Instructor: Piotr Hajlasz
Office Hours: By appointment.
Office: 420 Thackeray Hall
Email: hajlasz@pitt.edu
Textbook: I will not use any particular textbook, but I will post my notes online. You can find here some hand written notes that I used in the past for the course, but I will be writing a new set of notes (in LaTeX) that will cover whole material of the course. For the paractice problems use the book Berkeley Problems in Mathemtics and study problems from the past preliminary exams.
Advanced Calculus I (pages 1-296)
Advanced Calculus I (supplementary materials)
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%)
Homework: Posted weakly on the course webpage to be submitted on Canvas. The problems must be typed in LaTeX. Late homework will not be accepted. Selected problems will be graded by Anthony Cappello.
Course content Logic and set theory. Axiom and construction of real numbers. Supremum, infimum, sequences, series and limits, upper and lower limits. Continuity and differentiability of functions. Riemann integral. Metric spaces, contraction principle with applications, Arzela Ascoli theorem, Stone-Weierstrass theorem.
Video 1 Video 2 Video 3 Video 4
Problems (LaTeX file) Problems (pdf file) No deanline, not required, Canvas
HW#1 (LaTeX file) HW#1 (pdf file) September 17, Canvas.
HW#2 (LaTeX file) HW#2 (pdf file) September 24, Canvas
HW#3 (LaTeX file) HW#3 (pdf file) October 8, Canvas
HW#4 (LaTeX file) HW#4 (pdf file) October 15, Canvas
HW#5 (LaTeX file) HW#5 (pdf file) November 1, Canvas
HW#6 (LaTeX file) HW#6 (pdf file) December 8, Canvas
HW#7 (LaTeX file) HW#7 (pdf file) Not available yet
HW#8 (LaTeX file) HW#8 (pdf file) Not available yet