock model 
• 9.8 Notes and further reading 
• 9.9 Exercises 

Chapter 10 Pension mathematics  

• 10.1 Summary 
• 10.2 Introduction 
• 10.3 The salary scale function 
• 10.4 Setting the DC contribution 
• 10.5 The service table 
• 10.6 Valuation of benefits 
• 10.6.1 Final salary plans 
• 10.6.2 Career average earnings plans 
• 10.7 Funding the benefits [Not covered on SOA exam MLC]
• 10.8 Notes and further reading 
• 10.9 Exercises 

Chapter 11 Yield curves and non-diversifiable risk  

• 11.1 Summary 
• 11.2 The yield curve 
• 11.3 Valuation of insurances and life annuities 
• 11.3.1 Replicating the cash flows of a traditional non-participating product 
• 11.4 Diversifiable and non-diversifiable risk 
• 11.4.1 Diversifiable mortality risk 
• 11.4.2 Non-diversifiable risk 
• 11.5 Monte Carlo simulation [Not covered on SOA exam MLC]
• 11.6 Notes and further reading 
• 11.7 Exercises  

Chapter 12 Emerging costs for traditional life insurance  

• 12.1 Summary 
• 12.2 Introduction 
• 12.3 Profit testing a term insurance policy 
• 12.3.1 Time step 
• 12.3.2 Profit test basis 
• 12.3.3 Incorporating reserves 
• 12.3.4 Profit signature 
• 12.4 Profit testing principles 
• 12.4.1 Assumptions 
• 12.4.2 The profit vector
• 12.4.3 The profit signature 
• 12.4.4 The net present value 
• 12.4.5 Notes on the profit testing method 
• 12.5 Profit measures 
• 12.6 Using the profit test to calculate the premium 
• 12.7 Using the profit test to calculate reserves 
• 12.8 Profit testing for multiple state models 
• 12.9 Notes 
• 12.10 Exercises 

Chapter 13 Participating and Universal Life insurance 

• 13.1 Summary 
• 13.2 Introduction 
• 13.3 Participating insurance 
• 13.3.1 Introduction 
• 13.3.2 Examples 
• 13.3.3 Notes on profit distribution methods 
• 13.4 Universal Life insurance 
• 13.4.1 Introduction 
• 13.4.2 Key design features 
• 13.4.3 Projecting account values 
• 13.4.4 Profit testing Universal Life policies 
• 13.4.5 Universal Life Type B 
• 13.4.6 Universal Life Type A 
• 13.4.7 No-lapse guarantees 
• 13.4.8 Comments on UL profit testing 
• 13.5 Comparison of UL and whole life insurance policies 
• 13.6 Notes and further reading 
• 13.7 Exercises 

Calculators

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.  

Actuarial Examinations

In conjunction with M 339V, M 339U covers the content of SOA Exam MLC and CAS Exam LC. Topics covered: life insurance, survival models, life tables, insurance benefits, annuities, and premium calculation. See https://www.soa.org/education/exam-req/edu-asa-req.aspx for further details regarding these exams.

Last Updated

Mark Maxwell August 2014

M 339W Financial Mathematics for Actuarial Applications

Prerequisite and degree relevance

Mathematics 339D with a grade of at least C-.  Moreover, the instructor also advises that students will need a thorough understanding and operational knowledge of (at least) classical calculus, calculus-based probability (with emphasis on the normal distribution), the term structure of interest rates, and the principles of risk-neutral pricing in the binomial asset-pricing model.

Text

Robert L. McDonald, Derivative Markets, 2nd Edition (2015) Prentice Hall, ISBN 9780321280305

Description of the Course

This course is intended to provide the mathematical foundations necessary to prepare for a portion of the SOA Exam MFE and the "financial economics" portion of the CAS Exam 3.

Additionally, the course is aimed at building up the vocabulary and the techniques indispensable in the workplace at current financial and insurance institutions. This is not an exam-prep seminar.

The material exhibited includes: an in-depth study of the normal and log-normal distributions, the simple random walk, basics of stochastic calculus, the Samuelson (geometric Brownian motion) stock-price model and the Black-Scholes formula, analysis of option Greeks, market making, non-deterministic interest rate models (both discrete, and continuous-time), bond pricing, Monte-Carlo simulations.  The remainder of the Exam MFE/3F curriculum is exhibited in course M339d (also offered by the Department of Mathematics). 

Topics

• Orientation.  Standing assumptions. Conventions.
• Binomial interest rate models.
• Black-Derman-Toy.
• Review of uniform distribution. Random number generation.
• Probability on the coin toss space.  Simulation of the random walk.
• Law of Large Numbers.  Risk-neutral pricing by simulation (the binomial case).
• Scaled random walk. 11.3: Proceeding to continuous time.
• Normal and log-normal distributions.
• Log-normal stock-price model.
• Brownian motion.
• Introduction to formal stochastic calculus for financial mathematics.
• Stochastic integral ("definition", obstacles). Itô-Doeblin Lemma. Itô processes.
• Samuelson's model for stock prices.  Portfolio representation.
• Sharpe ratio. The risk-neutral probability measure.
• Black-Scholes PDE.  Risk-neutral pricing.
• Black-Scholes pricing formula.  Price curve prior to expiration.
• Black-Scholes pricing for options of futures, currencies, discrete-dividend-paying stocks.
• Correlated assets.  Exchange options.  Black-Scholes pricing and exotic options.
• Forward prices for powers of the underlying.
• Implied volatility.
• Greeks in the Black-Scholes pricing.  A detailed look on the ∆.  Option elasticity.
• “Greeks” in the binomial tree.  Market making and ∆−hedging.
• Self-financing portfolios.  Overnight profit/loss.  Γ−hedging.
• Market-making and bond-pricing.  Duration-hedging.
• The Ornstein-Uhlenbeck process.  Continuous-time interest rate models.
• Black formula.
• Monte Carlo valuation.
• Variance reduction methods. Control variate method.

