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

*Announcements*

- This page is in its early stages and will likely change during the next month.

*Course Information*

We'll study of the geometry of 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

*Course Events*

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

**Section:**

**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

### Important Dates

- Drop deadline:
- Midterm 1: Wednesday, Feb 28
- Midterm 2: Wednesday, April 4
- Final exam:

*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.

- Jan 31:
- Feb 7:
- Feb 14:
- Feb 21:
- Feb 28: no homework due (midterm)
- Mar 7:
- Mar 14: no homework (spring break)
- Mar 21:
- Mar 28:
- Apr 4: no homework (midterm)
- Apr 11:
- Apr 18:
- Apr 25:

*Course materials*

- 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

*Topic schedule *

*Subject to change.*

- Week 1
- Mon: Introduction, functions and limits
- Wed: Limits and continuity
- Fri: Continuity theorems

- Week 2
- Mon: Least upper bound property
- Wed: Compactness
- Fri: Heine-Borel theorem

- Week 3
- Mon: Derivative in 1-dimension
- Wed: Computing derivatives
- Fri: What does Df know about f?

- Week 4
- Mon: Derivatives in R^n
- Wed: Partial derivatives
- Fri: Chain rule

- Week 5
- Mon: Presidents' day (no class)
- Wed: Inverse function theorem
- Fri: Implicit function theorem

- Week 6
- Mon: Least upper bound property revisted
- Wed: Midterm
- Fri: Quadratic forms

- Week 7
- Mon: Manifolds and tangent spaces
- Wed: Lagrange multipliers
- Fri: Taylor polynomials

**Spring break**

- Week 8
- Mon: The integral
- Wed: Integrability criteria
- Fri: Computing integrals (fundamental theorem of calculus)

- Week 9
- Mon: Fubini's theorem
- Wed: Change of variables
- Fri: Partitions of unity

- Week 10
- Mon: Diffeomorphisms
- Wed: Midterm 2
- Fri: Change of variables (proof)

- Week 11
- Mon: Alternating algebra
- Wed: Differential forms
- Fri: Chains

- Week 12
- Mon: Stokes' theorem
- Wed: Stokes' theorem on manifolds
- Fri: Application of Stokes to vector calculus in R^3

- Week 13
- Mon: Stokes applications
- Wed (last class): Stokes applications