MAT 32201/MAT 52301  Analysis in Several Dimensions  Spring 2021
The class is a hybrid course meeting Monday and Wednesday 2:404:00pm in JavitsLectr 102.
For virtual participants, the Zoom meeting link for the course is:
https://stonybrook.zoom.us/j/92738277763?pwd=M05aVTV5T1B6NzhFRXB5aTF3eDhzUT09
Meeting ID: 927 3827 7763
Password: 927611
You must use your Stony Brook account to access the meeting room.
Course information
Office hours:  Monday 12:001:00pm 
Wednesday 4:005:00pm (give me enough time to get back to my office)  
or by appointment  
Office hours will be held on Zoom:  
https://stonybrook.zoom.us/j/2482038879?pwd=TDVGcUU4UXhlNHEvdllnckdBb2VpZz09  
You also have the option to come to my office (Math Tower 4101B) by appointment.  
Textbook:  James R. Munkres, Analysis on Manifolds, Westview Press, 1991. 
Course links
Course schedule and assignments
Each week’s homework assignment is due at the beginning of Monday's lecture (2:40pm) of the following week.
Week  Date  Sections  Assignment 
1  Feb. 1 Feb. 3 
1 Linear algebra; 2 Matrix inversion and determinants 3 Review of topology of R^n 
HW 1 (Due Wed., Feb. 10) p. 9 #1,4; p. 24 #1,4; p. 30 #2,6,8 
2  Feb. 8 Feb. 10 
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. 15 Feb. 17 
7 The chain rule 8 The inverse function theorem 
HW 3 p. 54 #1,4,5; p. 63 #2,3 
4  Feb. 22 Feb. 24 
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  Mar. 1 Mar. 3 
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  Mar. 8 Mar. 10 
16 Partitions of unity MIDTERM 1 (Sections 114) 
HW 6 p. 143 #1,3 
7  Mar. 15 Mar. 17 
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. 22 Mar. 24 
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. 29 Mar. 31 
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. 5 Apr. 7 
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. 12 Apr. 14 
29 Tangent vectors and differential forms 30 The differential operator 
HW 11 p. 243 #1,2,4; p. 251 #1,3,4 
12  Apr. 19 Apr. 21 
31 Application to vector and scalar fields; 32 Action of a differentiable map MIDTERM 2 (Sections 1629) 
HW 12 p. 260 #2,5 
13  Apr. 26 Apr. 28 
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 3 May 5 
37 The generalized Stoke's theorem 39 The Poincare lemma; 40 The deRham groups of punctured Euclidean space 

May 13  Final Exam
due Monday, May 10 at midnight 
Cumulative All sections included in Midterms 1, 2 plus 1924 