# Course Offerings - Catalog 2013-14

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## Mathematics

### Division of Science and Mathematics

The Mathematics Program seeks to give students an understanding and appreciation of the beauty and utility of mathematics. Quantitative and analytic skills are increasingly important in economics, biology, and the social sciences, as well as in the physical sciences and engineering. The study of mathematics at Centre provides the opportunity for the development of clear, logical, and creative thinking that may be applied to a wide variety of problems and interests. In addition to these important problem-solving skills, the mathematics major will learn to present concise, logical arguments in writing and orally. Emphasis is placed on mathematical thinking and precise communication of these thoughts.

Students completing the mathematics major at Centre have a broad range of interests. Many also complete a major in another field. For example, recent mathematics majors have second majors in chemistry, computer science, economics, English, history, physics, and Spanish. Some majors earn secondary teacher certification. Our graduates often decide to continue their academic studies, entering law school, medical school, engineering programs, and M.B.A. programs, as well as graduate school programs in a wide variety of disciplines. Others choose to join the workforce immediately, taking jobs with companies and agencies such as the Census Bureau, the military services, computer companies, public or private schools, and financial institutions. Our students find that the problem-solving and communication skills learned in the mathematics major serve them well in whatever career paths they follow.

### Faculty

Jeffrey Heath (chair), Rasitha Jayasekare, Joel Kilty, Alex McAllister, Sarah Murray, Christine Shannon, Forrest Stonedahl, Ellen Swanson, Lesley Wiglesworth, John Wilson

Students

Jack Jackson, Matt McCurdy## Requirements for the Major

MAT 170 or both MAT 140 and 141, or equivalent;MAT 171, or equivalent (this requirement should be completed by the end of the freshman year to complete the major in the normal four-year period);

MAT 230, 240, 290, 330, 380;

One of CSC 261, MAT 190, or PHY 330;

Four MAT courses numbered 300 or higher (excluding MAT 420). A CentreTerm math course or CSC 117 may substitute for one of these courses.

## Requirements for the Minor

MAT 170 or both MAT 140 and 141, or equivalent;MAT 171, or equivalent;

MAT 230;

MAT 240 or 290;

Any three MAT courses numbered 300 or higher (excluding MAT 420). CSC 117 may substitute for one of these.

## Mathematics Courses

**MAT 110 Mathematics in Our Society**

An introduction to applied mathematics devoted to solving contemporary problems from diverse disciplines. This course helps students develop logical thinking skills and improve quantitative skills, particularly with linear equations (in the context of decision-making) and with exponential and logarithmic models (in the context of finance). Further topics will be chosen from graph theory, geometry, symmetry, coding, game theory, social issues, and logic. Not open to students who have established basic skills in math.

**MAT 130 Introduction to Statistics**

An investigation into the mathematical techniques for analyzing and interpreting data with the goal of understanding our world and facilitating informed decision-making processes. The course includes the study of random variables, descriptive statistics, basic probability theory, and inferential statistics. Specific topics include frequency distributions, mean, median, variance, probability distributions, hypothesis testing, confidence intervals, correlation, and regression analysis. Prerequisite: Basic skills in mathematics or permission of the instructor.

**MAT 140 Differential Calculus with Review**

This is the first course in a two-course sequence that provides both an in-depth review of functions and an introduction to differential calculus. In particular, limits and derivatives are introduced as tools used to analyze the behavior of algebraic, exponential, and logarithmic functions. This course is not available to students with credit for MAT 160 or 170. Prerequisite: MAT 110 or placement.

**MAT 141 Integral Calculus with Review**

A continuation of MAT 140, this course begins with an in-depth review of trigonometric functions and their derivatives. The majority of the course focuses on the definition of the integral, the Fundamental Theorem of Calculus, and applications of the integral. Prerequisite: MAT 140.

**MAT 170 Calculus-I**

An introduction to single variable calculus reviewing the real number line, inequalities and absolute value, and discussing functions and graphing, limits, continuity, the derivative, rules of differentiation, the Mean Value Theorem, applications of the derivative, antiderivatives, Riemann sums and the definite integral, the Fundamental Theorem of Calculus, and applications of the integral. Prerequisite: placement. Note: This course is not available to students with credit for MAT 140.

**MAT 171 Calculus-II**

A continuation of MAT 170. The techniques of single variable calculus are applied to inverse trigonometric, exponential, and logarithmic functions. Also included are further techniques of integration, indeterminate forms, improper integrals, and infinite series. Prerequisite: MAT 141 or 170 or placement.

**MAT 190 Discrete Mathematics**

An introduction to the study of “discrete” mathematical objects and number systems, in contrast to the study of the continuous real number line. The course explores many topics at the analytical level of calculus: relations, logic, techniques of proof, counting techniques, algorithms, graph theory, number systems, Boolean algebra, and set theory. Prerequisite: MAT 141 or 170 or permission of the instructor.

**MAT 210 Mathematics in the Elementary School**

A study of the language and structure of mathematics including problem solving, set and number theory, integers, rational and real numbers, probability and statistics, and geometry. NOTE: This course is open to early elementary education majors only.

