Matthew Romney

MAT 322-01/MAT 523-01 - Analysis in Several Dimensions - Spring 2022

This course meets MW 2:40-4:00 in Mathematics P131.

Course Syllabus

Course information

Office hours:   Monday 1:00-2:00pm
Wednesday 12:00-1:00pm
or by appointment
 
  You have the option to come to my office (Math Tower 4101B) or to meet on Zoom:
https://stonybrook.zoom.us/j/2482038879?pwd=TDVGcUU4UXhlNHEvdllnckdBb2VpZz09
 
Textbook:   James R. Munkres, Analysis on Manifolds, Westview Press, 1991.
 

Course links

  Tips for Learning Math
 
  Course Outline, Part 1
 
  Midterm 1 Guide   Sample proof problems (key)
 
  Midterm 2 Guide   (key)
 
  Last year's final exam  
 

Course schedule and assignments

Each week’s homework assignment is due at the beginning of Wednesday's lecture (2:40pm) of the following week. Homework should be submitted on Gradescope. The class entry code is 5VGJKB.
 

Week   Date Sections Assignment
1   Jan. 24  
Jan. 26  
1 Linear algebra; 2 Matrix inversion and determinants
3 Review of topology of R^n
HW 1
p. 9 #1,4; p. 24 #1,4; p. 30 #2,6,8
2 Jan. 31  
Feb. 2  
4 Compact and connected subspaces
5 The derivative; 6 Continuously differentiable functions
HW 2
p. 39 #3; p. 48 #1,2,3,4
3 Feb. 7  
Feb. 9  
7 The chain rule
8 The inverse function theorem
HW 3
p. 54 #1,4,5; p. 63 #2,3
4 Feb. 14  
Feb. 16  
9 The implicit function theorem
10 The integral over a rectangle; 11 Existence of the integral
HW 4
p. 70 # 1,5; p. 78 #1,4,6; p. 90 #1,5
5 Feb. 21  
Feb. 23  
12 Evaluation of the integral; 13 The integral over a bounded set
14 Rectifiable sets
HW 5
p. 97 #1,6,9; p. 103 # 2,3; p. 111 # 2,4,7
6 Feb. 28  
Mar. 2  
16 Partitions of unity
MIDTERM 1 (Sections 1-14)
HW 6
p. 143 #1,3
7 Mar. 7  
Mar. 9  
17 The change of variables theorem
18 Diffeomorphisms in R^n
HW 7
p. 151 #3,4,5; p. 160 #1,3,4
8 Mar. 21  
Mar. 23  
19 Proof of change of variables; 20 Applications of change of variables
21 Volume of a parallelopiped; 22 Volume of a parametrized manifold
HW 8
p. 167 #5ab, 6; p. 177 #4; p. 187 #1,5
9 Mar. 28  
Mar. 30  
23 Manifolds in R^n; 24 The boundary of a manifold
25 Integrating a scalar function over a manifold
HW 9
p. 193 #2; p. 202 #3,4; p. 208 #3,5
10 Apr. 4  
Apr. 6  
26 Multilinear algebra; 27 Alternating tensors
28 The wedge product
HW 10
p. 217 #3,8; p. 226 #2,7; p. 236 #1,2
11 Apr. 11  
Apr. 13  
29 Tangent vectors and differential forms
30 The differential operator
HW 11
p. 243 #1,2,4; p. 251 #1,3,4
12 Apr. 18  
Apr. 20  
31 Application to vector and scalar fields; 32 Action of a differentiable map
MIDTERM 2 (Sections 16-29)
HW 12
p. 260 #2,5
13 Apr. 25  
Apr. 27  
33 Integrating forms over parametrized manifolds; 34 Orientable manifolds
35 Integrating forms over orientable manifolds; 36 A geometric interpretation of forms and integrals
HW 13
p. 265 #4; p. 273 #3; p. 280 #1,3
14 May 2  
May 4  
37 The generalized Stoke's theorem
39 The Poincare lemma; 40 The deRham groups of punctured Euclidean space
Due May 11 (on Gradescope)   Final Exam Cumulative
All sections included in Midterms 1, 2
plus 19-24