# Analysis II

 Analysis II, MAT321, 4 credits Meetings: Mon Wed (see below) Prerequisite: MAT 320 Text: Elementary Classical Analysis Edition 2 Marsden and Hoffman Professor C. Sormani Office Hours: Monday Wednesday 10-11 am, 3-4 pm Office: Gillet Hall 200B Email: sormanic (at) member.ams.org Webpage: http://comet.lehman.cuny.edu/sormani has a link to this course's webpage

Course Description: Metric Geometry, Compactness, C([0,1]), Contraction Mapping Principle, Proofs of the Fundamental Theorem of ODE, of Curves, Implicit and Inverse Function Theorems, Sets of Measure Zero, Lebesgue Integration

Grading Policy: 5 Projects: 6% each... Exam I: 20%... Exam II: 20%... Final:30%...

In the following syllabus, homework is written below the lesson when it is assigned and should be completed before the next meeting. Starred problems may form parts of the projects. Students should come to the 10 am office hour to ask questions on homework before class.

Meeting with Differential Geometry: For most of the course we will be meeting in the same room as Differential Geometry. We will be having quizes at the same time. If you are taking both courses you will need to take your quiz at 10:30 am Monday. Later on in the semester we will start meeting seperate from Differential Geometry, only 10-11:00 am Mon Wed, and the course will be conducted like an independant study.

Syllabus: (will be updated regularly)

• Mon 1/28 11-12:50: Exam for 320
• Wed 1/30 11-12:50: Open and Closed Sets in Metric Spaces 2.1-2.4
Read each section before doing problems in that section, the proofs of theorems should always be read (see pages 130-131). If an exercise asks a question, be sure to give a proof or counter example.
Do 2.1/1*,6*; 2.2/3,4,5*; 2.3/1,5; 2.4/1,2*,3 (no proof), 4, 6
• Mon 2/4 11-12:50: Closure and Boundary 2.5-2.6
No proofs on tonights homework, just think and give the answer,
Read 2.5, Do 2.5/1,2,3, Read 2.6, Do 2.6/1-5.
• Wed 2/6 11-12:50: Continuous Maps 4.1
Read 2.7 (review of analysis 1), Read 4.1,
Do 4.1/1,2*,3,5
Project 1: the five starred problems assigned 1/30-2/6 is due Mon 2/11
• Mon 2/11 11-12:50: Completeness 2.8
Read 2.8, Do 2.8/1,2*,3* (remember the discrete metric is the metric where everything is a distance 1 apart)
• Wed 2/13 11-12:50: Compactness 3.1-3.2
Read 3.1 and proofs, do 3.1/2*, 3*, 5
Read 3.2 and proofs, do 1, 4*,
• Wed 2/20 11-12:50: Mapping Compact Sets 4.2-4.5 (4.6-4.8)
Read Theorem 4.2.1 and its proof, Read examples on page 183, Do 4.2/1,2,3 (ignoring anything related to "connected" and only do "compact")
Project 2 on Compactness (*ed problems 2/11-2/13) is due 2/25
• Mon 2/25 11-12:50: Contraction Mapping Principle (5.7)
• Wed 2/27 11-12:50: Review
This exam is on 2.1-2.4, 4.1, 3.1-3.2 and Projects 1 and 2. You will need to know proving techniques including quantifiers, first lines of proofs by contradiction, how to prove something is an open set, how to prove one set is inside another, how to prove a map is continuous using balls and using epsilon deltas. There will not be full length proofs on the exam, just questions regarding first and last steps and such things, how to draw a diagram that will help you select a ball and so on. You should also be able to identify sets in a plane which are closed, bounded, open, compact, and homeomorphic without providing proofs. Sample Exam. We will not test completeness, compactness, sequences, closures, accumulation points, or the contraction mapping principle.
• Mon 3/3 11-12:50: Exam I
• Wed 3/5 11-12:50: C([0,1]) 5.1, 5.5-5.6 (handout)
Project 3: (b)-(j) on the handout. Note (a) is in Marsden.
• Mon 3/10 11-12:50: Smooth curves 6.1
Just read 6.1.
Project 3 on C([0,1]) is due (try a little bit of each letter and leave space).
• Wed 3/12 11-12:50: Smooth Functions 6.1-6.3
• Mon 3/17 10-11am: C([0,1]) and 11-12:50: Ordinary Differential Equations 7.5
• Wed 3/19 10-11am: C([0,1]) and 11-12:50: Fundamental Theorem of Curves
Resubmission of Project 3 is due Wed 3/26
• Wed 3/26 10-11am: Compactness and 11-12:50: Chain and Product Rules in Vector Calculus 6.5-6.6
Resubmission of Project 2 is due Mon 3/31
• Mon 3/31 11-12:50: Inverse Function Theorem 7.1
• Wed 4/2 11-12:50: Smooth Surfaces 5.1-5.3
Resubmission of Project 3 is due Mon 4/7
• Mon 4/7 11-12:50: Implicit Function and Rank Theorem 7.2-7.4
• Wed 4/9 10-11 am : Implicit Function and Rank Theorem 7.2-7.4
• Mon 4/14 10-11am : Resubmission of Projects 2-3 and review of Projects 2 and 3.
• Wed 4/16 11-12:50: Review for Exam II
Know closed sets, bounded sets, accumulation points, interiors, closures, Heine-Borel, compactness, sequential compactness, continuous functions, preimages of open sets, Lipschitz functions, Contraction mappings, C([0,1]), and projects 1-3. Sample Exam and Sample Exam Solutions. After thoroughly memorizing all key definitions, try the sample exam without looking anything up, then try it for an extra hour looking up a couple things, then check the solutions. Then study again and try the sample exam again in test conditions and try to finish it in an hour. You might try making up similar problems. In fact similar problems are in the textbook.
• Spring Recess
• Mon 4/28 11am-1pm : Exam II
• Wed 4/30 10-11am: Review of Exam II and Lipschitz
• Wed 4/30 1-2pm: Review of Ptwise Convergence 5.1
• Mon 5/5 10-11am: Review of Uniform Convergence 5.1-5.5
• Mon 5/5 3-4pm: Completeness of C([0,1]) 5.5-5.6
• Wed 5/7 10-11am: Arzela Ascoli and Contraction Mapping, 5.6
• Wed 5/7 3-4pm: Measure Zero (handout)
Project 4 on Measure Zero is due on Mon 5/19 (no resubmissions):
Lateness on this project will lead to an incomplete in the course.
• Mon 5/12 10-11am : A hint at Lebesgue Measure and Integration
• Mon 5/19 10-4pm : Office Hours and deadline for Project 4
• Wed 5/21 3-4 pm Analysis II Review
The final will be 4 proofs selected from the following: (a) a proof that a set is open, (b) a proof that a function is continuous, (c) a proof that a set is compact, (d) a proof that a set is not compact, (e) a proof that a space is complete (f) a proof that a set has measure 0
If you are not ready for the final, do not take it. I will give this exam only once and the same exam to everyone. Those of you who take the exam on time, do not talk about it. I will post the solutions when all students have taken the exam.
• Final TBA (after May 21): Maybe Fri 5/23 10-12