Home page for Math 509, Advanced Analysis
Spring 2019
Instructor: Charles L. Epstein
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The Course
In this course we continue our study of the foundations of analysis.
The goal
of this course is to further develop your instincts as a mathematical
analyst.
This
entails understanding the basic concepts and techniques of analysis, as
well as
developing intuition and proficiency in the construction of rigorous
proofs. In this semester we continue our discuss of approximating
and representing functions, and the topology of the space of continuous
functions with the sup-norm. After a short discussion of linear algebra
and the geometry of Euclidean space we then introduce basic concepts of
Fourier analysis. We then turn to differential and integral calculus in
several variables.
Several textbooks will be used this semester: The Way of Analysis, by Strichartz; A First Course in Analysis, by Conway; Calculus on Manifolds, Spivak.
A problem set will be assigned every week, on Tuesdays. We may have a midterm and final exam.
- The class meets from 3:00 to 4:30 on Tuesdays and Thursday in DRL A4.
- Each student is required to register for and attend an evening lab session.
-
My office hour is 3:00-5:00 on Monday. Send e-mail if you have a question or would like to come see me at some other time.
-
email: cle@math.upenn.edu.
Send e-mail or call if you need to see me at some other time.
-
My office is DRL 4E7, my telephone number is: 8-8476.
- The TA is Dominick Villano. His email is dvillano@math.upenn.edu and his office is 4C13 in DRL.
Syllabus for the course
- More on functions of a single variable
- Approximation versus Interpolation
- Norm topology of Rn
- Arzela-Ascoli Theorem
- Linear analysis and Geometry in Rn and Cn
- Fourier Series
- Power series
- Elementary transcendental functions
- Basic properties of Fourier series
- Convergence of Fourier Series
- Applications of Fourier Series
- Differential Calculus in Several Variables
- Linear approximation
- Higher derivatives and Taylor's theorem
- Lagrange multipliers
- Vector valued maps and the inverse function theorem
- Riemann Integration in the Rn
- Basic definitions
- Fubini's Theorem
- Change of Variable Formula
- Differential forms in R2
and R3
- Integration by parts and Green's Theorem in the plane and R3
Lecture Notes
Announcements
- We will have a midterm exam on February 28, 2019
- Your lowest homework score will be dropped.
- We will have an extra class on Tuesday, April 2 from 4:30-5:30 in DRL
4C8. We will go over the solutions to the Midterm Exam.
Problem Sets
- Problem set 1 due January 29, 2019.
- Problem set 2 due February 5, 2019.
- Problem set 3 due February 12, 2019.
- Problem set 4 due February 19, 2019.
- Problem set 5 due February 26, 2019.
- Problem set 6 due March 26, 2019.
- Problem set 7 due April 2, 2019.
- Problem set 8 due April 9, 2019.
- Problem set 9 due April 16, 2019.
- Problem set 10 due April 25, 2019.