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 1011 am, 34 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 1011:00 am Mon Wed,
and the course will be conducted like an independant study.
Syllabus: (will be updated regularly)

Mon 1/28 1112:50: Exam for 320

Wed 1/30 1112:50: Open and Closed Sets in Metric Spaces 2.12.4
Read each section before doing problems in that section,
the proofs of theorems should always be read (see pages 130131).
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 1112:50: Closure and Boundary 2.52.6
No proofs on tonights homework, just think and give the answer,
Read 2.5, Do 2.5/1,2,3, Read 2.6, Do 2.6/15.

Wed 2/6 1112:50: 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 1/302/6 is due Mon 2/11

Mon 2/11 1112:50: 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 1112:50: Compactness 3.13.2
Read 3.1 and proofs, do 3.1/2*, 3*, 5
Read 3.2 and proofs, do 1, 4*,

Wed 2/20 1112:50: Mapping Compact Sets 4.24.5 (4.64.8)
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 2/112/13)
is due 2/25

Mon 2/25 1112:50: Contraction Mapping Principle (5.7)

Wed 2/27 1112:50: Review
This exam is on 2.12.4, 4.1, 3.13.2 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 1112:50: Exam I

Wed 3/5 1112:50: C([0,1]) 5.1, 5.55.6 (handout)
Project 3: (b)(j) on the handout. Note (a) is in Marsden.

Mon 3/10 1112:50: 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 1112:50: Smooth Functions 6.16.3

Mon 3/17 1011am: C([0,1])
and 1112:50: Ordinary Differential Equations 7.5

Wed 3/19 1011am: C([0,1]) and
1112:50: Fundamental Theorem of Curves
Resubmission of Project 3 is due Wed 3/26

Wed 3/26 1011am: Compactness and
1112:50: Chain and Product Rules in Vector Calculus 6.56.6
Resubmission of Project 2 is due Mon 3/31

Mon 3/31 1112:50: Inverse Function Theorem 7.1

Wed 4/2 1112:50: Smooth Surfaces 5.15.3
Resubmission of Project 3 is due Mon 4/7

Mon 4/7 1112:50: Implicit Function and Rank Theorem 7.27.4

Wed 4/9 1011 am : Implicit Function and Rank Theorem 7.27.4

Mon 4/14 1011am : Resubmission of Projects 23
and review of Projects 2 and 3.

Wed 4/16 1112:50: Review for Exam II
Know closed sets, bounded sets, accumulation points, interiors,
closures, HeineBorel, compactness, sequential compactness,
continuous functions, preimages of open sets, Lipschitz
functions, Contraction mappings, C([0,1]), and projects 13.
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 11am1pm : Exam II

Wed 4/30 1011am: Review of Exam II and Lipschitz

Wed 4/30 12pm: Review of Ptwise Convergence 5.1

Mon 5/5 1011am: Review of Uniform Convergence 5.15.5

Mon 5/5 34pm: Completeness of C([0,1]) 5.55.6

Wed 5/7 1011am: Arzela Ascoli and Contraction Mapping, 5.6

Wed 5/7 34pm: 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 1011am : A hint at Lebesgue Measure and Integration

Mon 5/19 104pm : Office Hours and deadline for Project 4

Wed 5/21 34 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 1012