**MAT 230 Calculus-III**

An extension of the concepts of function, limit, derivative, and integral to three-dimensional space and vector spaces; the course describes many applications and their historical significance, such as planetary motion and magnetic fields. Topics include vector algebra, elementary differential geometry of curves and surfaces, limits, continuity, partial derivatives, directional derivatives, multiple integrals, line integrals, surface integrals, Green's Theorem, Stokes' Theorem, and the Divergence Theorem. Prerequisite: MAT 171 or placement.

**MAT 240 Linear Algebra**

The abstract study of systems of linear equations: the determination of whether a system has no, one, or infinitely many solutions and the techniques for obtaining such solutions. The topics include the algebra of matrices, Gaussian elimination, vector spaces, spanning, linear independence, basis, dimension, inner products, Gram-Schmidt orthogonalization, determinants, linear transformations and their matrix representations, eigenvalues and eigenvectors. Prerequisite: MAT 171.

**MAT 290 Foundations of Mathematics**

This course develops the abstract thinking and the writing skills necessary for proof-oriented mathematics courses and surveys various areas of mathematics. Fundamental concepts and questions are studied from mathematical logic, abstract algebra, number theory, and real analysis. Further topics include complex analysis, statistics, graph theory, and/or other areas of mathematics according to the interests of instructor and students. Prerequisite: MAT 171.

**MAT 310 Probability Theory**

A mathematical study of chance, this course uses counting techniques and topics from calculus to develop a mathematical approach that describes the likelihood of events happening. Specific topics include an introduction to the theory of probability, random variables, discrete and continuous probability distributions, expected values, moments and moment-generating functions, distributions of functions of random variables, and multivariate distributions. Prerequisite: MAT 230 or permission of instructor.

**MAT 311 Mathematical Statistics**

A calculus-based course in statistics devoted to techniques for analyzing and interpreting data with the goal of understanding our world and facilitating informed decision-making processes. This course is a continuation of MAT 310 that studies applications of sampling distributions related to the normal distribution. These include estimation of parameters, confidence intervals, hypothesis testing, regression analysis and least-squares estimators, correlation, design of experiments, analysis of variance, chi-square tests, and nonparametric statistics. Prerequisite: MAT 310 or permission of the instructor.

**MAT 330 Abstract Algebra-I**

This course defines and investigates the key properties of the mathematical structure called an algebraic group, studying many examples, including groups of numbers, groups of functions, and groups of matrices, with the goal of determining the common properties of all of these mathematical systems. Topics include the basic properties of the integers, permutation groups, subgroups, Lagrange’s Theorem, quotient groups, isomorphisms and homomorphisms, and Cayley's Theorem. Prerequisite: MAT 290.

**MAT 331 Abstract Algebra-II**

A continuation of MAT 330, in which key properties from the integers and the real numbers are used as models for the algebraic structures known as rings and fields. Students construct and examine a rich collection of examples including rings of polynomials, Gaussian integers, and finite fields. Topics include prime factorization, integral domains, ideals, ring homomorphisms, and extension fields. Prerequisite: MAT 330.

**MAT 340 Complex Variables**

A study of functions of one complex variable, the course extends notions from the calculus of real-valued functions. Topics include complex numbers, limits, continuity, differentiability, Cauchy-Riemann equations, analytic functions, elementary transformations, complex integration, Cauchy's Theorem, the annulus theorem, Cauchy's Integral Formula, Morera's Theorem, complex power series, Laurent series, and the theory of residues. Prerequisite: MAT 230.

**MAT 360 Differential Equations**

This course describes the many physical and social phenomena that involve a change in some quantity with respect to time and are described mathematically via differential equations. Topics include techniques for solving first-order differential equations (exact, separable, linear, integrating factors, homogeneous), solving higher order linear differential equations (constant coefficients, undetermined coefficients, variation of parameters), and the Laplace transform methods and series solutions of differential equations. Prerequisite: MAT 230 or permission of the instructor.

**MAT 380 Introduction to Real Analysis**

A systematic exploration of how calculus provides profound insights into explaining and understanding our world and its phenomena. The study of real analysis discusses the theoretical foundations of single variable calculus to arrive at a deep understanding of why calculus works. Topics include properties of the real numbers, limits, continuity, differentiation, and integration. Prerequisite: MAT 230 and MAT 290.

**MAT 407 Mathematical Logic**

The course is dedicated to studying the reasoning processes and the relational systems common to all fields of mathematics and the relation between truth and proof in mathematics. The course begins by developing sentential logic and first-order logic, considers the soundness and completeness of these systems, and studies the properties of first-order theories and models, including the consequences of compactness. Finally, the Gödel Incompleteness Theorems are proved. Philosophical and practical implications of these results are discussed throughout the course. Prerequisite: MAT 290 or PHI 310.

**MAT 415 Numerical Differential Equations**

A study of how computers can be used to find approximate solutions to differential equations. We will develop algorithms for solving both initial-value and boundary-value problems for first- and second-order differential equations, as well as second-order partial differential equations as time permits. For each method we will address issues related to implementation and perform appropriate error and stability analysis of the algorithm. Prerequisite: MAT 230. MAT 240 and CSC 117 would be helpful but are not required.

**MAT 420 Putnam Seminar (one credit hour)**

The Putnam Exam is a notoriously challenging annual mathematical competition. This is a course on problem-solving focusing on Putnam-style problems. Students learn strategies for tackling such problems and become familiar with the style by working out problems from past Putnam exams, discussions, and presentations to the group. The semester culminates in taking the Putnam exam. Prerequisite: MAT 230.