Mathematical and Computational Methods

I am writing the following review having completed the course except for the epilogue and the final. It's difficult for me to write a review because I was familiar with most of the material, and so I must try and write from the point of view of a sophomore level physics major whom the course is aimed at - with prerequisites including 3 semesters of calculus (including multivariable) and knowledge of introductory physics including classical mechanics and electromagnetism, and a small amount of relativity and quantum mechanics. I will also add some of my thoughts based on my lifetime experience as a physics student and wannabe theoretical physicist. Apologies in advance for what will be probably a long review.

The course lasts 18 weeks (3/4 of the weeks were exam weeks) and costs $250 for a verified certificate. I thought the price was on the high side (I don't know who determines the price, but I don't think it is Professor Freericks). In comparison the Quantum Mechanics MITx 8.05x, and 8.06x courses - each 15 week courses charge $99 for a certificate. Most of the material was generously made available even to the non-verified students except the problem sets (which were available to non-verified students in the form of pdf downloads) and the exams. This has the downside of making the sometimes excellent Labs and In class problems/exercises ungraded. I think it was overall a good tradeoff.

The Professor of the course Jim Freericks was practically available 24/7 in moderating the course, and in making corrections to the grader when necessary. He is to be commended for both the amount of time putting together the course as a MOOC, and for the devotion to the course while it was running. I don't think you can get better service for a course live or online.

The course is titled Mathematical and Computational Methods, but it is really only a mathematical methods of physics course - there was a time when the professor did attempt to incorporate computational and programming aspects into the course at Georgetown, but apparently that did not work out well, and so the computational material was removed - I will discuss this later.

The only MOOC course that I am aware of that covers similar material is the MITx 18.03x Differential equations Series which is a one semester course that has been broken up into 5 mini courses and is even more expensive than this course at $382.50 and which covers ODE's, linear algebra, NxN systems, Fourier series, partial differential equations, and the Laplace Transform. The MITx course is a nice course (it could and should be more challenging) but is a little narrower in scope not covering vector calculus, or complex analysis, and little theory with no proofs in the course. I would probably recommend this course over that series.

The professor decided to use five relatively inexpensive books for the course, instead of using one of the standard textbooks like Boas. This saves the student some money, but also deprives them of a useful reference book on mathematical physics, as books like Boas/Arfken usually cover more material (a years worth at least) including special functions, PDE's, probability and statistics, calculus of variations and tensor analysis. So another tradeoff, which I think worked well as the coverage of the material by the professor was less encyclopedic and more idiosyncratic.

The course is designed to cover six areas - calculus review, vector calculus, complex analysis, linear algebra, differential equations, and Fourier series. I will give comments on each in turn.

Calculus review: This part was a long 3 weeks at the start of the course. About half was standard calculus material that I would recommend be dropped, and half was interesting new material that students will definitely profit from. Thus I would recommend dropping some of the historical material on sine tables and logarithms - interesting but no loner vital for students. I would also definitely drop much of the multiple variable integration material (moments of inertia) as it is a prerequisite anyway. There was excellent material on summing inverse powers of integers, Frullani and Gaussian integrals, Feynman parameters/integration, and hyperspheres. I think this part of the course should be geared towards developing more tricks and techniques in calculus rather than a review. If the material I recommended be dropped it could be replaced by adding material on Stirling's formula, Lagrange multipliers, Gamma function, Bernoulli numbers, Euler-Maclaurin formula, evaluating symmetric integrals in 3 dimensions (the type need for cross sections in QM), and more coverage of the Dirac delta function, and even perhaps a little variational calculus.

Vector calculus - While the standard material was covered, I think there were too many problems on calculating stuff like surface areas and volumes for which this material was not really designed - if there were less of the plug and chug for calculating line integrals etc. more time could have been devoted to more theoretical material like expressing Div, Curl, Grad in general orthogonal coordinate systems, and even introducing tensors and some beginning tensor analysis. . There were excellent labs on scattering, and some excellent material on Laplace's equation, and relaxation methods.

Complex analysis - this seemed to me the weakest part of the course, which is a shame as this material is probably the most valuable for a physics student. For some reason the material was compressed into 1/2 weeks, and covered way too fast in my opinion. And there were too few problems on computing integrals, with all the sorts of various contours you see in the complex analysis textbooks neglected. In this part more so than in the other parts more examples and problems would have been quite useful.

