Department of Mathematics and Computer Science Hobart and William Smith Colleges Mathematics Course Descriptions

This page contains a description of each Mathematics course taught in the department. Descriptions are taken from the Colleges' catalog. See also the course descriptions for Computer Science courses.

100 Elementary Functions Designed to help students prepare for the calculus sequence, this course involves the study of basic functions: polynomial, rational, exponential, logarithmic, and trigonometric. Topics covered include: a review of the real number system, equations and inequalities, graphing techniques, and applications of functions. With laboratory. Permission of instructor is required. This course does not normally count toward the major or minor in mathematics. (Fall, Spring) Typical reading: Flanders and Price, Elementary Functions and Analytic Geometry 110 Discovering in Mathematics A study of selected topics dealing with the nature of mathematics, this course has an emphasis on its origins and a focus on mathematics as a creative endeavor. This course does not normally count toward the major or minor in mathematics. (Fall, Spring) Typical reading: Miller, Heeren, Hornsby, Mathematical Ideas 130 Calculus I This course offers a standard introduction to the concepts and techniques of the calculus of functions of one variable and is especially designed for students contemplating further study in mathematics. A problem-solving lab is included as an integral part of the course. Required for the major in mathematics. May be exempted with departmental approval. (Fall, Spring) Typical reading: Stewart, Calculus 131 Calculus II This course is a continuation of the topics covered in MATH 130. A problem-solving lab is an integral part of the course. Required for the major in mathematics. (Fall, Spring) Typical reading: Stewart, Calculus 135 First Steps into Advanced Mathematics This course emphasizes the process of mathematical reasoning, discovery, and argument. It aims to acquaint students with the nature of mathematics as a creative endeavor, as well as with the nature of mathematical methods and the phenomenon of mathematical proof, and focuses on the development of problem-solving skills. Specific topics covered vary from year to year. MATH 135 is strongly recommended as an entry course to a major or minor in mathematics; it is also appropriate for a student who has already taken calculus. (Fall) 204 Linear Algebra This course is an introduction to the concepts and methods of linear algebra. Among the most important topics are general vector spaces and their subspaces, linear independence, spanning and basis sets, solution space for systems of linear equations, linear transformations and their matrix representations, and inner products. It is hoped that the student will develop an appreciation for the process of mathematical abstraction and the creation of a mathematical theory. Prerequisite: MATH 131. Required for the major in mathematics. (Offered annually) Typical reading: Anton, Elementary Linear Algebra 214 Applied Linear Algebra A continuation of linear algebra with an emphasis on applications. Among the important topics are eigenvalues and eigenvectors, diagonalization, and linear programming theory. The course explores how the concepts of linear algebra are applied in various areas, such as, graph theory, game theory, differential equations, Markov chains, and least squares approximation. Prerequisite: MATH 204. (Offered alternate years) Typical readings: Anton, Elementary Linear Algebra; Rorres and Anton, Applications of Linear Algebra 232 Multivariable Calculus A study of the concepts and techniques of the calculus of functions of several variables, this course is required for the major in mathematics. Prerequisites: MATH 131 and MATH 204. (Offered annually) Typical reading: Stewart, Calculus 237 Differential Equations This course offers an introduction to the theory, solution techniques, and applications of ordinary differential equations. Models illustrating applications in the physical and social sciences are investigated. The mathematical theory of linear differential equations is explored in depth. Prerequisites: MATH 232 and MATH 204. (Offered annually) Typical reading: Zill, A First Course in Differential Equations with Applications 278 Number Theory Over the centuries mathematicians, both amateur and professional, have been fascinated by questions and problems about integers. Attempts to solve these problems have led to significant developments in various branches of mathematics. This course couples reason and imagination to consider number theoretic problems, some solved and some unsolved. Topics to be covered include: divisibility, primes, congruencies, number theoretic functions, primitive roots, quadratic residues, and quadratic reciprocity, with additional topics selected from perfect numbers, FermatŐs Theorem, sums of squares, and Fibonacci numbers. Prerequisites: MATH 131 and MATH 204 or permission of instructor. (Offered alternate years) Typical reading: Burton, Elementary Number Theory 320 Seminar for Mathematics Teachers Designed for students intending to teach mathematics at the secondary level, this seminar focuses on the phenomenon of mathematical problem solving and explores ways teachers can shape the classroom experience so their students develop good problem-solving skills. After analyzing their own problem-solving processes, students in the seminar study several formal theories of mathematical problem solving and conduct an independent research project in which they analyze problem-solving behavior in other students. Prerequisites: MATH 331 and enrollment in the secondary mathematics education program, permission of instructor. (Offered alternate years) Typical readings: Polya, How to Solve It; Schoenfeld, Mathematical Problem Solving 331 Foundations of Analysis I This course offers a careful treatment of the definitions and major theorems regarding limits, continuity, differentiability, integrability, sequences, and series for functions of a single variable. A problem-solving lab is an integral part of the course. Required for the major in mathematics. Prerequisites: MATH 204. (Offered alternate years) Typical reading: Belding and Mitchell, Foundations of Analysis 332 Foundations of Analysis II This course begins with a generalization