

Course Offerings  Mathematics
Division of Science and Mathematics
Alex McAllister (chair)Marian Anton, Jeffrey Heath, Joel Kilty, Sarah Murray, Christine Shannon, Lesley Wiglesworth, John Wilson; students: Brian Bowles, Lauren Gates, Paige Hebard
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 problemsolving 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 problemsolving and communication skills learned in the mathematics major serve them well in whatever career paths they follow.
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 fouryear period);
MAT 230, 240, 290, 330, 380
;
One of CSC 261, MAT 190, MAT 270, or PHY 330;
Four MAT courses numbered 300 or higher (excluding MAT 420 and 499). 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 and 499). 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 decisionmaking) 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 decisionmaking 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 twocourse sequence that provides both
an indepth 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 indepth 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 or 141.
MAT 170 CalculusI
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 CalculusII
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 CalculusIII
An extension of the concepts of function, limit, derivative, and integral
to threedimensional 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, GramSchmidt orthogonalization, determinants,
linear transformations and their matrix representations, eigenvalues and
eigenvectors. Prerequisite: MAT 171.
MAT 250269 Centre Term Special
Topic Courses: 20082009 Topic:
MAT 256 Mathematics, Aesthetics and the Arts
Mathematics is embedded more deeply in the arts than many realize, both naturally and intentionally. Moving beyond the familiar themes of M.C. Escher and the Golden Ratio, we will explore connections between mathematics and the arts across a variety of genres, eras, cultures, and dimensions. Topics include tessellations, fractals, origami, constructions, perception, perspective, alternative geometries, and music. We will also look at a number of contemporary artists whose work is heavily influenced by mathematics. The course includes a trip to Washington, D,C. to attend the Math and Art sessions of the Joint Mathematics Meetings and to visit our national museums. Prerequisite: MAT 171.
MAT 270 Mathematical Methods of Economics
An introduction to mathematical tools used for modeling in economics. The tools studied include multivariate calculus with special emphasis on constrained optimization, as well as matrix algebra, diffential and difference equations. Applications focus on the use of marginal analysis, comparative statics, elasticities, isoquants, and CobbDouglas functions. Among the models discussed are the ISLM macroeconomic model, the Solow growth model, and Leontief inputoutput systems. The course includes an introduction to the computer algebra system Maple. Prerequisite: MAT 141 or 170, and ECO 220.
MAT 290 Foundations of Mathematics
This course develops the abstract thinking and the writing skills necessary for prooforiented 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 momentgenerating
functions, distributions of functions of random variables, and multivariate
distributions. Prerequisite: MAT 230 or permission of instructor.
MAT 311 Mathematical Statistics
A calculusbased course in statistics devoted to techniques for analyzing
and interpreting data with the goal of understanding our world and facilitating
informed decisionmaking 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 leastsquares estimators,
correlation, design of experiments, analysis of variance, chisquare tests,
and nonparametric statistics. Prerequisite: MAT 310 or permission of the
instructor.
MAT 330 Abstract AlgebraI
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 AlgebraII
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 realvalued functions. Topics include complex numbers,
limits, continuity, differentiability, CauchyRiemann 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 firstorder 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 370 Numerical Methods
A study of how computers obtain numerical estimates of solutions when
people apply mathematics to diverse disciplines (e.g. physics, economics,
medicine, etc.). In this course we discuss and develop various algorithms
that form the basis for computer applications including root finding,
interpolation, differentiation and integration, differential equations,
and systems of equations. Prerequisite: MAT 240 and CSC 117.
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 409 Number Theory
While number theory, which is the study of integers, can be accomplished with techniques in algebra, analysis and combinatorics, Elementary Number Theory does so without the use of other mathematical disciplines such as calculus or abstract algebra. This course covers such topics as the binomial theorem, the division algorithm, prime and composite numbers, Fibonacci numbers, the Euclidean algorithm, the Fundamental theorem of Arithmetic, congruences, divisibility tests and several encryption algorithms. Prerequisite: MAT 290.
MAT 420 Putnam Seminar (one credit hour)
The Putnam Exam is a notoriously challenging annual mathematical competition. This is a course on problemsolving focusing on Putnamstyle 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.
MAT 499 Research Seminar in Mathematics (one credit hour)
Students meet weekly to discuss and present undergraduate research topics in math ematics. Students are individually mentored by mathematics faculty in the study and in the oral and written presentation of a research topic at increasing levels of sophistication. Prerequisite: MAT 290.
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