# Math 25b: Honors Linear Algebra and Real Analysis II, Spring 2018

*Announcements*

- Thanks for the wonderful year!
- Once grades are released, if you have questions or would like to see your exam, please email me (it's best to do this sooner rather than later).

*Course Information*

We'll study functions of several variables, broadly referred to as mathematical analysis. This will include differentiation, integration, and rigorous proofs of analogues of the classical theorems of calculus in a broad setting. The plan is to work through Spivak's *Calculus on manifolds*,* *using Munkres' *Analysis on manifolds* as a guide. One of the central goals of the course is to introduce you to abstract mathematical thinking. Emphasis will be placed on careful reasoning and on learning to understand and construct proofs. Prerequisites: any of Math 23a, 25a, 55a. Many students find 25b more difficult than 25a, so if you took 25a, I would recommend taking 25b only if you got B- or above.

**Grading**: Weekly homeworks (30%), two midterms (40%), final exam (30%)

**Contact Info**

- Bena Tshishiku (Lecturer) bena at math.harvard.edu
- Joshua Lam (GCA) ylam at math.harvard.edu
- Ellen Li (CA) eyli at college.harvard.edu
- Charles O'Mara (CA) comara at college.harvard.edu
- Natalia M. Pacheco-Tallaj (CA) pachecotallaj at college.harvard.edu
- Michele Tienni (CA) micheletienni at college.harvard.edu

*Course Events*

**Class:** MWF 10-11am in SC 507

**Section: **Thursday, 6-7pm in SC 221. If you have special requests for what you'd like to be discussed during section, you can voice that here.

**Math Night (collaborative homework session): **Monday 8-10pm, Leverett House Dining Hall (more info)

**Office Hours. **

- Bena: Monday and Friday 3-4 in SC 232, or by appointment in SC 525
- Michele: Monday 8-10p at math night in Leverett Dining Hall
- Ellen: Monday 9-11p at math night in Leverett Dining Hall
- Josh: Tuesday 8-9p in Lowell Dining Hall
- Charlie and Natalia: sometime between 8-10ish pm in Lowell Dining Hall

### Important Dates

- Drop deadline: Monday, Feb 12
- Midterm 1: Wednesday, Feb 28
- Midterm 2: Wednesday, April 4
- Final exam: May 8, 9am-12

*Homework*

There will be 10 assignments posted below as the semester progresses. Homework will be due each Wednesday before 10am in the CA's mailboxes on the 2nd floor of the Science Center.

**Late homework policy:** For your homework grade, I will drop the score from your lowest assignment. View this as a one-time "get out of jail free card" in the event that you oversleep, have a midterm, etc. As a rule late homework will not be accepted. If you have a medical circumstance that prevents you from turning in homework, you will need a doctor's note to receive an extension.

**Collaboration: **You are encouraged to work together on the assignments (c.f. Math Night). In your solutions, you should acknowledge the students with whom you worked. For any solution you submit, you should understand it well enough that you can explain it to someone else and answer questions about it. If you find yourself writing down things that you can’t explain, you should go back and think more about the problem. Otherwise you’re doing yourself a disservice, and you might suffer more when it comes time to take the midterms/exam.

**LaTeX: **It is required that you type your solutions in LaTeX. You can either download LaTex or use sharelatex, which allows you download, edit, and compile LaTeX files online. See here for a guide to notation in LaTeX. Also Detexify is a useful tool for finding the commands for various symbols. The source code for the assignments should be a helpful guide.

- Due Jan 31: HW1, solutions, Extra Credit 1
- Due Feb 7: HW2, solutions
- Due Feb 14: HW3, solutions, Extra Credit 2
- Due Feb 21: HW4, solutions
- Feb 28: no homework due (midterm). Midterm 1 review page. Midterm 1 solutions.
- Due Mar 7: HW5, solutions, student solutions
- Mar 14: no homework (spring break)
- Due Mar 21: HW6, solutions, student solutions
- Due Mar 28: HW7, solutions, Extra Credit 3, student solutions
- Apr 4: no homework (midterm). Midterm 2 review page.
- Due Apr 11: HW8, solutions, student solutions
- Due Apr 18: HW9, solutions, student solutions
- Due Apr 25: HW10, solutions, student solutions

*Course materials*

- Lecture notes by Michele from the course.
- Main texts: Spivak
*Calculus on manifolds*and Munkres*Analysis on manifolds* - Supplementary texts. Spivak's book is dense. It can be slow going for beginners, but can be worth the effort. Munkres covers the same topics, but with more detail.
- The first three weeks of the course will be a brisk warm-up with chapters 4-11 of Spivak's
*C**alculus*. There's a pdf here. - Extra reference (no need to buy it): Hubbard-Hubbard,
*Vector calculus, linear algebra, and differential forms.*

- The first three weeks of the course will be a brisk warm-up with chapters 4-11 of Spivak's
- Review for Midterm 1.
- Review for Midterm 2.
- Review for the final exam.
- Learn more about planimeters.

