MAT 322-01/MAT 523-01 - Analysis in Several Dimensions - Spring 2022
This course meets MW 2:40-4:00 in Mathematics P131.
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 |