Last Updated

Milica Cudina March 2015

M 340L Matrices and Matrix Calculations

Prerequisite and degree relevance

Mathematics 408C, 408K, or 408N with a grade of at least C-. Only one of the following may count: Mathematics 340L, 341, Statistics and Data Sciences 329C, or Statistics and Scientific Computation 329C.

Text

David C. Lay, Linear Algebra and its Applications, 6th ed.

Background

M 341 (Linear Algebra and Matrix Theory) and M 340L (Matrices and Matrix Calculations) cover similar material. However, the emphasis in M 340L is much more on calculational techniques and applications, rather than abstraction and proof. (M 341 is the preferred linear algebra course for math majors and contains a substantial introduction to proof component.)

Course Content

Read the "Note to the Instructor" at the beginning of the book. The core of M340L is indeed the "core topics" listed on pages ix-x, plus sections 3.1 and 3.2. Various faculty members disagree strongly about which of the remaining "supplementary topics" and "applications" are most important; use your own judgment. You will probably have time for about half a dozen of these supplementary topics and applications.

The syllabus covers the essentials of all seven chapters of Lay, namely

1. Linear Equations in Linear Algebra,
2. Matrix Algebra,
3. Determinants,
4. Vector Spaces,
5. Eigenvalues and Eigenvectors,
6. Orthogonality and Least Squares, and
7. Symmetric Matrices and Quadratic Forms.

Each section is designed to be covered in a single 50-minute lecture. However, in practice chapters 1 - 3 should be covered more quickly (a bit slower on the last 3 sections of chapter 1), allowing more time for chapters 4-7. Most incoming M340L students are already quite adept at solving systems of equations, and it is important to move quickly at the beginning of the term to material that does challenge them, reserving time to tackle the more difficult vector space concepts of chapter 4. Many of the essential concepts, such as linear independence, are covered twice: once in chapter 1 for Rn, then again in chapter 4 for a general vector space.

Computers

Linear algebra lends itself extremely well to computerization, and there are many packages that students can use. Once students have learned the theory of row-reduction and matrix multiplication (which they pick up very quickly), they should be encouraged to use Maple, Matlab, Mathematica, or a similar package. There are also many "projects" on the departmental computers that students can learn from. Many concepts in the book, especially in the later chapters (e.g., understanding the long-time behavior of a dynamical system from its eigenvalues), can be absorbed quite easily through numerical experimentation.

Last Updated

Gary Hamrick, June 2003

M 341 Linear Algebra and Matrix Theory

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C-. Restricted to mathematics majors. Only one of the following may count: Mathematics 340L, 341, Statistics and Data Sciences 329C, or Statistics and Scientific Computation 329C. Majors with a 'math' advising code must register for M 341 rather than for M 340L; majors without a 'math' advising code must register for M 340L. Math majors must make a grade of at least C- in M 341.

Primary Text

Andrilli & Hecker, Elementary Linear Algebra 5th edition

Overview

This course has three purposes and the instructor should give proper weight to all three. The students should learn some linear algebra - for most of them, this will be the only college linear algebra course they take. This is one of the first proof courses these students will take and they need to develop some proof skills. Finally, this is, for almost all students, the introductory course in mathematical abstraction and provides a necessary prerequisite for a number of our upper-division courses. To teach this course successfully, the instructor should establish modest goals on all three fronts. On one hand, a student should not be able to pass this course simply by doing calculational problems well, but on the other hand, overly ambitious proof and abstraction goals simply discourage teacher and student alike.

To teach proofs, the instructor should cover Section 1.3 thoroughly to introduce various proof techniques. Afterward, a liberal (but not overwhelming) number of proofs should be sprinkled in the lectures, homework, and tests.

In teaching abstraction, it is critical to remember that almost no students are capable of becoming truly comfortable with it in a single semester; it is self-defeating to establish this as a goal. The study of abstract vector spaces is a unified treatment of various familiar vector spaces and students in this course should never be taken very far from the concrete. Linear algebra is the perfect subject for teaching students that abstraction can be a friend. For example, it underlines nicely how the solutions to a homogeneous system are better behaved than the solutions to a non-homogeneous system. However, amusing examples of unnatural algebraic systems that may or may not be vector spaces should be avoided.

A warning should be given concerning the calculational homework problems. The authors, intending the students to take full advantage of technology, have made no effort to make problems come out neatly.

Suggested Coverage

• Chapter 1 - Nine or ten lectures. The first two sections provide necessary definitions for Section 1.3. The entire chapter should be covered. Generally, move quickly but cover 1.3 meticulously. Three or four lectures should be devoted to this section.
• Chapter 2 - Six or seven lectures. Cover all sections but again move reasonably to have enough time for Chapters 4 and 5. 
• Chapter 3 - Three lectures. Row operations are easy for them and you can go quite quickly here.  Cover Sections 3.1 and 3.2. Section 3.3 is optional - you might also choose to cover parts of this section. Section 3.4 is a fairly reasonable attempt to introduce eigenvalues before introducing linear transformations. It is an interesting and important part of this chapter, at least in my opinion.  The instructor should cover at least part of this section, all if desired.
• Chapter 4 - Fourteen or fifteen lectures. This chapter is the meat of the course and the instructor should plan to take a good deal of time here. Sections 4.1-4.6 should be covered thoroughly. Section 4.7 is optional and should probably be skipped to provide more time for Chapter 5.
• Chapter 5 - About five lectures. In a perfect world, the entire chapter should be taught, but 5.5 is probably too much to hope for. Realistically, at least Sections 5.1 and 5.2 should be covered.

Last Updated

Ray Heitmann, July 2014

M 343K Introduction to Algebraic Structures

Prerequisite and degree relevance

Either consent of Mathematics Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. This course is designed to provide additional exposure to abstract rigorous mathematics on an introductory level. Students who demonstrate superior performance in M 311 or M 341 should take M 373K instead of 343K. Those students whose performance in M 311 or M 341 is average should take M 343K before taking M 373K. Credit for M 343K can NOT be earned after a student has received credit for M 373K with a grade of at least C-.

