MAT 322/523: Analysis in Several Dimensions, Spring 2024

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


Tentative schedule

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 Stokes' theorem and divergence theorem §37–38 (notes) None
W May 1 De Rham cohomology §40 (notes)
W May 15 Final Exam: 2:15–5:00 pm, Frey Hall 224