### Current Course Descriptions

Honors Calculus : Students who are interested in math or science might also want to consider a more challenging Honors version of Calculus II and III, Math 1610 and Math 2600 (the analogues of math 1410 and math 2400, respectively). These courses will cover essentially the same material as 1410 and 2400, but more in depth and involve discussion of the underlying theory as well as computations.

**1070. Mathematics of Change, Part I.** Staff. Prerequisite(s): None.

Limits, orders of magnitude, differential and integral calculus; Taylor polynomials; estimating and bounding; probability densities. Mathematical modeling and applications to the social, economic and information sciences.

**1080. Mathematics of Change, Part II.** Staff. Prerequisite(s): Math 1070 or permission of instructor.

Multivariate calculus; optimization; multivariate probability densities. Introduction to linear algebra; introduction to differential equations. Mathematical modeling and applications to the social, economic and information sciences.

**L/R 1300. Introduction to Calculus. (C)** Staff.

Introduction to concepts and methods of calculus for students with little or no previous calculus experience. Polynomial and elementary transcendental functions and their applications, limits, derivatives, extremum problems, curve-sketching, approximations; integrals and the fundamental theorem of calculus.

**L/R 1400. Calculus, Part I. (C)** Staff.

Brief review of High School calculus, applications of integrals, transcendental functions, methods of integration, integration applications, sequences, infinite series, Taylor's theorem. Use of symbolic manipulation and graphics software in calculus.

Review of exponents, logs, graphing, limits, differential and integral calculus; integration techniques; Taylor polynomials and series; differential equations; elements of multivariable calculus and optimization. Emphasis on modeling, number sense, verbal skills and applications.

**L/R 1410. Calculus, Part II. (C)** Staff. Prerequisite(s): Math 1400.

Functions of several variables, vector-valued functions, partial derivatives and applications, double and triple integrals, conic sections, polar coordinates, vectors, Vector calculus: functions of several variables, vector fields, line and surface integrals, Green's, Stokes' and divergence theorems. Applications to physical sciences. Use of symbolic manipulation and graphics software in calculus.

**L/R 1610. Honors Calculus. (C)** Staff. Prerequisite(s): Math 1400.

This is an Honors level version of Math 114 which explores the mathematics more deeply.

**L/L 1234. Community Math Teaching Project. (M)** Staff.

This course allows Penn students to teach a series of hands-on activities to students in math classes at a University City high school. The semester starts with an introduction to successful approaches for teaching math in urban high schools. The rest of the semester will be devoted to a series of weekly hands-on activities designed to teach fundamental aspects of geometry. The first class meeting of each week, Penn faculty teach Penn students the relevant mathematical background and techniques for a hands-on activity. During the second session of each week, Penn students will teach the hands-on activity to a small group of UCHS students. The Penn students will also have an opportunity to develop their own activity and to implement it with the UCHS students.

**L/R 1700. Ideas in Mathematics. (C) ** Natural Science & Mathematics Sector. Class of 2010 and beyond. Staff. May also be counted toward the General Requirement in Natural Science & Mathematics.

Topics from among the following: logic, sets, calculus, probability, history and philosophy of mathematics, game theory, geometry, and their relevance to contemporary science and society.

Elementary applications of decision analysis, game theory, probability and statistics to issues in accounting, contracting, finance, law, and medicine, amongst others.

**L/L 2020. Proving Things: Analysis. (C)** Staff. Corequisite(s): Math 1400, 1410 or 2400.

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying real and complex numbers, sequences, series, continuity, differentiability and integrability. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

**L/L 2030. Proving things: Algebra. (C)** Staff. Corequisite(s): Math 1400, 1410 or 2400.

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying arithmetic, algebra, linear algebra, groups, rings and fields. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

**2100. Mathematics in the Media. (C)** Staff. Prerequisite(s): Math 1410 or equivalent.

This course counts as a regular elective for both the Mathematics Major and Minor.

