Math 25b: Theoretical real analysis, Spring 2019


Announcements

  • This page is under construction.

  • If you want to read ahead, I recommend perusing Pugh’s Real mathematical analysis. This will be a central reference, especially at the beginning of the course. There is a tentative topic schedule below.


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, 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.

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

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: TBD. Submit questions here.

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

Office Hours (draft: subject to change)

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

  • Tuesday: (Bena) 3-4p

  • Wednesday: (Bena) 10:30-11:30a in SC 530

  • Thursday:

  • Friday: (TBD) 9-10:15a

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

  • Midterm 2: Fri, April 5

  • FInal exam: TBD


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.

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

  • Due Feb 15: HW2

  • Due Feb 22: HW3

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

  • Due Mar 8: HW4

  • Due Mar 15: HW5

  • No HW due Mar 22 (spring break)

  • Due Mar 29: HW6

  • 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 several extra credit opportunities, which will allow you to explore topics beyond the lectures/textbook. These assignments are meant to be fun, and the extra points might help you to not stress too much about your grade. The due dates will be on the Friday after each midterm and also the last week of class. 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.

  • EC1


Course Materials

  • Main text:

  • Supplementary texts. When learning a topic, it's always a good idea to have multiple sources! Some sources we will use:

    • Pugh, Real mathematical analysis

    • Munkres, Analysis on manifolds

    • Spivak, Calculus on manifolds

    • Hubbard-Hubbard, Vector calculus, linear algebra, and differential forms

  • 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.

  • Math extracurricular:

    • Math Table, Tuesdays at 6p with free food.

    • Open Neighborhood Seminar, some Mondays 4:30p.


Topic schedule (tentative)

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

Week 1:

  • Mon. Lecture 1: Continuity, the real numbers, least upper bound property

  • Wed. Lecture 2: Skeleton of calculus, continuity theorems, Dedekind cuts

  • Fri.

Week 2:

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

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

  • Fri. Homework 1 due.

Week 3:

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

  • Wed. Lecture 6: Differentiability, mean value theorem, Taylor polynomials

  • Fri. Homework 2 due.

Week 4 :

  • Mon. President’s Day (no class)

  • Wed. Lecture 7: Polynomial approximation, second derivative test

  • Fri. Homework 3 due.

Week 5:

  • Mon. Lecture 8: Integrability, Riemann integral

  • Wed. Lecture 9: Continuous implies integrable, Fundamental theorem of calculus, measure

  • Fri. Midterm 1, extra credit due (least upper bound property, metric spaces/Cauchy sequences/R, series)

Week 6:

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

  • Wed. Lecture 11: Multivariable derivative, chain rule

  • Fri. Homework 4 due.

Week 7:

  • Mon. Lecture 12: Continuous partials theorem, multivariable MVT, higher derivatives

  • Wed. Lecture 13: Inverse function theorem, implicit function theorem

  • Fri. Homework 5 due.

Week 8:

  • Mon. Spring break

  • Wed. Spring break

  • Fri. Spring break

Week 9:

  • Mon. Lecture 14: Inverse function theorem, manifolds in R^n and implicit function theorem, Lagrange multipliers

  • Wed. Lecture 15: Implicit function theorem, Lagrange multipliers, more manifolds

  • Fri. Homework 6 due.

Week 10:

  • Mon. Lecture 16: Change of variables, Lagrange multipliers and spectral theorem

  • Wed. Lecture 17: Stokes’ theorem, Linear forms on vector spaces

  • Fri. Midterm 2, extra credit due

Week 11:

  • Mon. Lecture 18: differential forms, vector fields, examples

  • Wed. Lecture 19: differential forms, exterior derivative, vector calculus

  • Fri. Homework 7 due.

Week 12:

  • Mon. Lecture 20: chains, boundaries, pullbacks

  • Wed. Lecture 21: Stokes’ theorem proof

  • Fri. Homework 8 due.

Week 13:

  • Mon. Lecture 22: Stokes’ theorem and vector calculus

  • Wed. Lecture 23: Stokes’ application: Green’s theorem

  • Fri. Homework 9 due.

Week 14:

  • Mon. Lecture 24: Stokes’ application: fundamental theorem of algebra

  • Wed. Lecture 25: Stokes’ application: Brouwer fixed point, HW10 due.

  • Fri. Extra credit due.