Course description

Elementary properties of the integers, groups, rings, and fields are studied.

The number of topics should be kept modest to allow adequate time to concentrate on developing the students' theorem-proving skills. Some instructors will prefer to introduce groups before rings and some will reverse the order. In any case, below are some reasonable choices of topics. One should not try to cover all of these topics. It is very important to avoid superficial coverage of too many topics. All potential graduate students will take M 373K, where it is possible to expect more and to do more.

Topics

• Groups: Axioms, basic properties, examples, symmetry, cosets, Lagrange's Theorem, isomorphism. Homomorphisms, quotient groups, and the Fundamental Homomorphism Theorem.
• Optional: Rings: Axioms, basic properties, examples, integral domains, and fields. rings and properties of fields.
• Optional: More about polynomial
• Other options: Groups acting on sets, characterization of the familiar number systems in terms of ring and field properties, and other applications of groups.

Last Updated

Durbin, July 2000

M 343L Applied Number Theory

Prerequisite and degree relevance

Mathematics 328K or 343K with a grade of at least C-.

Text

An Introduction to Mathematical Cryptography by Jill Pipher, Jeffrey Hoffstein, Joseph H. Silverman.

Topics

Basic properties of integers. Prime numbers and unique factorization. Congruences, Theorems of Fermat and Euler, primitive roots. Primality testing and factorization methods. Cryptography, basic notions. Public key cryptosystems. RSA. Implementation and attacks. Discrete log cryptosystems. Diffie-Hellman and the Digital Signature Standard. Elliptic curve cryptosystems. Symmetric cryptosystems, such as DES and AES.

Last Updated

Kathy Davis and Felipe Voloch, Spring 2010

M 346 Applied Linear Algebra

Prerequisite and degree relevance

The prerequisite is M 341 (or M 311) or M 340L with a grade of C- or better.

Text

Lorenzo Sadun, Applied Linear Algebra; the Decoupling Principle, second edition. 

We expect students to have a good feel for manipulating matrices, especially row reduction, but also taking determinants. We also expect students to have seen abstract vector spaces and linear transformations, but some rustiness is expected, and those topics should be reviewed. It is not assumed that students have seen eigenvalues and eigenvectors; those should be done from scratch.

This is a course in serious mathematics, not a cookbook. As such, results in lecture, and in the book, should generally be proved rigorously. However, it's not an intro-to-proof class and is aimed at an audience of engineers, economists, and physicists (as well as mathematicians), so *writing* proofs should only play a minor role in the problem sets and exams.

Detailed Syllabus

This number of days in this syllabus is based on a TTh class. 

 Chapter 1. The Decoupling Principle (one day)

•     Exploration: Beats

Chapter 2. Vector Spaces and Bases (two days)

•     Vector Spaces
•     Linear Independence, Basis, and Dimension
•     Properties and Uses of a Basis
•     Exploration: Polynomials
•     Change of Basis
•     Building New Vector Spaces from Old Ones
•     Exploration: Projections

Chapter 3. Linear Transformations and Operators (three days)

•     Definitions and Examples
•     Exploration: Computer Graphics
•     The Matrix of a Linear Transformation
•     The Effect of a Change of Basis
•     Infinite Dimensional Vector Spaces
•     Kernels, Ranges, and Quotient Maps

Chapter 4. An Introduction to Eigenvalues (four days)

•     Definitions and Examples
•     Bases of Eigenvectors
•     Eigenvalues and the Characteristic Polynomial
•     The Need for Complex Eigenvalues
•     Exploration: Circles and Ellipses
•     When is an Operator Diagonalizable?
•     Traces, Determinants, and Tricks of the Trade
•     Simultaneous Diagonalization of Two Operators
•     Exponentials of Complex Numbers and Matrices
•     Power Vectors and Jordan Canonical Form

Chapter 5. Some Crucial Applications (five days)

•     Discrete-Time Evolution: x(n) = Ax(n   1)
•     Exploration: Fibonacci Numbers and Tilings
•     First Order Continuous Time Evolution
•     Second order Continuous Time Evolution   
•     Reducing Second Order Problems to First Order
•     Exploration: Difference Equations
•     Long Time Behavior and Stability
•     Markov Chains and Probability Matrices
•     Exploration: Random Walks
•     Linear Analysis near Fixed Points of Nonlinear Problems
•     Exploration: Nonlinear ODEs

Chapter 6. Inner Products (four days)

•     Real Inner Products: Definitions and Examples
•     Complex Inner Products
•     Bras, Kets, and Duality
•     Expansion in Orthonormal Bases: Finding Coefficients
•     Projections and the Gram Schmidt Process
•     Orthogonal Complements and Projections onto Subspaces
•     Least Squares Solutions
•     Exploration: Fourier Series
•     Exploration: Curve Fitting
•     The Spaces l² and L²
•     Fourier Series on an Interval

Chapter 7. Adjoints, Hermitian Operators, and Unitary Operators (three days)

•     Adjoints and Transposes
•     Hermitian Operators
•     Quadratic Forms and Real Symmetric Matrices
•     Rotations, Orthogonal Operators, and Unitary Operators
•     Exploration: Normal Matrices
•     How the Four Classes are Related Exploration: Representations Of SU2

Chapter 8. The Wave Equation (four days)

•     Waves on the Line
•     Waves on the Half-Line: Dirichlet and Neumann Boundary Conditions
•     The Vibrating String

Last Updated

Charles Radin and Lorenzo Sadun, May 2008. 

M 348 Scientific Computation in Numerical Analysis

Prerequisite and degree relevance

Computer Science 303E or 307, and Mathematics 341 or 340L with a grade of at least C-.

