MAT 322 Analysis in Several Variables  Spring 2017

Syllabus,  schedule and homework.

Slides and other materials can be found here.

This schedule will be regularly updated. It is your responsibility to check it often.

Homework is due every Thursday. Underlined problems in should be handed in.

When Topics  # Homework, Exams, Remarks.
1/24
Administrative stuff and general information
Chapter 1 - Sections 1.1-1.2-1.3
0 Fill this form.
If you have the book: Read Chapter 0 and prepare a list questions to ask in class.
1/31
Section 6.1
Discussion about differential forms.
1 Section 1.1: 4, 6, 9
Section 1.2: 2, 4, 8,10, 14, 16
2/7
Section 6.1
Discussion about differential forms.
Snow
2 Section 6.1:1a, 1c, 2a, 6, 15

2/14
Discussion about integrating differential forms
Section 1.5 (up to Subsection Convergence) and
Section 3.1: Manifolds 
-Smooth Manifolds in Rn.
-Parametrization of manifolds.

 

3 Following the procedure we discussed in class (that is, dividing the curve in small arcs and considering vectors joining consecutive the endpoints of these arcs), find the integral of the 1-form    in R^2 2xdy+3y dx, against a  curve of your choice from the point (0,2) to the point (1,3). IMPORTANT: Explain and justify every step.
Extra credit: Try to find the integral along another curve with the same endpoints.

Section 6.1: 12, 15
Interesting linkages
Peaucellier linkage
Watt's linkage
Ellipse linkage
Elliptical trainer.
2/21
Section 2.10 The implicit and inverse function theorem (statements only. We will discuss the proofs later).
Section 3.1 Manifolds.
4 Section 1.5: 1, 2, 3, 5, 7.
Section 3.1: 2, 3, 12, 13.
2/28
Section 3.1 Manifolds.
Review

5 Section 2.10: 1, 2, 5a, 5b, 6,8, 9, 10 (Prove only that a continuous strictly monotonous function has an inverse. You do not need to prove that the inverse is continuous... but of course you can if you wish)
Section 3.1: 5, 6, 7, 8.
3/7
Section 3.2: Tangent spaces
6 Midterm 1: Tuesday 3/7 In class. Topics: Up to Section 3. (including Section 3.1).
No homework this week.
3/14


Spring break
3/21
Review

 
3/28
Section 4.1 Defining the integral
Section 4.3 What functions can be integrated?
7-8 Section 3.2: 1, 2, 3, 4a,b,c,5a, 6, 7, 8, 12,
4/4
Section 4.4 Measure zero
Review
9 Section 4.1: 1, 2, 3.  5a,b,c,d, 6, 8, 9, 10, 14
Section 4.3: 1, 2, 3, 5.
4/11
Midterm II
Section 4.5 Fubbini's theorem
Section 6.2 Integrating form fields
10 No homework this week but you should know how to solve the following problems.
Section 4.4: 1, 2, 5, 7.
Section 4.5: 2, 3, 7,8

Midterm 2: Thursday 4/13  In class. Midterm Topics: Sections 3.2, 4.1, 4.3, 4.4.

Sample problems for the midterm:
Section 3.2: 4, 6, 8
Section 4.1: .9, 11, 14, 21
Section 4.3: 2, 3, 5, 
Section 4.4: 2, 7

4/18-
Section 6.3 Orientation of manifolds
Section 6.4. Integrating forms over manifolds
11 Section 4.4: 2, 7
Section 4.5: 3, 7, 812, 15
4/25
Stokes theorem
Section 6.6 Boundary orientation
Section 6.7 The exterior derivative
Section 6.9 The pullback
12 Section 6.2: 1a,  2a, 3b
Section 6.3: 1, 2, 3, 4, 5, 12
6.4: 1, 4
5/2
Section 6.10 The generalized Stokes theorem
Review
13
Section 6.6:  2,   6a,b. (In problem 2 and 6a, you only need to show that the boundary of each region is a finite union of one dimensional manifolds.
Section 6.7: 1, 2, 6, 7b.
Secton 6.10: 2, 4, 5.
Sample problems for the final



Final Exam: Wed. May 10, 5:30pm-8:00pm in our usual classroom, Harriman 112