Analysis I, Math 4317

                   Lectures: MWF 1-2 pm in Skiles 256
                                      by Michael Loss (loss@math.gatech.edu)
                                      Office: Skiles 214 B
                                      Office hours: MW 3-4 or by appointment

    This page will be frequently updated,
           so, please, consult it often.
                          
                            Final Exam
    Dec 9th (Tue)    2:50 - 5:40 in Skiles 256

                       Review Problems


Analysis, it is sometimes said, is the art of taking limits! Many interesting mathematical objects arise as limits and one is faced with the challenge of
calculating with them. The square root of two, the elementary transcendental
functions are elementary examples, however, solutions of partial differential equations,
most of the time, appear as limits as well. It is therefore important to understand how
to compute with limits. Analysis provides a number of notions and tools for dealing with limits and this course is a gentle introduction to them.  Since the notions in analysis
are subtle, it is sometimes difficult to get an intuitive feeling of what these things mean.
The only way to stay on firm ground in this field is by understanding and doing proofs. You have to be able to prove certain statements, i.e., to either derive a statement through logical steps from another one, or to find a counterexample. Thus, this course is not
so much about solving particular problems. Its main goal is to convey an important
part of mathematics in a rigorous fashion.

Among the topics to discuss are elementary set theory, the real numbers,
metric spaces, continuous functions as well as differentiation and integration.
The course is based on the text `Introduction to Analysis' (IA) by Maxwell Rosenlicht,
Dover 1986, ISBN 0-486-65038-3. Please read the assigned text before coming to class.
Moreover, for some other topics I supply some notes on the web. You will find them
on this link.

  Practice is important. You will have to do homework that is regularly graded. In addition there will be two tests and a final exam.

  Your grade will be based on how well you solve the homework and the tests. The lowest test score will be dropped. Homework counts 30%, the test counts 30% and the final counts 40% towards your grade.

Now a few words about the homework. Every week I assign 10 problems and five of them will be graded, but I am not telling you which ones. It goes without saying that you to do the homework yourself, i.e., you do not copy from others. I do not mind that, should you get stuck, you ask your fellow students for help, but you have to try first yourself. Let me also remind you to uphold the honor code at all times. You find
more information at  http://www.deanofstudents.gatech.edu/Honor/

Please turn in the home work always on Mondays in class.

Here is a tentative syllabus. I do not want ot stick to it in a rigid fashion.
There may be a topic that for some reason will require that we spend more time
on it. I would like to have this option.  It is better to understand fewer topics
very well than lots of topics badly.

  Please note that weeks 3-14 have an exercise group assigned to the homework.
  This group solves the homework in detail, submits it to me on Wednesday of the
  following week in class. This groups solutions with my corrections will be posted
  on the web.

  Week 1: August 18 - 22.  Assigned Text: Chapter 1 in IA
  Homework: Do all the problems at the end of Chapter 1 in IA.

  Week 2: August 24 - 29.  Assigned Text: Chapter 2 in  IA
  Homework: Do problems  2, 4, 6, 7, 9, 10, 11, 12, 13, 16 on pages 29-31

  Week 3: September 1 - 5.  Assigned Text: Chapter 3, Section 1- 3, in IA
  Homework: Do problems 1, 2, 5, 6, 7, 10, 11, 12, 13, 14 on pages 61-62 of IA.
  Exercise Group: Fox, Michael; McBride, Chris;Yang Yang.   MWF 11am  - 1 pm

  SOLUTIONS Part 1

  SOLUTIONS Part 2

  Week 4: September 8 - 12.  Assigned Text: Chapter 3, Section 4 - 6, in IA
  Homework: Do problems 15, 16, 18, 20, 24, 26, 28, 30, 32, 33, on pages 62-64 of IA.
  Exercise Group: Beck, Yokhanan Z.; Lineberger, Kimberly A.; Mize, Christopher L. ; Korjenic Dijana. T Th 3pm -5pm
 
SOLUTIONS

  Week 5: September 15 - 19.  Assigned Text: Chapter 3, Section 4-6,  in IA
   Homework: Do problems 17, 19, 21, 27, 29, 31, 34, 35, 37, 38 on pages 62-65 in IA.
  Exercise Group:  Anderson, Joshua C.; Brandt, Patrick D.; Chen, Leo C.      MWF 5pm-7pm

SOLUTIONS
COMMENTS TO THE SOLUTIONS

Comments on the solutions will be posted soon.

