Math 25b: Theoretical real analysis, Spring 2019
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.
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%)
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
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
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
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.
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.
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.
Topic schedule (tentative)
This schedule will be updated as we go along. I’ve included pointers to the textbook(s).
Mon. Lecture 1: Continuity, the real numbers, least upper bound property
Wed. Lecture 2: Skeleton of calculus, continuity theorems, Dedekind cuts
Mon. Lecture 3: Convergence, sequences, limits, continuity
Wed. Lecture 4: Topology, open/closed sets, compactness/Heine-Borel, continuity
Fri. Homework 1 due.
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.
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)
Mon. Lecture 10: Integrability and measure, Cavalieri and Fubini
Wed. Lecture 11: Multivariable derivative, chain rule
Fri. Homework 4 due.
Mon. Lecture 12: Continuous partials theorem, multivariable MVT, higher derivatives
Wed. Lecture 13: Inverse function theorem, implicit function theorem
Fri. Homework 5 due.
Mon. Spring break
Wed. Spring break
Fri. Spring break
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.
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
Mon. Lecture 18: differential forms, vector fields, examples
Wed. Lecture 19: differential forms, exterior derivative, vector calculus
Fri. Homework 7 due.
Mon. Lecture 20: chains, boundaries, pullbacks
Wed. Lecture 21: Stokes’ theorem proof
Fri. Homework 8 due.
Mon. Lecture 22: Stokes’ theorem and vector calculus
Wed. Lecture 23: Stokes’ application: Green’s theorem
Fri. Homework 9 due.
Mon. Lecture 24: Stokes’ application: fundamental theorem of algebra
Wed. Lecture 25: Stokes’ application: Brouwer fixed point, HW10 due.
Fri. Extra credit due.