# Real Analysis 4155/9063 Fall 2013

### Instructor:

Rasul Shafikov, MC 112, (all email inquiries will be answered within 48 hours).
Office hours: Mondays 2:30 - 4:30 PM.
Announcements: There will be no class on Nov 1, 4, and 6. Class meets Nov 8 at the regular time.

### Course Coordinates:

Course meets MWF 1:30 - 2:30 PM in MC 107. OWL Login: MATH 4155A or MATH 9063A

### Prerequisites:

Multivariate Calculus, Linear Algebra, Metric Spaces.

### Evaluation:

Homework = 40%, Midterm = 20%, Final = 40%

### Content:

We will cover the following topics
1. Functions of several variables: continuity, differentiability, Implicit/Inverse function theorems, Rank theorem, higher order derivatives.
2. Integration: Riemann integral, Lebesgue integral (quick introduction), Fubbini's theorem, basic \$L^p\$ spaces, H\"older inequality, integrals depending on parameter.
3. Surface integrals: line and surface integrals, Divergence theorem, Green's formula, integration by parts, spherical coordinates, surface area, partition of unity.
4. Introduction to ODEs and PDEs: basic definitions and examples.
5. Distributions: test functions, basic properties of distributions, convolution, fundamental solutions of differential operators, Magrange-Ehrenpreis theorem.

### Lecture Notes

Lecture notes for most of the material covered in class will be provided. A new set of notes will be posted at the end of each week here.
1. Week 1 Differentiation of functions of several variables; (v. Sept 13)
2. Week 2 Inverse Function theorem and friends; (v. Sept 20)
3. Week 3 Integration: from Riemann to Lebesgue; (v. Oct 9)
4. Week 4 L^p spaces and their relatives; (v. Oct 8)
5. Week 5 Divergence theorem and consequences (v. Oct 11)
6. Week 6 Differential Equations (v. Oct 17)
7. Week 7 The space of test functions (v. Nov 16)
8. Week 8 - Exam
9. Weeks 9 and 10 Basic Theory of Distributions (v. Nov 9)
10. Weeks 11 Differentiation of distributions (v. Nov 16)
11. Weeks 12 Structure theorems and convolution of distributions (v. Nov 22)
12. Weeks 13 Fundamental Solutions of Differential Operators (v. Nov 29)
13. Weeks 13 Fundamental solutions of classical operators. (v. Dec 2)

### Homework

Homework will be assigned on a regular basis and will include routine exercises as well as some challenging problems. Students registered in Math 9063 will be given additional problems.
1. Assignment 1 Due September 27 in class.
2. Assignment 2 Due October 11 in class.
3. Assignment 3 Due October 25 in class.
4. Assignment 4 Due November 22 in class (Updated on Nov 18 to correct problem 4.2)
5. Assignment 5 Due December 6 in class.

### Exams

In-class Midterm on October 30,
Final Exam on December 11, 9:30 AM - 12:30 PM in MC 108

### Senate Regulations on Scholastic Offences

Please note the following points, which are required to be stated in this outline by the Senate regulations.

Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at the following Web site: http://www.uwo.ca/univsec/handbook/