**L/R 2400. Calculus, Part III. (C)** Staff. Prerequisite(s): Math 1410.

Linear algebra: vectors, matrices, systems of linear equations, eigenvalues and eigenvectors. Series solutions of ordinary differential equations, Laplace transforms and systems of ordinary differential equations. Use of symbolic manipulation and graphics software.

**L/R 2410. Calculus, Part IV. (C)** Staff. Prerequisite(s): MATH 2400.

Sturm-Liouville problems, orthogonal functions, Fourier series, and partial differential equations including solutions of the wave, heat and Laplace equations, Fourier transforms. Use of symbolic manipulation and graphics software.

**L/R 2600. Honors Calculus, Part II. (C)** Staff. Prerequisite(s): Calculus II.

This is an honors version of Math 240 which explores the same topics but with greater mathematical rigor.

**2900. Undergraduate Mathematics Research** Staff. Prequisite(s): Math 2400 (can be taken concurrently) or consent of instructor. Math 2410 would be helpful.

This is a project oriented mathematics research class that teaches students to solve real world problems by constructing an analyzing a mathematical model. Typically the the problems that the projects solve can come from mathematics, chemistry, biology, and material science but sometimes we also have problems from economics, finance, and social sciences. The research problems in the course can vary from year to year.

This course is meant to provide students with some of the background skills needed to successfully engage in mathematical and/or scientific research, acquaints students with some of the famous problems which mathematicians work on, and also provide students with the experience of working on a research project of their choice involving mathematics.

**3120. Linear Algebra. (M)** Staff. Prerequisite(s): MATH 2400. Students who have already received credit for either Math 3700, 3710, 5020 or 5030 cannot receive further credit for Math 3120 or Math 3130/5130. Students can receive credit for at most one of Math 3120 and Math 3130/5130.

Linear transformations, Gauss Jordan elimination, eigenvalues and eigenvectors, theory and applications. Mathematics majors are advised that MATH 312 cannot be taken to satisfy the major requirements.

**3130. Computational Linear Algebra.** Staff. Prerequisite(s): Math 1410, and some programming experience. Students who have already received credit for either Math 3700, 3710, 5020 or 5030 cannot receive further credit for Math 3120 or Math 3130/5130. Students can receive credit for at most one of Math 3120 and Math 3130/5130.

Many important problems in a wide range of disciplines within computer science and throughout science are solved using techniques from linear algebra. This course will introduce students to some of the most widely used algorithms and illustrate how they are actually used.

Some specific topics: the solution of systems of linear equations by Gaussian elimination, dimension of a linear space, inner product, cross product, change of basis, affine and rigid motions, eigenvalues and eigenvectors, diagonalization of both symmetric and non-symmetric matrices, quadratic polynomials, and least squares optimization.

Applications will include the use of matrix computations to computer graphics, use of the discrete Fourier transform and related techniques in digital signal processing, the analysis of systems of linear differential equations, and singular value decompositions with application to a principal component analysis.

The ideas and tools provided by this course will be useful to students who intend to tackle higher level courses in digital signal processing, computer vision, robotics, and computer graphics.

**3140. Advanced Linear Algebra. (M) ** Staff. Prerequisite(s): MATH 2400. Math 3140/5140 covers Linear Algebra at the advanced level with a theoretical approach. Students can receive credit for at most one of Math 3120 or Math 3140.

Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and Unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra. MATH 3140 satisfies the Mathematics major requirements.

**3200. Computer Methods in Mathematical Science I. (A)** Staff. Prerequisite(s): MATH 2400 or concurrent and ability to program a computer, or permission of instructor.

Students will use symbolic manipulation software and write programs to solve problems in numerical quadrature, equation-solving, linear algebra and differential equations. Theoretical and computational aspects of the methods will be discussed along with error analysis and a critical comparison of methods.