Course description

Introduction to mathematical properties of numerical methods and their applications in computational science and engineering. Introduction to object-oriented programming in an advanced language. Study and use of numerical methods for solutions of linear systems of equations, non-linear least-squares data fitting, numerical integration of multi-dimensional, non-linear equations, and systems of initial value ordinary differential equations.

M 349P Applied Statistical Estimates

Prerequisite and degree relevance

Mathematics 339J, and 341 or 340L, with a grade of at least C- in each. Please note that thorough knowledge of calculus, probability, and statistics will be assumed.

Textbook

Klugman, S.A., Panjer, H.H. and Willmot, G.E., Loss Models: From Data to Decisions, Fifth Edition, 2019

Some material is covered in online, open-source texts, instead of the Loss Models textbook.

Description of the Course

M 349P Probability Models with Actuarial Applications covers statistical estimation procedures for random variables and related quantities in actuarial models.

Meets with M 389P, the corresponding graduate-course number. Offered every fall semester only. This is a 3-credit course.

PART II ACTUARIAL MODELS

• 5 Continuous Models
  • 5.2 Creating new distributions

PART III MATHEMATICAL STATISTICS  

• 11 Maximum Likelihood Estimation 
  • 11.5 Variance and interval estimation for maximum likelihood estimation 
  • 11.6 Functions of asymptotically normal estimators
  • 11.7 Non-normal confidence intervals
• 13 Bayesian estimation 
  • 13.1 Definitions and Bayes’ theorem 
  • 13.2 Inference and prediction 
  • 13.3 Conjugate prior distributions and the linear exponential family 

PART IV CONSTRUCTION OF MODELS

• 15 Model selection 
  • 15.1 Introduction 
  • 15.2 Representations of the data and model 
  • 15.3 Graphical comparison of the density and distribution functions 
  • 15.4 Hypothesis tests 
  • 15.5 Selecting a model 

PART V CREDIBILITY 

• 16 Introduction and Limited Fluctuation Credibility 
  • 16.1 Introduction 
  • 16.2 Limited fluctuation credibility theory 
  • 16.3 Full credibility 
  • 16.4 Partial credibility 
• 17 Greatest accuracy credibility
  • 17.1 Introduction 
  • 17.2 Conditional distributions and expectation 
  • 17.3 The Bayesian methodology 
  • 17.4 The credibility premium 
  • 17.5 The Buhlmann model 
  • 17.6 The Buhlmann-Straub model 
  • 17.7 Exact credibility 
• 18 Empirical Bayes parameter estimation 
  • 18.1 Introduction 
  • 18.2 Nonparametric estimation 
  • 18.3 Semi-parametric estimation

Calculators

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.  

Actuarial Examinations

In conjunction with M 339J, M 349P covers the majority of content on SOA Exam ASTAM.  Students are expected to be familiar with survival, severity, frequency and aggregate models, and use statistical methods to estimate parameters of such models given sample data. Students are further expected to identify steps in the modeling process, understand the underlying assumptions implicit in each family of models, recognize which assumptions are applicable in a given business application, and appropriately adjust the models for the impact of insurance coverage modifications.

Last Updated

Alisa Havens Walch and Mark Maxwell, August 2014

M 349R Applied Regression and Time Series

Prerequisite and degree relevance

Mathematics 358K or 378K, with a grade of at least C- in each.

Textbook

Bowerman and Koehler, Forecasting, Time Series, and Regression, Fourth Edition

Description of the Course

The purpose of this course is to provide students in actuarial science, statistics, and applied disciplines with an introduction to simple and multiple regression methods for analyzing relationships among several variables, and to elementary time series analysis.  The emphasis will be on fitting suitable models to data, evaluating models using numerical and graphical techniques, and interpreting the results in the context of the original problem, as opposed to the derivation of mathematical properties of the models.  At the end of this course, students will be able to analyze many kinds of data in which one variable of interest is thought to depend on, or at least be related to, several other measured quantities, and some kinds of data collected over time or in some other serial manner.

Course Goals and Overview

Incoming Students should be very familiar with descriptive statistics, simple regression, the logic of statistical inference, hypothesis tests, and confidence intervals for means and proportions. M 349R is a computer-intensive course starting with an introduction to R and gradually moving towards SAS.  The focus of the course is on hands-on data analysis.  Students will work on projects with real data, identifying and stating the problems, planning/solving, and concluding/reflecting.  The textbook will be supplemented with R/SAS code and additional topics.

Timing

A typical semester has 42-44 MWF days.  The syllabus contains topics for 35 class days and an additional 6 class days with Optional Topics.  There are 3 class days for midterms or review.

Calendar

Approximate calendar with 38 days three times a week and 6 days for optional topics.

Last Updated

Mark Maxwell and Gustavo Cepparo, Date

M 358K Applied Statistics

Prerequisite and degree relevance

M 362K with a grade of C- or better. This course is intended for students in the Probability and Statistics math major specialization, students planning to teach secondary mathematics, students working for a BA in mathematics, and (as space permits) students in the natural sciences. Students preparing for graduate work in mathematical statistics should take M 378K instead of or after taking this course.

Text

The standard textbook is Moore and McCabe, Introduction to the Practice of Statistics, Fifth Edition. This will be supplemented with additional material. OpenIntro Statistics by David Diez, 4th Edition

Resources

Instructors should contact Martha Smith for more details on the project, pacing, and supplemental material.

Project

Students will be expected to do a term project to apply the material studied in the course.

Computer use

Students are expected to use software (typically, Minitab) to create graphs and do statistical calculations. They should also be able to interpret software output. 

