# Arts & SciencesMathematics

##### The First Two Years

Most students start (or continue) their study of calculus during their first semester at the College. For most, Math 111 (Calculus I) is the right choice, but for others, Math 112 (Calculus II) or perhaps a sophomore level course is the best choice. For students with AP credit from high school, the choice is explained here. Others, including transfer students, should consult the department about proper placement.

Calculus I, II, and III.

The first college mathematics course taken by most of our students is Math 111 (Calculus I).

Math 111 is a four credit hour course, meeting three hours per week with a faculty member in classes limited to about 35 students. The fourth hour is a required calculus lab, in which students meet with lab assistants and work of additional workbook style material. Both in class and in the laboratory, the course uses graphing calculators as tools for experimentation and visualization. The laboratory assignments complement the lectures and examine themes that run throughout the course. Key ideas in the course include rates of change, continuity, derivatives, approximation, numerical integration via Riemann sums and associated error analysis, techniques of differentiation, and applications of derivatives to graphing and mathematical modelling problems.

Math 112 completes the study of single variable calculus. Like Math 111, it is a four credit hour course with an attached calculus laboratory.  The course focuses primarily on definite integration, using both numerical techniques and integration via anti-derivatives. The course studies applications of the integral to problems in economics and finance, as well as to science and engineering. A special chapter is devoted to a simple class of ordinary differential equations and explains how useful they are for various applied problems. The course closes by returning to the theme of approximation, studying series approximations and their applications.

Math 212 (Multivariable Calculus) is a three credit hour course on the calculus of several variables. It studies surfaces in three-space, vectors, partial differentiation, multiple integration, and line integrals.

Linear Algebra.

Math 211 (Linear Algebra) is a three-credit course taken by students interested in mathematics, chemistry, physics, and economics, among others. Linear algebra begins with the study of systems of linear, or first degree, equations. The course then introduces matrices, i.e., rectangular arrays of numbers that provide a compressed way to write down and manipulate systems of linear equations. Matrices can also represent linear transformations, functions of a special type that act on the set of all n-tuples, the standard Euclidean space. In this context, the concepts of eigenvalues, eigenvectors, and canonical forms are introduced and used in the application of linear algebra to topics such as differential equations, geometry, etc. Depending upon the instructor, the course may focus on computational techniques, theoretical aspects, or applications of linear algebra, and the use of computer software such as Matlab or Maple may be integrated into the course. At William and Mary, linear algebra (also called "matrix analysis") and the closely related subject "operator theory" are the department's strongest research fields, and there is a follow-up course (Math 408, Advanced Linear Algebra) offered each year.

Foundations of Mathematics.

Math 214 (Foundations of Mathematics) was created a few years ago after students requested a course to prepare them for the more abstract flavor of upper division courses, whether pure or applied. Students report finding Math 214 to be quite challenging, in part because the things that it emphasizes are quite different from what students see in earlier mathematics courses. Topics in this "bridge course" can vary from term to term, but whatever topic is the course's organizing theme, the real purpose of the course is to introduce students to the reading and writing of proofs. Somewhere in the course, students will encounter elementary logic (de Morgan's laws, quantifiers, negations, implication, etc.), direct and indirect proofs, proofs by induction, introduction to the various number systems and their inter-relations, equivalence relations, functions of various kinds and their application to cardinality. Students find Math 214 to be very different from mathematics courses that they have taken before and the course should be taken by the end of the second year. Math 214 is a prerequisite for Math 307 and Math 311, two required courses in the mathematics major. It can also give students a glimpse of the spirit of modern mathematics.

Computer Science.

Every mathematics major needs to be proficient in computer programming, at least at the level of CS 141. Any programming experience that you already have will be helpful (but not required) in this course.  CS 141 provides crucial tools for students who plan to study or apply mathematics. Other recommended courses include CS 241 (Data Structures), which is an integral part of the applied mathematics track within the mathematics major. CS 303 (Algorithms) is also highly recommended for mathematics students.