Instructor: Johan Asplund, johan.asplund[at]stonybrook.edu, Office: Math Tower 3-116
Office hours: MW 5:30–6:30pm
MLC hours: Tu 5:00–6:00pm, via Zoom
Grader: Shuhao Li, shuhao.li[at]stonybrook.edu, Office: Math Tower 2-109
Office hours: W 1:00–2:00pm
MLC hours: W 9:00–10:00am, via Zoom and W 12:00–1:00pm in Math Tower S-235
Class time and location: MW 2:30–3:50pm Frey Hall 224
Syllabus: Here
Textbook: Analysis on Manifolds (Advanced Books Classics) by James R. Munkres
Homework: Homework will be available at 12:00pm on Mondays and is due the same time the following week.
Gradescope: We will use Gradescope for assignment submission. Entry code: ZW84R6
| Date | Topic | Read | HW |
|---|---|---|---|
| M Jan 22 | Review of linear algebra | §1–2 (notes) | #1 |
| W Jan 24 | Metric spaces and topology of ℝⁿ | §3 (notes) | |
| M Jan 29 | Compactness and connectedness | §4 (notes) | #2 |
| W Jan 31 | The derivative | §5 (notes) | |
| M Feb 5 | Chain rule | §6–7 (notes) | #3 |
| W Feb 7 | Inverse function theorem | §8 (notes) | |
| M Feb 12 | Implicit function theorem | §9 (notes) | #4 |
| W Feb 14 | Integrals | §10–11 (notes) | |
| M Feb 19 | Existence of integrals | §12 (notes) | #5 |
| W Feb 21 | Integrals over bounded sets | §13 (notes) | |
| M Feb 26 | Rectifiable sets, review | §14 (notes) | None |
| W Feb 28 | Midterm I (In class) (solutions) | ||
| M Mar 4 | Partitions of unity, change of variables | §15–17 (notes) | #6 |
| W Mar 6 | Change of variables | §17–19 (notes) | |
| M Mar 11 | No class (Spring Recess) | None | |
| W Mar 13 | No class (Spring Recess) | ||
| M Mar 18 | Manifolds in ℝⁿ | §23–24 (notes) | #7 |
| W Mar 20 | k-dimensional volumes | §21–22 (notes) | |
| M Mar 25 | Integrating a scalar function on a manifold | §24–25 (notes) | #8 |
| W Mar 27 | Multilinear algebra and tensors | §26–27 (notes) | |
| M Apr 1 | Wedge product | §27–28 (notes) | #9 |
| W Apr 3 | Tangent vectors and differential forms | §28–29 (notes) | |
| M Apr 8 | The exterior derivative, review | §29–30 (notes) | None |
| W Apr 10 | Midterm II (In class) (solutions) | ||
| M Apr 15 | Scalar and vector fields | §31–32 (notes) | #10 |
| W Apr 17 | Integrating differential forms | §32 (notes) | |
| M Apr 22 | Orientable manifolds | §34–35 (notes) | #11 |
| W Apr 24 | Stokes' theorem | §37 (notes) | |
| M Apr 29 | Poincaré lemma | §39 | None |
| W May 1 | De Rham cohomology, review | §40 | |
| W May 15 | Final Exam: 2:15–5:00 pm, Frey Hall 224 | ||