Mathematics

This course teaches basic computational methods for solving simple and common mathematical problems through computers and numerical software. This includes the construction, application, and analysis of basic computational algorithms. Mathematical models are often written as systems of linear and nonlinear equations and differential equations. Students discretize such equations by constructing computable approximations, and are expected to implement and apply such algorithms independently.

The course presents basic optimization theory, and gives an overview of the most important methods and their practical use.

This course studies geometric properties of curves and surfaces in the 3-dimensional Euclidean space, applying tools of multivariable and vector calculus. Topics include Euclidean Space, Tangent Vectors, Directional Derivatives, Curves in R^3, Differential forms, Mappings, Dot Products, Curves, The Frenet Formulas, Arbitrary-Speed Curves, Isometries of R^3, The Tangent Map of an Isometry, Euclidean Geometry, Congruence of Curves, Surfaces in R^3, Patch Computations, Differential Functions and Tangent vectors, Differential Forms on a Surface, Mappings of Surfaces, Integrations of forms, and Topological Properties of Surfaces.

This course covers the theorems usually used for numerical sequences and real functions, and their proofs. It discusses the main idea behind the construction of the integral in Riemann's sense, as well as how to write proofs, use the various notions, and independently study a numerical sequence or a given function. Topics include numerical sequences: theorems of monotonic convergence, adjacent and Cauchy sequences, notions of adherence values, upper/lower bounds and the Bolzano-Weierstrass theorem; local behavior of a function: theorems of extension by continuity and sequential characterization of continuity, applying this characterization to the limit of recurring sequences (a result accepted in advanced math), calculating derivatives, the Taylor-Young theorem, and the limited developments of reference functions, calculating limited developments to find limits and relative positions of curves; global behavior of a function: restoring and using the theorems of intermediate values, Heine, bijection, local extrema, Rolle and finite increments, Taylor with integral remainder and Taylor-Lagrange; Riemann integral: retaining the guiding idea behind the construction of the integral in the Riemann sense, demonstrating general results on the integral of functions, calculating integrals using primitives, integration by parts or change of variables, using the notion of comparison between Riemann integral and sum.