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


  • meme submissions. Thanks for a fun year!

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

  • Yu-Wei Fan (GCA) ywfan at

  • Joseph Feffer (CA) jrfeffer at

  • Davis Lazowski (CA) dlazowski at

  • Beckham Myers (CA) bmyers at

  • Laura Zharmukhametova (CA) lzharmukhametova at

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 12, due at 11:59pm

  • FInal exam: May 16, 9am in SC Hall E


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.

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 14, 11:59pm:

  • EC4

  • EC5 you will also need this

  • EC6 you will need to include this image in the same folder as the TeX file.

Turn in before May 12, 11:59pm:

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.

  • Valentine’s day fun here and here.

  • Midterm 1 materials

  • Midsemester feedback form.

  • Mathematica code from class 4/8.

  • Midterm 2 materials

  • Submit questions for section here.

  • Final exam review material (topic list and practice problems).

  • 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.

Week 1: Pugh 1.2, 1.3, 1.6

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

  • Wed. 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 polynomials

  • Fri. 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 integral

  • Wed. 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 rule

  • Wed. 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. Homework 7 due.

Week 11: Pugh 5.6, 5.7

  • Mon. Lecture 18: manifolds and Lagrange multipliers

  • Wed. Lecture 19: Stokes’ theorem, Linear and differential forms

  • Fri. Midterm 2

  • Sun: extra credit due (Taylor polynomials, log/exp, pictogram)

Week 12:

  • Mon. Lecture 20: integration of differential forms

  • Wed. Lecture 21: exterior derivative and pullbacks

  • Fri. Homework 8 due.

Week 13:

  • Mon. Lecture 22: chains and boundaries

  • Wed. Lecture 23: Stokes’ theorem proof, Brouwer fixed point theorem

  • Fri. Homework 9 due.

Week 14:

  • Mon. Lecture 24: Stokes’ applications: Green’s theorem, FTA

  • Wed. Lecture 25: Stokes’ applications: FTA, planimeters

  • Fri. HW10 due.