Topics

Aligned with Moore and McCabe

• Chapter 1: Looking at Data - Distributions
  • Sections 1 - 3, supplemented with additional activities and material.
• Chapter 2:  Looking at Data - Relationships
  • Sections 1 – 5, supplemented with additional material.
• Chapter 3: Producing Data
  • Sections 1 – 4, supplemented with additional material, including the project proposal.
• Chapter 4: Probability: The Study of Randomness.
  • Sections 1 – 5 (Mostly review from M362K.)
• Chapter 5: Sampling Distributions
  • Sections 1 and 2, supplemented with class activities and material.
• Chapter 6: Introduction to Inference
  • Sections 1 – 4, supplemented with class activities.
• Chapter 7: Inference from Distributions
  • Sections 1 – 3, supplemented with additional material.
• Chapter 8: Inference for Proportions
  • Sections 1 – 2
• Chapter 9: Analysis of Two-Way Tables
  • Sections 9.1 – 9.3 (optional topics in 9.2 may be omitted), possibly supplemented with additional material. (Instructor may substitute Chapter 12: One-Way Analysis of Variance.)
• Chapter 10: Inference for Regression
  • Sections 1 – 2, supplemented with derivations of formulas.

Last Updated

Martha Smith, August 2008

M 360M Mathematics as Problem Solving

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C- and written consent of instructor.

This is a course in problem-solving in mathematics, geared primarily toward prospective math teachers.  The goal of the course is to improve problem-solving skills.   Students will be solving problems in class and at home, in groups and individually.  The focus of the course is on the problem-solving process.  Students will gain familiarity with commonly used heuristics, learn to maintain good control of the problem-solving process, and will gain proficiency in presenting solutions in both oral and written form.

Last Updated

Kathy Davis and Altha Rodin, Spring 2010

M 361 Theory of Functions of a Complex Variable

Prerequisite and degree relevance

Mathematics 427J, 427K, or 427L with a grade of at least C-.

Course description

M 361 consists of a study of the properties of complex analytic functions. Students are mainly from physics and engineering, with some mathematics majors and joint majors. Representative topics are Cauchy's integral theorem and formula, Laurent expansions, residue theory and the calculation of definite integrals, analytic continuation, and asymptotic expansions. Rigorous proofs are given for most results, with the intent to provide the student with a reliable grasp of the results and techniques.

Text

A reasonable text is Brown and Churchill, Complex Variables and Applications, 8th edition.

Topics

• Complex Numbers
• Analytic Functions
• Elementary Functions
• Integrals
• Series
• Residues and poles
• Applications of Residues

Last Updated

John Dollard 2001

M 361K Introduction to Real Analysis

Prerequisite and degree relevance

Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who have received a grade of C- or better in Mathematics 365C may not take Mathematics 361K.

Course description

This is a rigorous treatment of the real number system, of real sequences, and of limits, continuity, derivatives, and integrals of real-valued functions of one real variable.

Text

A reasonable text is Introduction to Real Analysis by Bartle and Sherbert 4th Edition. The course might cover the bulk of chapters one through six in that book.

Topics

• The real number system: the axiomatic description of the real number system as the unique complete ordered field, with special emphasis on the completeness axiom; the elementary topology of the real line.
• Real sequences: the definition and elementary properties of sequential limits; subsequences and accumulation points; monotone sequences; inferior and superior limits; the Bolzano-Weierstrass theorem.
• Limits and continuity of functions: the definition and elementary properties of limits of functions, including the usual variations on the basic theme (e.g.,one-sided limits, infinite limits, limits at infinity); continuity; the fundamental facts concerning continuous functions on intervals (e.g., Intexmexliatc Value Theorem, Maximum-Minimum Theorem, continuity of inverse functions, uniform continuity on closed intervals).
• Differentiation: the definition and geometric significance of the derivative; differentiation rules; the Mean Value Theorem and its consequences; Taylor's Theorem; L'Hospital's rules; convexity.
• Riemann Integration: the definition and elementary properties of the Riemann integral; the integrability of continuous functions and monotone functions; the Fundamental Theorems of Calculus.

Last Updated

March 1989

M 362K Probability I

Prerequisite and degree relevance

Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 362K and Statistics and Scientific Computation 321 may not both be counted.

Course description

This is an introductory course in the mathematical theory of probability, thus it is fundamental to further work in probability and statistics. Principles of set theory and a set of axioms for probability are used to derive some probability density and/or distribution functions. Special counting techniques are developed to handle some problems. Properties associated with a random variable are developed for the usual elementary distributions. Problem-solving is required, and some theorem proving can be done, but the course emphasizes computation and intuition. 

Suggested Textbook

A First Course in Probability, 10 Edition, by Sheldon Ross.

The following course outline refers to section numbers in Ross' book and assumes a MWF lecture format (it must be modified for TTh classes) 

Some Alternate Textbooks:

1. Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd revised ed., AMS 1977. This book has an interesting style that is different from the more standard format of Ross. It introduces some important ideas in examples and exercises, so the instructor needs to know what not to omit. There is too much emphasis on computation for this course, but otherwise, it is very well written, with many good examples and exercises.
2. Saeed Ghahramani, Fundamentals of Probability. Prentice Hall, 1996. Similar to Ross' text.

Background

M 362K is required of all undergraduate mathematics majors, and it is a prerequisite for courses in statistics. However, many of the students are majoring in other subjects(e.g., computer science or economics), and have little preparation in abstract mathematics. Calculus skills (integration and infinite series) tend to be weak, even at this level. Similarly, you cannot expect students to have any background in proofs, and should not expect competence in this. The course tends to be relatively easier for the first three to four weeks, so some students get the wrong impression as to its difficulty. Clarifying this early for the students can avoid unpleasant surprises later.

Course Content

Emphasize problem solving and intuition. Some advanced concepts should be presented without proof, so as to devote more attention to the examples. Basic combinatorics: Counting principle, permutations, combinations. Basic concepts: Sample spaces, events, basic axioms and theorems of probability, finite sample spaces with equally likely probabilities. Conditional probability: Reduced sample space, independence, Bayes' Theorem. Random variables: Discrete and continuous random variables, discrete probability functions and continuous probability density functions, distribution functions, expectation, variance, functions of random variables. Special distributions: Bernoulli, Binomial, Poisson, and Geometric discrete random variables. Uniform, Normal, and Exponential continuous random variables. Approximation of Binomial by Poisson or Normal. Jointly distributed random variables: Joint distribution functions, independence, conditional distributions, expectation, covariance Sums of independent random variables: expectation, variance. Inequalities and Limit theorems: Markov's and Chebyshev's inequalities, Weak and Strong Law of Large Numbers, Central Limit Theorem.