Linear Algebra - While one needs to cover things like row operations and reduction, I think more of an emphasis on theoretical material (which was covered well) rather than some of the endless calculation exercises on determinants and inverses (and normalizing algebraic expressions for eigenvectors) would be more useful. Coverage of Pauli matrices was good as was material on certain special matrices.

Differential equations - I think this was the strongest part of the course. Theoretical material like the Wronskian were well covered. Perhaps too many problems for my taste, once you solved one linear first order differential equation - you have solved them all. I think undetermined coefficients should have been covered before variation of parameters.

Fourier analysis - This was the most enjoyable part of the course for me. The short book assigned for the reading did a nice job of proving some theorems that are difficult to prove in their generality. Often courses in mathematical physics neglect these convergence issues, but this was well covered here. An expansion to Fourier transforms would have been a nice addition.

Professor Freericks believes in a lot of practice, and sometimes too long algebra problems. While I don't shy away from such problems (I wrote a paper for 8.06x which required considerable algebra), I am not in favor of endless practice and computation unless it meets a specific purpose. As physics students progress, the medical model of see one, do one, teach one becomes more useful in my opinion. Thus I would eliminate the normalizing eigenvectors algebraic problems ( or at least limit it to one). The overall workload in the course was too high in my opinion. Eliminating many of the plug and chug problems and keeping the more interesting insightful exercises, would I think improve the course. The exams were on the easy side - but that's fine, the problem sets were good, but the In class problems were often painful for me. The labs were quite good, and often the best part of the course.

If I were designing a math methods course (and of course I am not a professor and far from one), I would definitely incorporate symbolic algebra programs like Maple, Mathematica, Sympy, Sage into the course. Half the problems would be done by hand (the computationally less intensive), and the other half would be done by a computer algebra system (designed to be close to impossible to do by hand). This would introduce students to the amazing developments in symbolic computation that have occurred over the past few decades.

Finally I would like to personally thank Professor Freericks for offering this and the other Quantum Mechanics courses that he has developed.

I want to make clear in advance that this course's main target is possibly not being a MOOC, but might be just an "I Want YOU", to dig up as many clever and interested oldies (maybe students too) as possible, and check via the course forum whether some might fit, getting recruited into some "Deep Citizen Science by Design" project (ad majorem gloriam?). Should not stop your enrollment!

My main interest in MOOCs of the Math-Physics spectrum is based on the educational and didactic methodology of these topics and it's been a while since I took in 2017 the last of about 50 edX-courses, and sadly, my impression of MOOCs (as genuine MOOCs) being dead is confirmed. Almost no(!) life in the forum, mostly only retirees (main target group of the project!) partaking, and in the finals: "< 10" ??? (not having asked yet for the number). And this course, imho!, will not bring us glory days MOOCs back. Neither the factual content, nor its presentation, nor the "open" exercises, nor the poor liveliness of the forum motivate me to a high ranking. I did not do any of the Khan's, but the worst Coursera I went through back then, possibly plagued by an "overmotivated" educator (too much of self-esteem in my judgement), like possibly in this course too, would rank similar like this. The "openness" got even more restricted: I did not pay $ 250.00 to see the exams and the problem sets and get their automated grading. Even just viewing the problems is locked by the promoting edX, even though the instructor tried to alleviate this. If you are into learning math-physics, I recommend to invest this amount in five or so pertinent books.

In details:

The title is fully misleading, there is nothing computational (beyond by hand) in it, and the instructor even explicitly expresses his resentment to CAS.
Any -even just slightly- mathematically inclined person can only look with utter disgust about the lack of any traces of rigor at the methodology "practiced" in this course. Trying to sell sloppily applied and sloppily formulated identities as "devious(!) tricks" looks more like immature sales argumentation than is really helpful; striving for being a "practitioner" instead of an "academic" fits the same juvenile category. Substituting "french cooking" (extensively long sequences of trivial manipulations) for structural precision is way to much primary guideline for this course.
The best of the course is the permanent -often helpful but not always to the point- presence of the instructor in answering the (mostly trivial) questions in the forum. If it were not for some crumpled warhorses among the attendants, there (almost?) would not be a single, interesting question in the forum. Of course, there are no CTAs -who would need them for the few students- anymore to bare the blame of ... disaster. You know. the eye of the lord makes sheep(le) wooly and fat.
The "lecture" videos are the most boringly read out *.ppt-slides ever. Imagine listening to spoken index-expressions, sometimes with plus's, sometimes with spoken sigma with bounds ... Reading the lecture notes for yorself might leave you more attentive.
There is a wealth of exercises, that I did not experience as remarkably paradigmatic or abundant, but they are enough to exhibit the still same detriments of the (unavoidable) automated edX-grading. There is no visible advance in presenting exercises, compared to the years before, and the "Explanations" (= solutions) are dire, if not even misleading.
I consider most of the mathematical content not only as too superficially, but also as rather old-fashionedly (especially the geometric parts) treated for an academic level. After some boring tour through ODE-routine don't hope for a compensating grande finale in the course: superficiality in Differential Geometry and Fourier matter (both condensed to one week's matter) can increase to almost arbitrary level.

