# Analysis II

### Syllabus

Summary:
This course is the second semester of a rigorous, proof-based course in Analysis. Topics include sequences and series of functions, curves in $$R^n$$, some basic topology, differentiation in $$R^n$$, the implicit and inverse function theorems, and an introduction to the Riemann integral in $$R^n$$.

Contact Information:
 Instructor: Marcel Oliver Email: m.oliver@jacobs-university.de Phone: 200-3212 Office hours: Mo 11:30, We 10:00 in Research I, 107

Time and Place:
 Lectures: Mo 9:45 and Tu 14:15 in East Hall 4 Tutorial: Th 19:30 in East Hall 4

Recommended Textbook:
W. Rudin, Principles of Mathematical Analysis, third edition

T. Tao, Analysis II, third edition

The final grade will be computed as a grade point average with the following weights:

 Homework: 20% Midterm Exam: 30% Final Exam: 50%

### Class Schedule (subject to change!)

 04/02/2019: Applications of the integral I: Taylor's formula; indefinite integrals; Uniform convergence revisited: exchange of integration and differentiation (see, e.g., this handout, Theorem 5.5) 05/02/2019: Applications of the integral II: Laplace's method and Stirling's formula (see O.A. Ivanov, Easy as $$pi$$? An introduction to higher mathematics, Springer, 1999, Section 2.5) 11/02/2019: Laplace's method (finish); Uniform convergence of series, Cauchy's criterion (Rudin, 7.7-7.10); uniform limits of continuous functions are continuous (Rudin, 7.11-7.12) 12/02/2019: Point-set topology in metric spaces I: open and closed sets (Rudin, 2.15-2.24) 18/02/2019: Point-set topology in metric spaces II: compact sets (Rudin, 2.31-2.40) 19/02/2019: Point-set topology in metric spaces II: more about compactness (Rudin, 2.41, 4.16 with alternative proof); continuous pre-images of open sets are open (Rudin, 4.8) 25/02/2019: Monotonically convergent sequences of functions on a compact set are uniformly convergent (Rudin, 7.13); convergence, integration, and differentiation revisited. 26/02/2019: Equicontinuity and the Arzela-Ascoli theorem 04/03/2019: Stone-Weierstrass theorem 05/03/2019: Power series, radius of convergence 11/03/2019: Curves in $$R^n$$ 12/03/2019: Midterm Review 18/03/2019: Midterm Exam 19/03/2019: Linear transformations, norm of a linear transformation, invertibility 25/03/2019: Total derivative 26/03/2019: Directional derivative, partial derivatives 01/04/2019: Chain rule 02/04/2019: Taylor's formula in $$R^n$$ 08/04/2019: Contraction mapping theorem 09/04/2019: Inverse function theorem 23/04/2019: Implicit function theorem I 29/04/2019: Implicit function theorem II 30/04/2019: Riemann integral in $$R^n$$ I: definition and elementary properties 06/05/2019: Riemann integral in $$R^n$$ II: iterated integrals and Fubini's theorem 07/05/2019: Riemann integral in $$R^n$$ III: change of variables and polar coordinates 13/05/2019: Riemann integral in $$R^n$$ IV: the divergence theorem with elementary applications 14/05/2019: Review for final exam TBA: Final Exam