• 1.1-1.4: 3 lectures, Limit this material to one week.
• 2.1-2.5; 2.7: 4 lectures, Do not get bogged down in 2.5; limit it to about one lecture.
• 3.1-3.4: 4 lectures, Students like tree diagrams for Bayes' Theorem, and need more help and examples to learn how to extract information from word problems.
• 4.1-4.5: 4 lectures.
• 4.6-4.7; 4.8.1: 3 lectures, Omit 4.6.2 and 4.7.1. One could delay 4.7 to 5.4.1. Sections 4.8.2 and 4.8.3 are optional.
• 5.1-5.5; 5.7: 7 lectures. Omit 5.5.1;  Section 5.6.1 is optional.
• 6.1-6.5: 4 lectures.
• 7.1-7.2; 7.4: 2 lectures, Omit 7.2.1, 7.2.2. Sections 7.5, 7.7, 7.8 are optional, as is correlation.
• 8.1-8.4: 3 lectures, Do not let any optional material crowd out the limit theorems. Emphasize intuitive understanding of the Central Limit Theorem by examples, and omit the proof, especially if optional 7.7 is not covered. One or two topics are optional.

There are a wealth of examples in the text, so the instructor has time to present only some of them. The outline above allows room for 34 lectures, 3 in-class exam days, and 3 review days, for a total of 40 days. A typical semester has 42 MWF class days in the fall and 44 in the spring, so a few days for make-up or optional material are provided. It is likely that an instructor will find no time for any of the optional material.

Last Updated

T. Arbogast, J. Luecke, and M. Smith, August 2008

M 362M Introduction to Stochastic Processes

Prerequisite and degree relevance

Mathematics 362K with a grade of at least B and Mathematics 341 (or 311) or Mathematics 340L with a grade of C- or better.

Course description

Introduction to Markov chains, birth and death processes, and other topics.

M 365C Real Analysis I

Prerequisite and degree relevance

Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who receive a grade of C- in one of the prerequisite courses are advised to take Mathematics 361K before attempting 365C. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.

Course description

This course is an introduction to Analysis. Analysis together with Algebra and Topology form the central core of modern mathematics. Beginning with the notion of limit from calculus and continuing with ideas about convergence and the concept of function that arose with the description of heat flow using Fourier series, analysis is primarily concerned with infinite processes, the study of spaces and their geometry where these processes act and the application of differential and integral to problems that arise in geometry, PDE, physics, and probability. This should be a course in analysis rather than point-set topology; the latter is covered in M376K. 

Text

An appropriate text is Rudin "Principles of Mathematical Analysis" and the course should roughly cover its first seven chapters. The main difference between M 361K and M 365C lies in the more abstract metric space point of view in the latter. A strong student should be able to handle M 365C without first taking M361K.

• The real number system and Euclidean spaces: the axiomatic description of the real number system as the unique complete ordered field; the complex numbers; Euclidean space IR.
• Metric spaces: elementary metric space topology, with special emphasis on Euclidian spaces; sequences in metric spaces - limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compactness in metric spaces; compact sets in R; connectedness in metric spaces; countable and uncountable sets.
• Continuity: limits and continuity of mappings between metric spaces, with particular attention to real-valued functions defined on subsets of R; preservation of compactness and connectedness under continuous mapping; uniform continuity.
• Differentiation on the line: the definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value Theorem and its consequences; Taylor's Theorem; L'Hospital's rules.
• Riemann integration on the line: the definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.
• Sequences and series of functions: uniform convergence, uniform convergence, and continuity, uniform convergence, and integration, uniform convergence and differentiation.

Last Updated

September 2008

M 365D Real Analysis II

Prerequisite and degree relevance

Mathematics 365C, with a grade of at least C-.

Course description

A rigorous treatment of selected topics in real analysis, such as Lebesgue integration, or multivariable integration and differential forms.

Possible Texts

Spivak, Calculus. Ross, Elementary Analysis; The Theory of Calculus. Fulks, Advanced Calculus.

This is a continuation of M 365C with emphasis on functions of several variables. The treatment should be reasonably simple (for example, it is inappropriate to use Banach space language). The teacher can select his own textbook and should weigh his/her choice carefully in light of the above remarks and not get too ambitious.

Last Updated

March 1989

M 367K Topology I

Prerequisite and degree relevance

Mathematics 361K or 365C or consent of instructor.

Course description

This will be a first course that emphasizes understanding and creating proofs; therefore, it provides a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to Concentrates on developing the students' theorem-proving skills. The syllabus below is a typical syllabus. Other collections of topics in topology are equally appropriate. For example, some instructors prefer to restrict themselves to the topology of the real line or metric space topology.

• Cardinality: 1-1 correspondence, countability, and uncountability.
• Definitions of topological space: basis, sub-basis, metric space.
• Countability properties: dense sets, countable basis, local basis.
• Separation properties: Hausdorff, regular, normal.
• Covering properties: compact, countably compact, Lindelof.
• Continuity and homeomorphisms: properties preserved by continuous functions, Urysohns Lemma, Tietze Extension Theorem.
• Connectedness: definition, examples, invariance under continuous functions.

Notes containing definitions, theorem statements, and examples have been developed for this course and are available. The notes include some topics beyond those listed above.

Last Updated

March 1989

M 367L Topology II

Prerequisite and degree relevance

Mathematics 367K with a grade of at least C- or consent of instructor.

Course description

Various topics in topology, primarily of a geometric nature.