I thoroughly valued and enjoyed MOOCs in their high times, but I do not dare to judge that they still deserve funding, when looking at this specimen: for an academic setting I consider this as: wrong target (practitioner?!), wrong staff (math indisposed), wrong methods ("french cooking"), ... Am I proven wrong by an unknown success? ... for whom?

I can not express my appreciation for this fantastic course using my limited English capability, but this course really intrigues me and gives me a stepping stone to higher professional career in the computational world:

* Highest quality of Math course I've ever taken, not limited to MOOC courses. It contains all the core modules you need to learn and master before going to dive into more advanced courses or research related to computational math/physics. It contains following modules:
- single variable calculus
- vector calculus
- complex analysis
- linear algebra
- differential equations (mostly ODE but a little bit PDE)
- Fourier analysis
- (A little bit of, but intriguing) differential geometry

You may worry about that depth of this course given its so broaden scope. Don't worry about that. The content is deep enough. The total course lasts 18 weeks, which is identical to the offline course. I've already got my bachelor degree in Engineering several years ago, but it still took me 4-8 hours per week to finish the weekly test (graded). On the other side, don't expect too much from a single course, this course can definitely prepare yourself to step into the more graduate-level advanced world. After this course, you will know what you should do next clearly. For example, after this course, based on my current career, I am eager to learn more on these topics: complex analysis, calculus of variation, differential geometry, numerical algebra and numerical methods in PDE. Compared to myself before Jan 2023, right now, I feel much confident to tackle the above obstacles one by one in the next 12-24 months. I won't feel disoriented any more.

* Most hard-working professor I've ever seen in terms of education and teaching. In the past 4 months, I raised many questions in the discussion forum and Professor Freericks always gave the response within 4 hours (usually within 2 hours) no matter it is 10 PM in weekdays or 9 AM in weekends. He is very patient to give the hints to guide me to the solution through a series of posts back and forth. I can not imagine anybody can do more on teaching students like this!

* Most sophisticated lecture note I've every seen. This course has a 280-page PDF note containing all the knowledge and examples it covers. Although it has not been officially published yet, I'd like to say it is a very valuable textbook you should have as a reference even if you don't have time to enroll.

At end, although I've passed the course and got the certificate, I still have some pending questions to investigate during the course since I was too busy to solve all questions popping up in my mind in past 4 months while working as a full-time employee at a tech startup. I will conclude this review by a sentence I learned from this course: "Be a practitioner, not a technician. Always think. Always ask why."

Full disclosure: I am a math teacher and I have been taking this course on audit. I am about a quarter of the way through and I am VERY impressed.

First, I have learned a ton. I actually have an MS in math but I have done things in this class that I have never done or seen before (1/n^4 infinite sum, Frullani integrals). I think the class strikes a wonderful balance between being insightful, and giving challenging problems to work on that are within reach. There is something here for everyone.

Second, I am working toward physics without much of a background so I really appreciate the addition of physical labs and derivations.

Third, and perhaps most impressive of all, the professor has been very active on the message boards responding to student questions, and adding clarifying points and references along the way. It's not a class where you don't have access to the professor. It's the opposite here.

If you are looking for a great post-calculus class on the math/theoretical physics side, you can't do better in the online era of 2023.

This course on mathematical methods teaches a fair amount of what you need to know for upper division undergraduate science courses. I previously had courses in calculus, linear algebra, and differential equations, and about half of the content of this course was new to me.

In particular, the Frullani integral, so-called "Feynman" integration, contour integration, solving non-linear first order differential equations, solving linear equations using the Wronskian matrix, and the Frenet-Serret formulas in differential geometry were entirely new.

The course also goes over vector analysis (which you'd get in a multivariable calculus course), linear algebra (an inevitable part of nearly any course beyond single-variable calculus), some elements of complex variables, linear differential equations, and Fourier series.

