Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

How can a candidate in a political race win the majority of votes yet lose the election? How can two competing candidates interpret the same statistic as being in their favor? How can the geometry of the voting district disenfranchise entire groups of voters? Can we quantify the power the President of the United States has? In this course, we will look at the mathematics behind these and related questions that arise in politics. We will study topics such as fairness, voting paradoxes, social choice, game theory, apportionment, gerrymandering, and data interpretation. The goal of the class will be to illustrate the importance of rigorous reasoning in various social and political processes while providing an introduction to some fascinating mathematics.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented. Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Number theory is the study of the most basic mathematical objects: the natural numbers (1, 2, 3, etc.). It begins by investigating simple patterns: for instance, which numbers can be written as sums of two squares? Do the primes go on forever? How can we be sure? The patterns and structures that emerge from studying the properties of numbers are so elegant, complex, and important that number theory has been called "the Queen of Mathematics." Once studied only for its intrinsic beauty, number theory has practical applications in cryptography and computer science. Topics include the Euclidean algorithm, modular arithmetic, Fermat's and Euler's Theorems, public-key cryptography, quadratic reciprocity. MATH 223 has a focus on learning to understand and write mathematical proofs; it can serve as valuable preparation for MATH 305.

Real analysis is the study of the rigorous theory of the real numbers, Euclidean space, and calculus. The goal is to thoroughly understand the familiar concepts of continuity, limits, and sequences. Topics include compactness, completeness, and connectedness; continuous functions; differentiation and integration; limits and sequences; and interchange of limit operations as time permits.

This course covers some basic notions of point-set topology, such as topological spaces, metric spaces, connectedness and compactness, Heine-Borel Theorem, quotient spaces, topological groups, groups acting on spaces, homotopy equivalences, separation axioms, Euler characteristic, and classification of surfaces. Additional topics include the study of the fundamental group (time permitting).

Linear algebra at this more advanced level is a basic tool in many areas of mathematics and other fields. The course begins by revisiting some linear algebra concepts from MATH 206 in a more sophisticated way, making use of the mathematical maturity picked up in MATH 305. Such topics include vector spaces, linear independence, bases, and dimensions, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products, multilinear forms, and modules. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical forms. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in MATH 206). We will also discuss the spectral theorem, the best example of successful diagonalization, and its applications.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.