# Math 25b: Theoretical multivariable calculus and real analysis, Spring 2019

*Announcements*

(3/17).

**HW6**comments (due March 29):Problem 4: Assume that f is defined on all of R. If f is only defined on [0,1], you’ll run into rouble defining f_n on [0,1].

Submit questions for

**section**here.Math extracurricular:

**Math Table**, Tuesdays at 6p with**free food**.**Open Neighborhood****Seminar**, some Mondays 4:30p.

*Course Information*

We'll study functions of several variables, broadly referred to as mathematical analysis. This will include differentiation, integration, and rigorous proofs of the classical theorems of calculus in a broad setting. The plan is to work through Pugh’s *Real mathematical analysis*. 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.

**Grading**: Weekly homework (30%), two midterms (20% each), final exam (30%)

*Remark on grading: homework is a relatively small percentage of your grade. The main reason for this is that the homework is collaborative, and I think that your final grade should be mostly based on what you can do by yourself. This semester the midterms will be take-home, so time constraints should be less of an issue. *

**Contact Info**

Bena Tshishiku (Lecturer) bena at math.harvard.edu

Yu-Wei Fan (GCA) ywfan at math.harvard.edu

Joseph Feffer (CA) jrfeffer at college.harvard.edu

Davis Lazowski (CA) dlazowski at college.harvard.edu

Beckham Myers (CA) bmyers at college.harvard.edu

Laura Zharmukhametova (CA) lzharmukhametova at college.harvard.edu

*Course Events*

**Class: **MW 9-10:15am

**Section: **Tuesdays 4:30-5:30 in SC 530. Submit questions here.

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

**Office Hours (draft: subject to change)**

Monday: (Davis) 8-10p in Leverett dining hall (math night)

Tuesday: (Bena) 3-4p in SC 304

Wednesday: (Bena) 10:30-11:30a in SC 530, (Laura) 9-10 in Lowell dining hall

Thursday: (Yu-Wei) 3-4 in SC 232, (Beckham) 8:30-10p in Lowell dining hall

Friday: (Joey) 9-10:15a in SC 411

Or by appointment (email me).

**Important dates (see also ****here****)**

Registration deadline: Fri, Feb 1

Drop deadline: Mon, Feb 11

Midterm 1: Fri, March 1, due at 11:59pm

Midterm 2: Fri, April 5

FInal exam: May 16, 9am

*Homework*

There will be 10 assignments posted below as the semester progresses. Homework will be due each Friday, submitted on Canvas.

**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. Furthermore, a note that merely says “Student visited HUHS today” is not very helpful; it’s better to have a note that says something like “Student may be unable to complete assignments for the next N days.” This will help me decide what length of extension is appropriate.

*Late submissions to Canvas*: Homework must be submitted to Canvas by 5pm each Friday. There will be a 3-point deduction for each minute after 5pm that the assignment is submitted. Don’t wait until the last minute!

**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, and you should write your solutions on your own. 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 (note that HW is only 30% of the final grade).

**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.

Homework assignments.

Due Feb 8: HW1, solutions, student solutions

Due Feb 15: HW2, solutions, student solutions

Due Feb 22: HW3, solutions, student solutions

No HW due Mar 1 (there will be a midterm)

No HW due Mar 22 (spring break)

Due Mar 29: HW6 and you will also need this file (in the same folder as your TeX file).

No HW due Apr 5 (there will be a midterm)

Due Apr 12: HW7

Due Apr 19: HW8

Due Apr 26: HW9

Due May 1: HW10

**Extra credit: **Throughout the semester there will be (at least?) 8 extra credit opportunities, which will allow you to explore topics beyond the lectures/textbook. These assignments are meant to be interesting/fun, and the extra points might help you to not stress too much about your grade. Please note that the **extra credit is “extra”**, i.e. optional, and if your HW/exam scores are good, you won’t need the extra credit to get a good grade in the course.

Extra credit assignments.