I found the problem sets and exams reasonably challenging but manageable. Depending on your previous preparation, you might find yourself spending considerably more time than the estimate of 7 to 10 hours in the course description. This is partly because the problems may take longer to do than you expect, partly because the video lectures will take longer (Professor Freericks talks fast and I had to keep pausing the video to follow what he was saying), and partly because of the assigned (optional) supplementary reading. He recommends that you buy four inexpensive paperback books and provides recommended readings in them, and also has class notes to supplement the lectures.

There's some overlap between between the linear algebra and differential equations part of this course and the five part MITx course in differential equations, but less than you might expect. It's definitely worth taking both if you want to study undergraduate physics.

Of course there's a lot more math to learn if you want to be proficient as a physics student, but this course is a good start, and it's the only online course I'm aware of that covers this material. If you want to learn more you'll need to work through a textbook or several. Recommended.

I enjoyed taking this course on the edx platform and learnt quite a lot but that does not make me a practitioner yet since that takes a lot of practice. The best features of the class were the in class problems, the problem of the week and the active participation of the instructor Prof. Freericks.

The problem sets and the exam were on the easier side and the pass percentage of 70% was quite easy to achieve. Another good feature about the course is that the instructor's notes are available in the pdf format along with the other free to use text books and coupled with an active discussion forum where fellow students and Prof. Freericks take part you are guaranteed to learn the concepts well. Even though I knew this material but still learnt a lot that I had missed when I self learnt this subject.

The one problem I had with the course is following the video materials since they are not blackboard or light screen format but powerpoint slides that the instructor reads from and so some of the algebraic manipulations that are presented are hard to follow but the trick is to first read the textbook and then watch the videos.

Overall I am happy that I took this course and learnt quite a lot and now it is time for me to practice, practice, practice and this is the only way I can become a practitioner.

I highly recommend this class. My expectations in taking this course were (a) to brush up on the engineering math courses I took more than 35 years ago and on which I performed poorly and (b) to obtain a foundation in mathematics which would help me understand advanced physics classes. This class met all my expectations and then some! The best part of the course was the selection of problems. The in-class (ungraded) problems immediately follow a lecture and a student who diligently solves these problems is guaranteed to master the lecture material. Similarly, the problems sets (graded) emphasize the course work and give the student a sense of satisfaction and confidence on getting the problem right. The multiple attempts provided at getting a problem right is a master stroke because it motivates the student to try our different approaches and this in itself foster learning and experimentation. The exams nicely summarize the material learnt over multiple lectures and forces the student to brush up on older material. I thought the exam problems a bit easier than those in the problem sets. I liked the labs which showed how the material learnt has practical application in physics. It is clear that Prof. Freericks sees beauty in the way physics problems are solved using math and, through his lectures, he does an excellent job of spreading his enthusiasm to his students. Prof. Freericks was always prompt (typically within an hour) in replying to questions and even the most minor comment was acknowledged. I invested roughly 10-12 hours a week on this class and that included reading the prescribed textbooks. My aim was to thoroughly understand the material and with this in mind I also solved some additional problems in the textbooks. I think at a minimum a 6-8 hours/week investment is needed to do justice to this class.

I am an adult learner with a Master's degree in Engineering and took this course via the audit option on EdX. This was a great sampling of myriad techniques I had not seen before in my studies. Had I known of some of these techniques when I was getting my BS and MS, I would have had an easier time in math-intensive physics courses. In particular the presentation on the Frullani and Gaussian integrals and Feynman integration was enlightening.

Without taking anything away from the course, I believe there could be great value in showing how correct answers could be shown to agree with simulated/numerical answers using Wolfram Alpha, Desmos, MATLAB, Excel, programming etc. This would be particularly valuable in Fourier series to show how the infinite sums of sinusoids approach the value of any function or how geometric sums could converge to a finite value.

This course covers a huge range of topics from "Devious math tricks", Feynman Integration, Infinitely dimensional Spheres, Vector Calculus, Complex Numbers, Linear algebra, Differential equations etc etc .

All topics are covered in lectures and full details are provided in the accompanying text book.

In addition to the lectures, there is plenty of course work with detailed answers. Some easy, some not so easy.

Also there are plenty of video labs to demonstrate the material taught.

Finally, Prof. Freericks promptly answers all questions posted by students on the course. This is really beneficial and was much appreciated by myself.

I have been fully occupied on this course for the past few months, and have learn a lot! I will definitely be looking at the two follow-on course on Quantum mechanics when they come around.