Physics and Mathematics Self-Learning Odyssey

July 2020 - November 2022

In July 2020, I started a big project, inspired by Scott H. Young's self-taught computer science journey. The goal? To cover the topics taught in an undergrad physics and math program. The journey was supposed to take about 18 months, but it took a bit longer - just over 2 years and 4 months. But, the experience was priceless. I based my study plan mostly on MIT's undergrad programs in physics and math, which are free on their MIT Open Courseware platform. Except for a few, I chose all my courses from this resource. Below, I've listed the courses, along with my notes and the problems I worked on.

Courses:

Trigonometry and Precalculus

Short Description: This was a course I mostly went through by going over several videos online. All in all, I would have spent more time doing problems, having lacked this made it more difficult when I went on to start calculus-but things got better.

Single Variable Calculus 18.01

Single Variable Calculus Notes

Single Variable Calculus Problems

Short Description: This is likely to be the most important math course you'll ever take, I had some difficulties at the start and I think that it was due to my weaknesses in Trigonometry and Precalculus coming into this, but surprisingly the struggle at the beginning highly improved my trigonometry and precalculus - more than I would’ve thought.

Multivariable Calculus 18.02

Multivariable Calculus Notes

Multivariable Calculus Problems

Short Description: Generalizes Calculus to 2 and 3 dimensions, working with multidimensional functions and generalizing many concepts along the way. The course content was really helpful as it had full recitations along with lectures containing wonderful intuitive explanations.

Differential Equations 18.03

Differential Equations are the mathematical creatures that model our daily life (Swings, Electrical Circuits, Transformers)…

Ordinary Differential Equations Notes

Differential Equations Problems

Short Description: The course content came with recitations that were very helpful and the lectures were nice but a little bit out of date in terms of the video quality. Most importantly this course had plentiful problems posted.

Linear Algebra 18.06

Linear Algebra Notes

Linear Algebra Problems

Short Description: Linear algebra forms the bulk of the math used in the applied sciences, it also proves to be very useful in the realm of pure math. In another course, I went through a professor who said “There is no such thing as learning too much linear algebra”. The main objects in this course were vector spaces since they serve to be a useful arena for realizing several concepts-such as geometrically describing systems of linear equations through matrices and interpreting matrices in more insightful ways. The course content was absolutely fantastic, along with the recitations the lecturer-(Gilbert Strang) was passionate, very humorous, and explained things simply and intuitively- without a doubt one of my favorite professors. Let’s just say that if you're studying Linear-Algebra, not watching this course would be a bit of a crime. On the other hand perhaps if someone is looking for a more formal introduction to Linear-Algebra, this might not be the first course to go through.

Introductory Classical Mechanics 8.01

Introductory Classical Mechanics Notes

Classical Mechanics 8.01 Problems

Short Description: This course worked through the elementary physics seen in mechanical systems. Projectile motion, harmonic oscillators, pendulums, rotating objects, and so forth. The lectures were truly amazing, professor Walter Lewin puts a lot of heart and passion into the lectures and tries to guide the viewers through physics by means of experiments as he himself says “Seeing is Believing”, the lectures come along with problems and exams all with solutions which made the learning process much easier.

Introductory Electromagnetism 8.02

Introductory Electricity and Magnetism Notes

8.02 Electricity and Magnetism Problems

Short Description: This course is the 2nd in the sequence of freshman physics courses. It was once again lectured by Professor Walter Lewin- very exciting. He layed out many interesting experiments, combining theoretical and experimental features into the lectures while leaving time for problem-solving in recitations. Like many of the other courses, I went through the course twice, having gone through it with a greater level of maturity the second time around filling in a lot of conceptual as well as practical gaps-but more importantly it lets me see things with fresh eyes.

Real Analysis 18.100B

Real Analysis Core Topics

Real Analysis Problems

Short Description: This course works through several of the same concepts as in calculus, now from a mathematically rigorous lens. This approach makes things so different that the courses share very few similarities. It gives you the necessary grounding that you will need as you go on to courses like Analysis 2, Measure theory, and Functional analysis- they will assume fluency. In so far as the resources I used, I mostly went through scattered resources online, fortunately, MIT graciously had me covered on the problem-solving side of things. The first time around I went through all the problems in the 18.100 B course, the second time around I went through several of the problem sets from 18.100C. This was the first proof-based course I’d ever taken, and it was indeed very challenging, especially so on a problem-solving level, I was definitely banging my head against the wall more than a few times with this course, nevertheless, I gained a lot not only in terms of the material but in how it molded my thinking.

