|
課程概述 |
1. Uniform convergence of sequence of functions: pointwise and uniform convergences, integration and differentiation of series, Arzela-Ascoli theorem, fixed point theorem, Stone-Weierstrass theorem, Dirichlet and Able test
2. Differentiable mappings, matrix representation, conditions for differentiability, Taylor's theorem and higher derivatives, maxima and minima
3. Contraction principle, inverse function theorem, implicit function theorem, rank theorem. Lagrange multipliers, existence theorem for ODE
4. Integration: integrable functions, volume, measure zero, Lebesgue's theorem, improper integral, Fubini's theorem, change of variables, polar, spherical, cylindrical coordinates, interchanges of limiting operations
5. Fourier analysis: Fourier series, inner product spaces, orthogonal families, completeness and convergence theorems, functions of bounded variation, Fejer's theorem |
|
課程目標 |
Advanced calculus is a critical course for students who are seriously interested in mathematics or students who need knowledge of more deep analysis to deal with problems in various fields. In this course we present a theoretical basis of analysis suitable for students who have completed a course in elementary calculus. In last semester, several fundamental topics such as real number system, topology of the Euclidean spaces and metric spaces, continuity, uniform convergence, and differentiation and integration of one variable functions have already been covered. In this semester, more topics including the Fourier series, the inverse function theorem and the implicit function theorem, differentiation and integration of functions of several variables, and differential form will be presented. |