MATH 127C — Real Analysis (Summer 2025)
This page mirrors announcements, policies, and a living schedule for the Summer 2025 offering. Lecture notes and problem sets reflect the topics we covered this term: metric spaces, compactness/connectedness, multivariable differentiability (Jacobian, chain/implicit/inverse theorems), $k$–volume and Gram determinants, change of variables, Fubini/Tonelli, and Green/Stokes/Divergence.
At a Glance
- Instructor: Shizhou Xu
Email: shzxu@ucdavis.edu
Office: Mathematics Building, Room 3134 - Lecture: MWF · 100 minutes · Wellman 212
Discussion/Section: T · 50 minutes · Wellman 212 - Office Hours: Wed 3–4pm
- Course Site: This page, Canvas
- LMS: Canvas (assignments, submissions, grades)
- Textbook: No required text. Curated lecture notes provided.
Suggested references (optional):- Folland, Gerald, Real Analysis
- Lax, Peter, Linear Algebra
- Munkres, James, Analysis on Manifolds
- John Hunter; Bruno Nachtergaele, Applied Analysis
- Software/Tools: LaTeX (Overleaf), Python (NumPy/SymPy/Jupyter) optional for checks/plots
Announcements
- See Canvas “Announcements” page.
Course Description
A rigorous treatment of multivariable analysis with emphasis on differentiability and integration in higher dimensions. Topics include metric spaces, continuity and compactness, differentiability in $\mathbb{R}^n$, Jacobians, $k$–volume via Gram determinants, change of variables, Fubini/Tonelli, and integral theorems (Green, Stokes, Divergence) with applications to data science, machine learning, and physics.
Learning Outcomes
By the end of the course, students will be able to:
- Topology of metric spaces: work with limits, Cauchy sequences, completeness, compactness, and connectedness; prove standard results.
- Differentiability in $\mathbb{R}^n$: compute and reason with Jacobians; apply chain, inverse, and implicit function theorems.
- $k$–volume & Gram determinants: compute volumes of $k$–parallelepipeds; use Pythagorean identities for $k$–volume.
- Measure theory: understand the construction of Riemann integration; Lebesgue measure, general measure space; motivation and construction of Lebesgue integration.
- Integration theory: state and apply Fubini/Tonelli; perform multivariable change of variables with $C^1$–diffeomorphisms.
- Integral theorems: state and apply Green’s, Stokes’, and the Divergence theorems; interpret flux/circulation and orientation.
- Communication: write clear, self-contained solutions with precise definitions and correct citations of theorems.
Prerequisites / Expected Background
- Prior courses in single-variable analysis and linear algebra (or equivalent).
- Comfort with $\varepsilon$–$\delta$ proofs and matrix calculus.
Materials & Resources
- Primary: Instructor’s lecture notes (posted weekly).
- Supplemental: Spivak; Marsden–Tromba; Hubbard–Hubbard (see above).
- Computation: Optional Jupyter notebooks for visualization/checks.
- Help: Office hours; Ed Discussion (linked from Canvas); TA hours; campus tutoring.
Assessments & Grading
- Homework : 00% — optional.
- Quizzes (weekly): 30% — weekly test.
- Attendance (weekly): 30% — participation.
- Final Exam (Week 6): 40% — comprehensive (100 minutes).
Letter Grades: Based on total percentage; small curve possible at instructor’s discretion.
Regrade Policy: Submit within 7 days of return via Canvas with a clear explanation.
Homework & Collaboration
- Discuss ideas with classmates; write up independently and cite collaborators/resources.
Exam Rules
- Closed book; 1 double-sided page of notes permitted; no electronic devices.
Tentative Schedule (accelerated 6-week summer term)
Topics reflect what we covered in Summer 2025. Readings = lecture notes section numbers; optional text references in parentheses.
| Week | Topics | Readings | Deliverables |
|---|---|---|---|
| 1 | Metric spaces: open/closed, limits, Cauchy, completeness; compactness (Heine–Borel in $\mathbb{R}^n$), connectedness; continuous maps | Notes §1–2 (Spivak Ch. 1–2; Hubbard Ch. 1) | HW1 out (Thu) |
| 2 | Differentiability in $\mathbb{R}^n$: linear maps, Jacobian, chain rule; inverse & implicit function theorems; directional derivatives vs. differentiability | Notes §3 (Spivak Ch. 2–3) | HW1 due Tue; HW2 out; Quiz 1 Thu |
| 3 | $k$–frames and $k$–volume: Gram matrix, Gram–determinant; Pythagorean identity for $k$–volume; orthogonal invariance | Notes §4 (Marsden–Tromba Ch. 4 App.) | HW2 due Thu; HW3 out |
| 4 | Parametrized manifolds & area element $J_\alpha$; change-of-variables in $\mathbb{R}^k$ (diffeomorphisms); Fubini/Tonelli (hypotheses & counterexamples) | Notes §5–6 (Spivak Ch. 5) | HW3 due Tue; Midterm Thu |
| 5 | Green’s theorem (circulation & flux forms), divergence theorem in plane; orientation, flux & circulation; examples (area via Green) | Notes §7 (Marsden–Tromba Ch. 6–8) | HW4 out; Quiz 2 Thu |
| 6 | Stokes’ theorem in $\mathbb{R}^3$; surface integrals, normal form, geometry; synthesis & review; exam prep | Notes §8 (Spivak Ch. 5) | HW4 due Tue; Final Exam Thu |
How to Succeed
- Practice steadily. Start problem sets early; work beyond the assigned minimum.
- Prove carefully. State definitions/theorems you use; justify each step.
- Check hypotheses. For results like Fubini/CoV/IFT, verify assumptions (e.g., $C^1$–diffeomorphism, absolute integrability).
- Reflect. After feedback, correct mistakes and summarize take-aways.
Downloadables
- Syllabus (PDF): (posted on Canvas)
- Lecture Notes (PDF): LaTeX & PDF on Canvas
Version History
- v2025.1 (06/24/2025): Summer 2025 launch; schedule & policies posted.
Page maintained by Shizhou Xu (Summer 2025).