Probabilistic Systems and Analysis

Probabilistic Systems and Analysis Core Topics

Probabilistic Systems and Analysis Problems

Short Description: Probability Theory is used all over the place- in the sciences and life. There’s a nice quote that captures this “Probability is common sense reduced to calculation”. The first time around I actually went through the 18.05 Introduction to Probability and Statistics course, but having gone and reviewed several courses in my challenge, I decided that going through 6.010 Probabilistic Systems and Analysis this time around would have been more beneficial as far as the topics it covered and the resources for it. In this course I mostly went through the Lecture videos by John Tsitsiklis along with his book on Probability theory, both were extremely insightful and you could really tell that he understood the material at a deep level. The course also came with recitation videos which were helpful, and more importantly, it came with a crap ton of insightful problems that helped get my problem-solving skills up to speed.

Complex Analysis 18.04

Complex Analysis Core Topics

Complex Analysis Problems

Short Description: This was one of my favorite math courses, despite having the word analysis in it, it didn’t feel nearly as ‘proofy’ as the course in Real Analysis- for better or worse. Fortunately, there were a decent bit of online resources going through the topics covered in this course, however the lecture notes posted were also quite rich in the examples they gave. The course also came with problems along with solutions which were of course very helpful.

Quantum Physics 8.04

Quantum Physics I Core Topics

Quantum Physics 8.04 Problems

Short Description: This course dealt with a lot of the foundational aspects of quantum mechanics, I would say that pretty much everything you learn in the course you will use later on in the more advanced physics courses. There were two lecture series associated with this course, one by professor Allan Adams and the other by Barton Zwiebach, the first time around I went through Barton’s course and then I went through Allan Adam’s Lectures, having done so benefited me more than I thought it would have, both courses overlapped for the most part except on their general focus and certain applications of quantum mechanics, Allan Adams often gives more physics insights and comparisons than Barton, while on the other hand Barton is more mathematical in his treatment and works more with the details, having gone through both courses was quite useful in seeing things from two different perspectives.

Introduction to Special Relativity 8.033

Introduction to Special Relativity Core Topics

Special Relativity Problems

Short Description: This course dealt with many of the core features of Special relativity as well as essential applications. The internet was rich with resources covering special relativity- at least at this level, which made it much easier to grasp the basics, the first time around I went through 8.033, but then having realized that sometime in 2021 MIT posted a new course on special relativity I went back and went through that course, solving the problems and watching several of the lectures.

Waves and Vibrations 8.03

Waves and Vibrations Core Topics

Waves and Vibrations Problems

Short Description: This course worked through much of the elementary wave-like and optical-like phenomena we experience in our daily lives. I first watched Walter Lewin’s lectures and then went through the more modern Yen Jie Lee’s course on MIT open courseware, I liked both treatments a lot, they overlapped a fair in the content they covered and the general philosophy they took in that they consistently combined both the experimental and theoretical aspects, however, every once in a while the tools they used and explanations they gave diverged. both perspectives were very useful to have and gave me a new means of seeing things.

Statistical Physics I 8.044

Statistical Physics Core Topics

Statistical Physics Problems

Short Description: Originally I covered this course by spending time with resources online and then went through some practice exams. Unfortunately, I didn’t find that many great resources online, I did however go through with the lecture notes thereafter. The second time I decided to go through every single problem set this course had with few exceptions, and also several other practice exams in order to improve my problem-solving skills.

Introduction to Topology 18.901

Introduction to Topology Core Topics

Introduction to Topology-Self-Made Notes

Introduction to Topology Problems

Short Description: This course was quite abstract in its concepts, the main motivation behind this course rested in generalizing many of the concepts that we find ourselves in Euclidean Space to more general spaces and the means by which we went about it was through open sets. This along with Algebra 1 were my favorite math courses in the whole challenge, I mostly studied from the book topology by James Munkres which is the standard text, the big downside of this course was that I could not find problems online with solutions, the problems I found most didn’t have solutions and the ones that did cover more advanced material. On the other hand during the beginning of May 2022 I made a trip to Canada and bought myself Schaum’s General Topology Outline which came with many problems and solutions, but also with wonderful explanations.