of the notions of limit, continuity, and differentiability (developed in MATH 331), and extends them to the two-dimensional setting. Next, the Fundamental Theorem of Calculus is extended to line integrals and then to GreenŐs Theorem. The course culminates with a brief introduction to analysis in the complex plane. Prerequisites: MATH 232 and MATH 331. (Offered occasionally) Typical reading: Belding and Mitchell, Foundations of Analysis 350 Probability This is an introductory course in probability with an emphasis on the development of the studentŐs ability to solve problems and build models. Topics include: discrete and continuous probability, random variables, density functions, distributions, the Law of Large Numbers, and the Central Limit Theorem. Prerequisite: MATH 232 or permission of instructor. (Offered annually) Typical reading: Ross, A First Course in Probability 351 Mathematical Statistics This is a course in the basic mathematical theory of statistics. It includes the theory of estimation, hypothesis testing, and linear models, and, if time permits, a brief introduction to one or more further topics in statistics (e.g., nonparametric statistics, decision theory, experimental design). In conjunction with an investigation of the mathematical theory, attention is paid to the intuitive understanding of the use and limitations of statistical procedures in applied problems. Students are encouraged to investigate a topic of their own choosing in statistics. Prerequisite: MATH 350. (Offered occasionally) Typical reading: Bain and Engelhardt, Introduction to Probability and Mathematical Statistics 353 Mathematical Models Drawing on linear algebra and differential equations, this course investigates a variety of mathematical models from the biological and social sciences. In the course of studying these models, such mathematical topics as difference equations, eigenvalues, dynamic systems, and stability are developed. This course emphasizes the involvement of students through the construction and investigation of models on their own. Prerequisites: MATH 237 or permission of the instructor. (Offered alternate years) Typical reading: Haberman, Mathematical Models 360 Foundations of Geometry An introduction to the axiomatic method as illustrated by neutral, Euclidean, and non-Euclidean geometries. Careful attention is given to proofs and definitions. The historical aspects of the rise of non-Euclidean geometry are explored. This course is highly recommended for students interested in secondary-school teaching. Prerequisite: MATH 331 or MATH 375. (Offered alternate years) Typical reading: Greenberg, Euclidean and Non-Euclidean Geometries: History and Development 371 Topics in Mathematics Each semester this course is offered, it covers a topic in mathematics that is not usually offered as a regular course. This course may be repeated for grade or credit. However, it cannot be counted more than once toward the fulfillment of requirements for a major in computer science or mathematics, except by permission of the department chair. Prerequisite: MATH 204. (Repeatable) (Offered occasionally) 375 Abstract Algebra I This course studies abstract algebraic systems such as groups, examples of which are abundant throughout mathematics. It attempts to understand the process of mathematical abstraction, the formulation of algebraic axiom systems, and the development of an abstract theory from these axiom systems. An important objective of the course is mastery of the reasoning characteristic of abstract mathematics. Required for the major in mathematics. Prerequisites: MATH 331 or (MATH 204 and CPSC 221) or permission of the instructor. (Offered alternate years) Typical reading: Fraleigh, A First Course in Abstract Algebra 376 Abstract Algebra II This course is a continuation of the study of algebraic systems begun in MATH 375. Among the topics covered are rings, fields, principal ideal domains, unique factorization domains, Euclidean domains, field extensions, and finite fields. The latter portion of the course emphasizes applications of group, ring, and field theory drawn from such areas as error-correcting codes, exact computing, crystallography, integer programming, cryptography, and combinatorics. Prerequisite: MATH 375. (Offered occasionally) Typical readings: Mackiew, Applications of Abstract Algebra; Hadlock, Field Theory and Its Classical Problems 380 Mathematical Logic First order logic is developed as a basis for understanding the nature of mathematical proofs and constructions and to gain skills in dealing with formal languages. Topics covered include propositional and sentential logic, logical proofs, and models of theories. Examples are drawn mainly from mathematics, but the ability to deal with abstract concepts and their formalizations is beneficial. Prerequisite: MATH 204, PHIL 240, or permission of instructor. (Offered alternate years) Typical reading: Enderton, A Mathematical Introduction to Logic 436 Topology This course covers the fundamentals of point set topology, starting from axioms that define a topological space. Topics typically include: topological equivalence, continuity, connectedness, compactness, metric spaces, product spaces, and separation axioms. Some topics from algebraic topology, such as the fundamental group, might also be introduced. Prerequisite MATH 331 or permission of the instructor. (Offered occasionally) 446 Real Analysis This course presents a careful study of various concepts of analysis. Such topics as convergence and continuity are briefly examined, first on the real line and then in more general metric spaces. Other topological properties of metric spaces are studied. An examination of different types of integrals concludes the course. Prerequisite: MATH 331 or permission of instructor. (Offered occasionally) Typical reading: Goldberg, Methods of Real Analysis 448 Introduction to Complex Analysis An introduction to the theory of functions of a complex variable. Topics include the geometry of the complex plane, analytic functions, series expansions, complex integration, and residue theory. When time allows, harmonic fuctions and boundary-value problems are discussed. Prerequisite: MATH 331 or permission of instructor. (Offered alternate years) Typical reading: Churchill and Brown, Introduction to Complex Variables 450 Independent Study 495 Honors