Mathematics

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.

The course introduces the basic theories and methods of stochastic analysis and its application in Finance. It discusses how to apply the basic theories and methods in stochastic analysis to financial pricing and derives the famous Black-Scholes formula. Other topics include Brownian motion, stochastic integral, and Ito formula. This course is taught in both English and Chinese.

The basic content of the course includes complex planes and extended complex planes, Cauchy-Riemann equations and holomorphic functions, fractional linear transformations and basic elementary functions, single-valued branches of complex integrals and multivalued functions, Cauchy integral theory, power series and Laurent levels Number, isolated singularity and residue theorem, biholomorphic mapping and Riemann mapping theorem, preliminary analysis and extension, etc.

The course describes how non-linear systems can be treated through analysis, simulation, and controller design. Lectures cover non-linear phenomena; mathematical modeling of nonlinear systems; stationary points; linearization around stationary points and trajectories; phase plane analysis; stability analysis using the Lyapunov method; circle criterion; small-gain and passivity; computer tools for simulation and analysis; effects of saturation; backlash and dead-zones in control loops; describing functions for analysis of limit cycles; high-gain methods and relay feedback; optimal control; and nonlinear synthesis and design. Laboratory exercises include analysis using the describing function and control design with dead-zone compensation for an air throttle used in car motors; energy-based design of a swing-up algorithm for an inverted pendulum; and trajectory generation using optimal control for the pendulum-on-a-cart process.