Spring Semester 2019

Analysis II


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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
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

Additional Reading:
T. Tao, Analysis II, third edition

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

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

Last modified: 2019/02/14
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/spring2019/math212/index.html
Marcel Oliver (m.oliver@jacobs-university.de)