MAT 322-01/MAT 523-01 - Analysis in Several Dimensions - Spring 2021
The class is a hybrid course meeting Monday and Wednesday 2:40-4:00pm 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 12:00-1:00pm |
| Wednesday 4:00-5:00pm (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 1-14) |
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 16-29) |
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 19-24 |