M 368K Numerical Methods for Applications

Prerequisite and degree relevance

Mathematics 348 with a grade of at least C-. Only one of the following may be counted: Computer Science 367, Mathematics 368K, Physics 329.

Course description

Continuation of Mathematics 348. Topics include splines, orthogonal polynomials, and smoothing of data, iterative solution of systems of linear equations, approximation of eigenvalues, two-point-boundary value problems, numerical approximation of partial differential equations, signal processing, optimization, and Monte Carlo methods.

M 371E Learning Assistant Experience in Mathematics

Prerequisite and degree relevance

Mathematics 408C, 408K, 408N, 408R, or equivalent, and consent of instructor.

Course description

Students assist instructors and TAs in mathematics courses. This is a hands-on experience in what it is like to teach and support students in the learning of mathematics in undergraduate courses. Students in M 371K must attend classroom training and discussions and work in Calculus discussion sections or undergraduate classrooms where mathematics is being taught.

The ultimate goal of the course is that you acquire a basic understanding of the fundamental principles of learning in our discipline, a realistic perspective of your own strengths and weaknesses as developing professionals, and a compelling interest in learning about and confronting the challenges that lie before you in the remainder of your education and in your future professional lives as mathematicians.  In this regard, this course will also expose you to ethical issues and to the process of applying ethical reasoning in real-life situations.

The intention is that by the end of this course you will be simultaneously effective and efficient teachers who have reflected upon the ethical issues related to supervising and instructing others and the leadership characteristics needed for this situation. To do this, you will develop a coherent framework for understanding human learning in the context of mathematics instruction, which you can articulate.  Furthermore, you will gain experience applying the framework while planning your own class lessons, presentations, and assessments.

Course Objectives

By the end of the course, you will be able to do each of the following in a limited context:

1. Explain fundamental principles of human learning and their application in the development of intellectual skills.
2. Articulate meaningful instructional goals for professionals, pre-professionals, and other students of mathematics.
3. Design effective learning sequences and lessons that focus on the development of (a) intellectual flexibility and depth and (b) excellent fundamental skills.
4. Develop a consistent and effective method for assessing students’ performance,
5. Speak, present, and write clearly and cogently. Give succinct instructions and direct, effective, positive and negative feedback.
6. Systematically analyze the effectiveness of your teaching on the basis of student accomplishment.
7. Contribute to the improvement of your own teaching and the teaching of your peers by providing thoughtful, informative analyses of instructional effectiveness.
8. Construct a philosophy of teaching.
9. Increase your awareness of factors that bear on ethical decision-making, and equip you to be your best self in difficult situations.

M 372K PDE and Applications

Prerequisite and degree relevance

Mathematics 427J or 427K with a grade of at least C-. One of M 361K or M 365C is also recommended.

Course description

Partial differential equations arise as basic models of flows, diffusion, dispersion, and vibrations. Topics include first- and second-order partial differential equations and classification, particularly the wave, diffusion, and potential equations, their origins in applications and properties of solutions, characteristics, maximum principles, Greens functions, eigenvalue problems, and Fourier expansion methods.

M 373K Algebraic Structures I

Prerequisite and degree relevance

Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who receive a grade of C- in one of the prerequisite courses are advised to take Mathematics 343K before attempting 373K. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.

Course description

M 373K is a rigorous course in pure mathematics. The syllabus for the course includes topics in the theory of groups and rings. The study of group theory includes normal subgroups, quotient groups, homomorphisms, permutation groups, the Sylow theorems, and the structure theorem for finite abelian groups. The topics in ring theory include ideals, quotient rings, the quotient field of an integral domain, Euclidean rings, and polynomial rings.

This course is generally viewed (along with 365C) as the most difficult of the required courses for a mathematics degree. Students are expected to produce logically sound proofs and solutions to challenging problems.

Text

Herstein, Topics in Algebra

Material to be covered

Chapters 1, 2, 3, and if time permits some topics in Chapters 4 and 5. This includes properties of the .integers, including divisibility and prime factorization; properties of groups, including subgroups, homomorphisms, permutation groups, the Sylow theorems; properties of rings, including subrings and ideals, homomorphisms, domains, especially Euclidean, principal ideal, and unique factorization domains, polynomial rings. If time permits: Fields, elementary properties of vector spaces including the concept of dimension, field extensions.

We will be glad to discuss any questions or listen to any comments which you may have now or during the term on the course, the text, or the syllabus.

M 373L Algebraic Structures II

Prerequisite and degree relevance

Mathematics 373K with a grade of at least C-. M373L is strongly recommended for undergraduates contemplating graduate study in mathematics. 

Course description

M 373L is a continuation of M 373K, covering a selection of topics in algebra chosen from field theory and linear algebra. Emphasis is on understanding theorems and proofs.

Text

Herstein, Topics in Algebra

Material to be covered

Chapters 4, 5, (sections 1-3), 6.

This includes elementary properties of vector spaces and fields, including bases and dimension, elementary properties of linear transformations, relations to matrices, change of bases, dual spaces, characteristic roots, canonical forms, inner product spaces, normal transformations, quadratic and bilinear forms.

If time permits: topics left up to the instructor.

M 374M Mathematical Modeling in Science and Engineering

Prerequisite and degree relevance

Mathematics 427J or 427K, and 340L or 341, with a grade of at least C- in each; and some basic programming skills.  Moreover, students will be expected to have some familiarity with the software package Matlab.

Text

J. David Logan, Applied Mathematics, 4th Edition

Description of the Course

This course is for students interested in mathematical modeling and analysis.  The goals are to develop tools for studying differential equation models that arise in applications and to illustrate how the derivation and analysis of models can be used to gain insight and make predictions about physical systems.  Emphasis should be placed on examples and case studies, and a broad range of applications from the engineering and physical sciences should be considered.