Turn in before March 8, 5pm:

Turn in before April 12, 5pm:

*Course Materials*

Main text: Pugh,

*Real mathematical analysis*(you can an online copy from Harvard’s library for free)Supplementary texts. We won’t use these directly very much (so you don’t have to buy them), but if you want an addition resource, one of these might help.

Munkres,

*Analysis on manifolds*Hubbard-Hubbard,

*Vector calculus, linear algebra, and differential forms*Spivak,

*Calculus on manifolds*

If you have topics you’d like to hear discussed in

**section**, please submit them here.**Homework comments/tips**from the CAs can be found here.Midterm 1 materials

Midsemester feedback form.

*Topic schedule (tentative)*

This schedule will be updated as we go along. I’ve included pointers to the textbook.

**Week 1: **Pugh 1.2, 1.3, 1.6

Mon. Lecture 1:

**Continuity**, the real numbers, least upper bound propertyWed. Lecture 2: Skeleton of calculus, continuity theorems, Dedekind cuts

Fri.

**Week 2: **Pugh 2.1, 2.2, 2.4

Mon. Lecture 3: Convergence, sequences, limits, continuity

Wed. Lecture 4: Topology, open/closed sets, compactness/Heine-Borel, continuity

Fri. Homework 1 due.

**Week 3: **Pugh 2.4, 3.1

Mon. Lecture 5: More topology, interior/exterior/boundary, compactness and coverings, the Cantor set

Wed. Lecture 6:

**Differentiability**, mean value theorem, Taylor polynomialsFri. Homework 2 due.

**Week 4 : **Pugh 3.1

Mon. President’s Day (no class)

Wed. Lecture 7: Polynomial approximation, second derivative test

Fri. Homework 3 due.

**Week 5: **Pugh 3.2

Mon. Lecture 8:

**Integrability**, Riemann integralWed. Lecture 9: Continuous implies integrable, Fundamental theorem of calculus, measure

Fri. Midterm 1 due at 11:59pm

**Week 6: **Pugh 3.2, 4.1, 4.5

Mon. Lecture 10: Integrability and measure, Cavalieri and Fubini

Wed. Lecture 11: Function spaces, ODEs

Fri. Homework 4 due, extra credit due (LUB property, Cantor’s construction of R, compactness)

**Week 7: **Pugh 4.3, 4.5

Mon. Lecture 12: Function convergence, equicontinuity, Arzela-Ascoli theorem

Wed. Lecture 13: Arzela-Ascoli theorem, ODE existence theorem

Fri. Homework 5 due.

**Week 8: **

Mon. Spring break

Wed. Spring break

Fri. Spring break

**Week 9: **Pugh 5.1, 5.2, 5.3

Mon. Lecture 14:

**Multivariable derivative**, chain ruleWed. Lecture 15: Continuous partials theorem, multivariable MVT, higher derivatives

Fri. Homework 6 due.

**Week 10: **Pugh 5.5, 5.6

Mon. Lecture 16: Implicit function theorem

Wed. Lecture 17: Implicit and inverse function theorems

Fri. Midterm 2, extra credit due

**Week 11: **Pugh 5.6, 5.7

Mon. Lecture 18: manifolds and Lagrange multipliers

Wed. Lecture 19:

**Stokes’ theorem**, Linear forms on vector spacesFri. Homework 7 due.

**Week 12: **

Mon. Lecture 20: differential forms, vector fields, examples

Wed. Lecture 21: differential forms, exterior derivative, vector calculus

Fri. Homework 8 due.

**Week 13: **

Mon. Lecture 22: chains, boundaries, pullbacks

Wed. Lecture 23: Stokes’ theorem proof

Fri. Homework 9 due.

**Week 14: **

Mon. Lecture 24: Stokes’ theorem and vector calculus, Green’s theorem

Wed. Lecture 25: Stokes’ applications: FTA, Brouwer fixed point, HW10 due.

Fri. Extra credit due. (planimeters)