Global Arc

Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems. Prerequisite: MAT103 or equivalent.

Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 216/218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or equivalent.

Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or equivalent.

Many societal issues benefit from precise mathematical analysis. In this course, three such issues will be studied: information theory, a 20th-century field of mathematics used in all modern means of communication; voting schemes, i.e. the different ways of deciding, from all the individual votes, what the opinion is of the whole population; and population growth models, so important in many discussions of our impact on the environment. For each, appropriate mathematical concepts will be introduced and then used to derive results in a mathematically rigorous way, followed by a discussion of their meaning and relevance.

An introduction to classical number theory to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions, and quadratic reciprocity. There will be a topic from more advanced or more applied number theory such as p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the mathematics department and for non-majors interested in exposure to higher mathematics.

An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include the rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel theorem, the Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's theorem.

Rigorous theoretical introduction to the foundations of analysis in one and several variables: basic set theory, vector spaces, metric and topological spaces, continuous and differential mapping between n-dimensional real vector spaces. Normally followed by MAT218.

A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms.

Continuation of Multivariable Analysis and Linear Algebra I (MAT 216) from the fall. A rigorous course in analysis with an emphasis on proof rather than applications. Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini. See the department website for details: http://www.math.princeton.edu.

Continuation of Single Variable Analysis (MAT215) and Honors Linear Algebra (MAT217) needed to prepare for further work in differential geometry, analysis and topology. Calculus on manifolds: Introduces the concept of differentiable manifold, develops the notions of vector fields and differential forms, Stokes' theorem and the de Rham complex. The basic existence theorem in ODEs is used to prove the Frobenius theorem on integrability of plane fields. The intent is to provide the preparation for the courses in differential geometry and topology.