*Topic schedule *

*Subject to change.*

- Week 1 (Spivak
*Calculus*Chapters 5-7)- Mon: Introduction, functions and limits
- Wed: Limits and continuity (algebra of limits)
- Fri: Continuity theorems (intermediate value, boundedness, maximum value theorems)

- Week 2 (Spivak
*Calculus*Chapter 8, Spivak*CoM*Chapter 1, Munkres Sections 1.3 and 1.4)- Mon: Least upper bound property (ordered fields, IVT, boundedness theorem)
- Wed: Compactness (subsets of Euclidean space)
- Fri: Heine-Borel theorem (onion rings, closed rectangles are compact, boundedness theorem)

- Week 3 (Spivak
*Calculus*Chapters 9-11)- Mon: Derivative in 1-dimension (definition, examples, chain rule)
- Wed: Computing derivatives (chain rule, Rolle's theorem, mean value theorem)
- Fri: Polynomial approximations (L'Hospital's rule, Taylor polynomials)

- Week 4 (Spivak
*CoM*Chapter 2, Munkres Sections 5-7)- Mon: Derivatives in R^n (directional derivatives, the derivative, examples)
- Wed: Partial derivatives, (continuous partials theorem, C^1 functions, higher-order derivatives)
- Fri: Chain rule, (applications: 1-d derivative rules, multivariable MVT, derivative of inverse)

- Week 5 (Spivak
*CoM*Chapter 2, Munkres Sections 8-9)- Mon: Presidents' day (no class)
- Wed: Inverse function theorem
- Fri: Inverse function theorem proof

- Week 6 (for manifolds: Hubbard Sections 3.0-3.2)
- Mon: Least upper bound property revisited
- Wed: Midterm
- Fri: Manifolds, definitions, linked rods in the plane, manifold recognition

- Week 7 (Hubbard 3.7)
- Mon: Tangent spaces (of a graph, of a level set), implicit function theorem
- Wed: Lagrange multipliers (intuition and proof)
- Fri: applications of Lagrange multipliers (proof of spectral theorem)

**Spring break**

- Week 8 (Spivak
*CoM*pages 46-56, Munkres sections 10-12)- Mon: The integral (Archimedes method of exhaustion, defining the integral over rectangles)
- Wed: Integrability criteria (measure 0 and content 0 sets)
- Fri: Computing integrals (fundamental theorem of calculus)

- Week 9 (Spivak
*CoM*pages 56-66, Munkres sections 16-17)- Mon: Fubini's theorem (proof and applications)
- Wed: Change of variables (introduction, 1-dimensional COV)
- Fri: Partitions of unity (existence theorem, bump functions, application to integration)

- Week 10 (Spivak
*CoM*66-74, Munkres sections 18-20)- Mon: Diffeomorphisms (properties, local decomposition into primitive diffeomorphisms)
- Wed: Midterm 2
- Fri: Change of variables (primitive diffeomorphisms, proof of COV, application)

- Week 11 (Hubbard sections 6.0-6.2, 6.4, 6.7-6.8)
- Mon: Alternating algebra (elementary k-forms, wedge product)
- Wed: Differential forms (examples in R^3, exterior derivative)
- Fri: Forms and integration (integrating a k-form on R^n over a k-cube, pullbacks)

- Week 12 (Spivak
*CoM*97-105)- Mon: Chains (examples, the boundary map, partial^2=0)
- Wed: Stokes' theorem (proof, examples)
- Fri: Stokes' application (winding numbers, fundamental theorem of algebra)

- Week 13 (Spivak
*CoM*122-137)- Mon: Application of Stokes (Green and divergence theorems; areas, volumes, surface areas)
- Wed (last class): Stokes application (Green's theorem and planimeters)

These images are related to Homework 8, problem 6 (Cavalieri's principle).