# 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 05/02/2019: Applications of the integral II: Laplace's method and Stirling's formula 11/02/2019: Sequences and series of functions I: uniform convergence 12/02/2019: Sequences and series of functions II: uniform limits of continuous functions are continuous 18/02/2019: Sequences and series of functions III: uniform convergence with integration and differentiation 19/02/2019: Sequences and series of functions IV: Equicontinuity and the Arzela-Ascoli theorem 25/02/2019: Sequences and series of functions V: Stone-Weierstrass theorem 26/02/2019: Sequences and series of functions VI: power series, radius of convergence 04/03/2019: Point-set topology in metric spaces I 05/03/2019: Point-set topology in metric spaces II 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