Instructor: Piotr Hajlasz

Office Hours: TBA & by appointment.

Office: 420 Thackeray Hall

Email: hajlasz@pitt.edu

This course will be substantially more difficult than Advanced Calculus I, because its main goal is to (a) prepare the graduate students for the Preliminary Exam, (b) prepare the undergraduate studnets for the graduate school. While the exams will be substantially easier than the Prelimnary Exam, they will also be substantilly more difficult than the exams in Advanced Calculus I.

Textbook:

Right now I am following metric spaces from my notes:

Current Lecture Notes

I will keep updting the notes and if I have time I will include analysis of several variables there. Below you can find hand written notes that I used in the past for the course. The material on the one variable analysis and metric spaces that find in the hand written notes is already covered in Current Lecture Notes. However, as of now, analysis of several variables is only available in the hand written notes.

Advanced Calculus I (pages 1-296)

Advanced Calculus I (supplementary materials)

Advanced Calculus II

For the paractice problems use the book Berkeley Problems in Mathemtics and study problems from the past Preliminary Exams. Note that a lot of examples in my notes are taken form the past Preliminary Exams.

Reccomended books:

1. Berkeley Problems in Mathemtics, Third Edition, by P. N. De Souza and J.-N. Silva

This is an excellent collection of problems for the course. Problems have full solutions. Chapters 1,2,4 cover much of the material needed for the Preliminary Exam in Analysis. Other chapters contain excelelnt problems for other graduate courses like Linear Algebra, Algebra, Analysis II and Ordinary Differential Equations. This book is a must for any Ph.D. student.

2. Elementary Classical Analysis , Second Edition, by J. E. Marsden and M. J. Hoffman

3. Principles of Mathematical Analysis, by W. Rudin

4. Mathematical Analysis, Second Edition, by T. M. Apostol

5. Real Mathematical Analysis by Ch. Ch. Pugh.

Course Grade: Homework (30%) + Two midterm exams (20% + 20 %) + Final exam (30%)

Course content Metric spaces, Banach contraction principle, Arzela-Ascoli theorem, Stone-Weierstrass theorem, differentiability of functions of several variables, inverse function theorem, implicit function theorem, Lagrange multipliers, integration (whatever it means) etc.

Homework

Homework problems will be posted this webpage. It has to be submitted on Canvas. The problems must be typed in LaTeX and submitted as a pdf file.

The online submissions will be checked for completeness. However, there will be tests during recitations based on the homework and the score from the homework will be determined from these tests. Well, if you submit solutions (and usually most of the students submit complete solutions), you have to understand them. If you are honest, you will have no difficulties with the tests. No excuse.

HW#1 (LaTeX file) HW#1 (pdf file) Due Sunday, January 25, 2026 (via Canvas).

HW#2 (LaTeX file) HW#2 (pdf file) Due Sunday, February 1, 2026 (via Canvas).

HW#3 (LaTeX file) HW#3 (pdf file) Due Sunday, February 8, 2026 (via Canvas).

HW#4 (LaTeX file) HW#4 (pdf file) Due Sunday, February 15, 2026 (via Canvas).

HW#5 (LaTeX file) HW#5 (pdf file) Due Sunday, February 22, 2026 (via Canvas).

HW#6 (LaTeX file) HW#6 (pdf file) Due Sunday, March 1, 2026 (via Canvas).

HW#7 (LaTeX file) HW#7 (pdf file) Due Sunday, March 15, 2026 (via Canvas).

HW#8 (LaTeX file) HW#8 (pdf file) Not available yet.

HW#9 (LaTeX file) HW#9 (pdf file) Not available yet.