  Week 6: September 22 - 26.  Assigned Text: Chapter 4, Section 1- 3,  in IA
  Homework: 1a), 1d), 2, 4, 9, 10, 11, 13, 14, 15 on page 91, 92 od IA.
  Exercise Group:  Spears, Kevin W.; Williams, Meoshe F.;  Lai, Gangbin; Lowry, David J.  MWF 2pm - 4 pm

SOLUTIONS

PRACTICETEST

   There will be a 50 min test on Wednesday September 24 in class.  Everything
 we have been talking about up and including week 5  is covered.

Week 7: September 29 - October 3.  Assigned Text: Chapter 4, Section 1-3  in IA
Homework: 3, 6, 7, 12, 16, 17, 18, 19, 20, 21 on pages 91-93 of IA.
Exercise Group:  Kinnaman, Kelsey M.;  Reese, Mary E.;  Sirow, Gabrielle S.    T Th before 11 am

SOLUTIONS

Week 8: October 6 - 10.  Assigned Text: Chapter 4, Section 4-6  in IA
Homework: 25, 27, 30, 32, 33b, 34, 35, 36, 37, 38 p. 93/94 of IA.
Homework is due on Friday October 17.

SOLUTIONS

  Week 9: October 13 - 17.  Assigned Text: Chapter 4, Section 4-6 in IA
  Homework: 23, 26, 28, 29, 41, 43, 44, 45, 46 p. 93-95 in IA.
  Exercise Group:  Hollis, David A.;  Rajendar, Ashok; Xu, Kexin; Matsuzawa, Takashi.    MWF 2pm -  4pm HOMEWORK is DUE ON FRIDAY OCT. 24.

SOLUTIONS

  Week 10: October 20 - 24.  Assigned Text: Chapter 5  in IA
  Homework: Problems 1-6 and 8 on p 108/109 in IA. Please do 8 instead of 7 which is too hard.
  Exercise Group:  Boyd, William R.;  Nichols, Diana R.; Villarreal Monica; Farhana Rahman; Wonki, Kim.   T Th 6pm - 8pm  HOMEWORK DUE ON FRIDAY OCT. 31

SOLUTIONS

Week 11: October 27 - 31.  Assigned Text: Chapter 5  in IA
  Homework:  Problems 9-14 on page 109/110
  Exercise Group:  Namsu Sunwou;  Ivan Kolesov;  Armour, Jonathan N.  T Th 5 pm - 7pm
 HOMEWORK DUE ON FRIDAY NOVEMBER  7.

PRACTICETEST

SOLUTIONS

   There will be a 50 min test on Wednesday October 29  in class.  Everything
 we have been talking about up and including week 10  is covered.


  Week 12: November 3 - 7.  Assigned Text: Chapter 6 Sections 1, 2 in IA
  Homework: problems 1-3, 7-11 on page 132/133 of IA
  Exercise Group:  Gallien, Brandy L.; Chung, Gi Yun; Khan Soniya; Perez Matios;
Krenek, Russell W MWF 4pm - 6pm
HOMEWORK DUE ON FRIDAY NOVEMBER 14.

SOLUTIONS

  Week 13: November 10 - 14.  Assigned Text: Chapter 6 Sections 3-4  in IA
  Homework: problems 12-17,19, 21and 22 on pages 133/134 in IA.
  Exercise Group:  Futral, Joanna E.; Milhose, Blacki R.; Kang, Hyo Jung.   MWF 5pm - 7pm
 HOMEWORK DUE ON FRIDAY NOVEMBER 21

SOLUTIONS

  Week 14: November 17 - 21.  Assigned Text: Chapter  6 Section 5 in IA
  Homework: problems 20, 23-28 on pages 134/135 in IA.
  Exercise Group:  Froehlich, Stefan M.; Santomauro, Giles A.; Shah, Sarang A.  TTh 5pm-7pm
HOMEWORK DUE ON MONDAY DECEMBER 1

SOLUTIONS

  Week 15: November 24 - 28.  Assigned Text: Chapter  in IA

  Week 16: December 1 - 5.  Assigned Text:

The Weierstrass Approximation Theorem