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 Precalculus: Elementary Functions
Intended for students who plan to continue in the calculus sequence, this course involves the study of basic functions: polynomial, rational, exponential, logarithmic, and trigonometric. Topics include a review of the real number system, equations and inequalities, graphing techniques, and applications of functions. Includes problem-solving laboratory sessions. Permission of instructor is required. This course does not count toward the major or minor in mathematics. (Offered annually)
Typical reading: Larson and Hostetler, Precalculus
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. (Offered each semester)Typical reading: Smith, The Nature of Mathematics; Averbach and Chein, Problem Solving Through Recreational Mathematics
130 Calculus I
This course offers a standard introduction to the concepts and techniques of the differential calculus of functions of one variable. A problem-solving lab is included as an integral part of the course. This course does not count towards the major in mathematics. (Offered each semester)Typical reading: Larson, Hostetler, and Edwards, Calculus
131 Calculus II
This course is a continuation of the topics covered in MATH 130 with an emphasis on integral calculus, sequences, and series. A problem-solving lab is an integral part of the course. Prerequisite: MATH 130 or permission of the instructor. (Offered each semester)Typical reading: Larson, Hostetler, and Edwards, 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, demonstrates the methods and structure of mathematical proof, and focuses on the development of problem-solving skills. Specific topics covered vary from year to year. MATH 135 is required for the major and minor in mathematics. Prerequisite: MATH 131 or permission of the instructor. (Offered each semester)Typical reading: Schumacher, Chapter Zero
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 designed to develop an appreciation for the process of mathematical abstraction and the creation of a mathematical theory. Prerequisite: MATH 131, and MATH 135 strongly suggested, or permission of the instructor. 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 every third year)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. Prerequisite: MATH 131. (Offered annually)Typical reading: Stewart, Multivariable 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 or permission of the instructor. (Offered annually)Typical reading: Edwards and Penney, Differential Equations and Boundary Value Problems
278 Number Theory
This course couples reason and imagination to consider a number of theoretic problems, some solved and some unsolved. Topics include divisibility, primes, congruences, number theoretic functions, primitive roots, quadratic residues, and quadratic reciprocity, with additional topics selected from perfect numbers, Fermat’s Theorem, sums of squares, and Fibonacci numbers. Prerequisites: MATH 131 and MATH 204 or permission of the instructor. (Offered every third year)Typical reading: Burton, Elementary Number Theory
313 Graph Theory
A graph is an ordered pair (V, E) where V is a set of elements called vertices and E is a set of unordered pairs of elements of V called edges. This simple definition can be used to model many ideas and applications. While many of the earliest records of graph theory relate to the studies of strategies of games such as chess, mathematicians realized that graph theory is powerful well beyond the realm of recreational activity. In class, we will begin by exploring the basic structures of graphs including connectivity, subgraphs, isomorphisms and trees. Then we will investigate some of the major results in areas of graph theory such as traversability, coloring and planarity. Course projects may also research other areas such as independence and domination. (Offered occasionally)Typical reading: Chartrand and Zhang, Introduction to Graph Theory
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. Prerequisites: MATH 135 and MATH 204. (Offered annually)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 Green's 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 student’s 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 alternate years)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 alternate years)Typical reading: Larsen and Marx, Mathematical Statistics and Its Applications
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 204 and MATH 237 or permission of the instructor. (Offered every third year)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 every third year)Typical reading: Greenberg, Euclidean and Non-Euclidean Geometries: History and Development
371 Topics in Mathematics
Each time 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. Recent topics include combinatorics, graph theory, and wavelets. Prerequisite: MATH 135 and MATH 204 or permission of instructor. (Offered alternate years)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. Prerequisites: MATH 135 and MATH 204 or permission of the instructor. (Offered annually)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)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 every third year)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)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 every third year)Typical reading: Churchill and Brown, Introduction to Complex Variables
450 Independent Study
495 Honors