**3400. (LGIC210) Discrete Mathematics I. (M)** Staff. Prerequisite(s): MATH 1410 or permission of the instructor.

Topics will be drawn from some subjects in combinatorial analysis with applications to many other branches of math and science: graphs and networks, generating functions, permutations, posets, asymptotics.

**3410. (LGIC220) Discrete Mathematics II.** Staff. Prerequisite(s): Math 3400/Logic 2100 or permission of the instructor.

Topics will be drawn from some subjects useful in the analysis of information and computation: logic, set theory, theory of computation, number theory, probability, and basic cryptography.

See also:Math 3410 web page (Spring 2007). **Spring 2009 web page**

**3500. Number Theory. (M)** Staff.

Congruences, Diophantine equations, continued fractions, nonlinear congruences, and quadratic residues.

**L/L 3600. Advanced Calculus. (C)** Staff. Prerequisite(s): MATH 2400.

Syllabus for MATH 360-361: a study of the foundations of the differential and integral calculus, including the real numbers and elementary topology, continuous and differentiable functions, uniform convergence of series of functions, and inverse and implicit function theorems. MATH 508-509 is a masters level version of this course.

**L/L 3610. Advanced Calculus. (C)** Staff. Prerequisite(s): MATH 3600.

Continuation of MATH 360.

**L/L 3700. Algebra. (C)** Staff. Prerequisite(s): MATH 2400 and MATH 3140 or permission of instructor. Students who have already received credit for either Math 3700, 3710, 5020 or 5030 cannot receive further credit for Math 3120 or Math 3130/5130. Students can receive credit for at most one of Math 3120 and Math 3130/5130.

Syllabus for MATH 3700-3710: an introduction to the basic concepts of modern algebra. Linear algebra, eigenvalues and eigenvectors of matrices, groups, rings and fields. MATH 5020-5030 is a masters level version of this course.

**L/L 3710. Algebra. (C)** Staff. Prerequisite(s): MATH 3700. Students who have already received credit for either Math 3700, 3710, 5020 or 5030 cannot receive further credit for Math 3120 or Math 3130/5130. Students can receive credit for at most one of Math 3120 and Math 3130/5130.

Continuation of MATH 3700.

**4100. Complex Analysis. (C)** Staff. Prerequisite(s): MATH 2400 or permission of instructor.

Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.

**4200. Ordinary Differential Equations. (C)** Staff. Prerequisite(s): MATH 2400 or permission of instructor.

After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.

**4250. Partial Differential Equations. (A)** Staff. Prerequisite(s): MATH 2400 or permission of instructor. Knowledge of PHYS 0150-0151 will be helpful.

Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.

**4320. Game Theory. (C)** Staff. Prerequisite(s): Math 2400.

A mathematical approach to game theory, with an emphasis on examples of actual games. Topics will include mathematical models of games, combinatorial games, two person (zero sum and general sum) games, non-cooperating games and equilibria.

**4600 (MATH 5000). Geometry-Topology, Differential Geometry. (M)** Staff. Prerequisite(s): Math 2400/2410, Math 3600 or 5080, or with the permission of the instructor.

Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.

**4650 (MATH 5010). Geometry-Topology, Differential Geometry. (M)** Staff. Prerequisite(s): Math 2400/2410, Math 3610 or 5080, or with the permission of the instructor.

Review of 2- and 3-dimensional vector calculus, differential geometry of curves and surfaces, Gauss-Bonnet theorem, elementary Riemannian geometry, knot theory, degree theory of maps, transversality.

**4800. (MATH 5500) Elementary Topics in Advanced Real Analysis. (M)** Staff. Prerequisites: A year of analysis at the 300 level or above (for example, Mathematics 3600-3610, or 5080-5090); a semester of linear algebra at the 3000 level or above (for example, Mathematics 3700) or Permission of Instructor.

**4990. Supervised Study. (C)** Staff. Prerequisite(s): Permission of major adviser. Hours and credit to be arranged.

Study under the direction of a faculty member. Intended for a limited number of mathematics majors.

*last updated 05/23/2022*