Topics

The following outline is a  list of relevant concepts for each core topic. Instructors should carefully choose and balance the concepts depending on the case studies they have in mind.  Note that a well-designed case study will likely occupy 1-2 class days, and can be continued in an associated homework assignment. The number of class days listed below is for a standard MWF schedule.  A typical semester has 43 MWF days and the schedule below contains material for 41 days, allowing time for two midterm exams. The suggested text provides some coverage of all the core topics; instructors may find it necessary to employ supplementary material to increase the depth of coverage in areas of interest, and to support their case studies. 

1. Dimensional analysis and scaling (6 days) 
   1. Fundamental physical dimensions, units 
   2. Dimensional vs dimensionless quantities 
   3. Unit-free equations and their properties 
   4. Buckingham Pi Theorem 
   5. Characteristic scales for a function 
   6. Transforming equations to dimensionless form 
   7. Scaling to expose dominant/small effects 
2. Dynamical systems in one dimension (4 days) 
   1. Properties of solutions 
   2. Phase line diagrams 
   3. Equilibrium solutions 
   4. Stability of equilibria 
   5. Classification via linearization 
   6. Classification via Lyapunov functions 
   7. Bifurcation of equilibria 
   8. Basic types of bifurcations, hysteresis 
3. Dynamical systems in two dimensions (9 days) 
   1. Properties of solutions 
   2. Phase plane diagrams, nullclines, direction fields 
   3. Equilibrium solutions, stability 
   4. Stability in linear systems, eigenvalues 
   5. Phase diagrams for linear systems 
   6. Stability in nonlinear systems, linearization theorem 
   7. Stability in nonlinear systems, Lyapunov theorem  
   8. Bifurcations in linear and nonlinear systems 
   9. Closed orbits and limit cycles, Hopf bifurcation 
   10. Poincare-Bendixson theorem 
4. Regular perturbation methods (6 days) 
   1. Perturbed equations, regular vs singular 
   2. Characteristics of regular problems 
   3. Approximation via asymptotic series 
   4. Regular method for algebraic equations 
   5. Typical error bounds 
   6. Regular method for differential equations 
   7. Typical error bounds, issue of uniformity 
   8. Poincare-Lindstedt method for oscillatory problems 
5. Singular perturbation methods (5 days) 
   1. Characteristics of singular algebraic problems 
   2. Rescaling method for algebraic problems 
   3. Characteristics of singular differential problems 
   4. Boundary layers, failure of regular method 
   5. Two-scale method for problems with boundary layers 
   6. Idea of inner, outer, and matched expansions 
   7. WKB method for oscillatory, exponential problems 
6. Calculus of variations (11 days) 
   1. Function spaces and functionals 
   2. Absolute extrema of a functional 
   3. Local extrema of a functional, issue of norms 
   4. Concept of admissible variations 
   5. Necessary conditions for local extrema 
   6. Fundamental lemma, Euler-Lagrange equations 
   7. Fixed-endpoint and free-endpoint problems 
   8. Multiple-function and higher-order problems 
   9. Isoperimetric constraints, multiplier rule 
   10. Convexity and sufficient conditions 

Last Updated

Oscar Gonzalez November 2015

M 378K Introduction to Mathematical Statistics

Prerequisite and degree relevance

Mathematics 362K with a grade of at least C-. Same as Statistics and Data Sciences 378. Only one the following may be counted: Mathematics 378K, Statistics and Data Sciences 378, Statistics and Scientific Computation 378. 

Students taking this course are usually majoring in mathematics, actuarial science, or one of the natural sciences. M 362K, 358K, and 378K form the core sequence for students in statistics.

Goals and level of course

Goals are to give students some insight into the theory behind the standard statistical procedures and to prepare continuing students for the graduate courses. Within the limits of the prerequisites, students are expected to derive and apply the theoretical results as well as carry out some standard statistical procedures.

Topics covered

• Moment-Generating Functions
• Gamma, Chi-squared, t- and F-distributions
• Sampling Distributions and the Central Limit Theorem
• Point Estimation (bias, mean square error, relative efficiency, consistency, sufficiency, Method of Moments, Method of Maximum Likelihood, Rao-Blackwell Theorem, and Minimum Variance Unbiased Estimation). Examples should include cases where more than one estimator is possible. Examples involving max and min are probably the easiest to do.
• Confidence intervals (concepts; small and large sample CIs for means and differences of means; large sample CIs for proportions and differences of proportions; selecting sample sizes)
• Hypothesis testing (concepts; small and large sample for means and differences of means; large sample for proportions and differences of proportions; small sample for proportions)
• Errors and power (type I and II errors, power and Neyman Pearson Lemma, calculating sample sizes for desired error level or power)
• Likelihood ratio tests
• If time permits: Some selection from Order statistics, Chi-squared tests, non-parametric tests, least squares regression.

Detailed syllabus based on Wackerly et al (fifth edition)

(Chapters 7 - 10 constitute the heart of the course)

• Review of Probability and Introduction to Statistics:
  • Chapter 1, Sections 2.1, 2.2, 2.12, 3.1, 3.11, 3.12, 4.1, 4.2, 4.3, 4.10, 4.12, 5.1, 5.12, 6.1, 6.7
• Additional Probability Topics: Gamma and Chi-Squared Distributions: Section 4.6
  • Moment Generating Functions: Sections 3.9, 4.9
  • Probability Distributions of Functions of Random Variables: Sections 6.4, 6.5
  • Probability Distributions of Max and Min: First part of Section 6.6
• Sampling Distributions and the Central Limit Theorem: All of Chapter 7
• Estimation All of Chapter 8
• Properties of point Estimators and Methods of Estimation Chapter 9, omitting Section 8
• Hypothesis Testing Chapter 10 (Section 9 optional)
• (Additional topics as time permits)

M 378N Generalized Linear Models

Prerequisite and degree relevance

Mathematics 378K with a grade of at least C- or consent of instructor. Mathematics 375T (Topic: Generalized Linear Models) and 378N may not both be counted.

Course description

Extensions to ordinary least-squares regression, including Poisson regression, the lasso, mixed models, and ridge regression.