Functional Analysis 18.102

Functional Analysis Core Topics

Functional Analysis Problems

Short Description: This course worked with abstract spaces mostly in infinite dimensions. This course was by far the hardest math course in the whole challenge, it was both abstract in its concepts and difficult in its problems. I think one of the big mistakes that would’ve helped me do better would’ve been to have taken a more advanced course in linear algebra from a mathematician’s point of view- such as the book linear algebra done right by Sheldon Axler. On the other hand, this course came with problems as well as solutions that were quite insightful. The resources for this course were what gave me the hardest time, most of the ones I found online worsened rather than improved my understanding, as they left out much of the core ideas and were not as thorough. However towards the end of the challenge I managed to encounter myself with Frederich Schuller’s Quantum theory course, the first 7 lectures were all functional analysis and were fantastic, unfortunately, there were several other topics in the course that Frederich Schuller didn’t go through, but nevertheless, it made the basics much more grounded- and as I write this MIT just released a completely new course on Functional Analysis with video lectures and amazing problems…. sigh….

Atomic and Optical Physics I 8.421

Superfluidity in Bose-Einstein Condensates.pdf

Atomic and Optical Physics Problems

Short Description: This course covered the atomic structure of atoms, and studied the interactions of atoms with external fields and light. Toward the end of the course, it went through more advanced concepts such as the concept of coherence. This course was the hardest physics course I encountered throughout the challenge, the lecturer provided great insights, but I personally found that he often does a lot of steps in his head or took things as obvious- which made me bang my head against the wall more than a few times. The course had seemingly amazing problems but didn’t have any solutions. However later on I found another atomic and optical physics course from the University of New Mexico that came with problems and solutions.

Abstract Algebra I 18.701

Algebra 1 Core Topics

Algebra 1 Selected Notes

Algebra 1 Problems

Short Description: The main topic for this course was group theory, which along with Calculus and Differential Equations forms the backbone for much of the math used in the sciences. This course was among my favorites. The first time around I mostly used random resources I found online, and the 2nd time around when I went back to review it I watched through all the Harvard Lectures on the subject. Besides this, I also happened to find a great book on Abstract Algebra which had several problems relevant to this course which I did. Along with this I also went through some problems on MIT-d-space and solve some problems on there as well.

Classical Mechanics II 8.223

Classical Mechanics 2 Core Topics

Classical Mechanics II Problems

Short Description: This course dealt with many standard concepts one would find in a 2nd course in Undergraduate Classical mechanics. The main tools I used for studying this course were a lecture series by Profesor Jacob Linder among other resources online. The second time around I managed to discover resources from a past MIT Classical mechanics II course that came with plenty of insightful problems and solutions.

Introduction to Partial Differential Equations 18.152

Introduction to Partial Differential Equations Core Topics

Introduction to Partial-Differential Equations

Introduction to Partial Differential Equations Problems

Short Description: This course worked through many of the strategies used in solving general PDEs and applied them in the context of most of the well-known PDE’s in the Sciences. what I found quite interesting was that although the course did have a fair bit of computation it also developed techniques that had a much more mathematical analysis-like flavor, I often found myself using estimates, and inequalities and in general going through things in a rigorous proof-based style. For this course, I mostly used online random resources to study as I found that the lecture notes skipped a fair amount of steps and left me hanging. The course came with problems, most of them without solutions. The second time around I went through another P.D.E course that was in similar spirit to this one, called Introduction to Linear P.D.E’s, which did have several problem sets with solutions.

Quantum Physics II 8.05

Quantum Physics II Core Topics

Quantum-Physics-8.05-Notes

8.05 Quantum Physics Problems

Short Description: This was the 2nd quantum physics course in the undergraduate sequence, it reviewed and expanded on essential topics from 8.04 as well covering several other courses. The course had a full set of lectures by Professor Barton Zwiebach. Unfortunately, though the problem sets that were posted did not come with solutions. What I did instead prepared notes-explaining several of the concepts and went through problems from past qualifying exams and some problem sets from another quantum mechanics course I found made by the University of New Mexico.

Quantum Physics III 8.06

Quantum Physics III Core Topics

Quantum-Physics 8.06 Notes

The Lamb Shift and It's Consequences

Einstein's A and B Coefficient Theory

 Bell's Theorem and Quantum Teleportation

Short Description: This was the last in the sequence of undergraduate quantum physics courses, it largely deals with approximation schemes and how they could be applied in useful real-world contexts. The course came with a full set of lectures by Professor Barton Zwiebach, Unfortunately, though, the problems put out didn’t come with solutions, which made it a little difficult for me. I decided to go through a few questions from past qualifying exams, but largely my focus was placed on projects and notes. I wrote three papers describing concepts related but still relatively outside what we covered in the course, them being “The Lamb Shift and Quantization of the Electromagnetic Field”, “Bell’s Theorem and Quantum-Teleportation” as well as “Einstein’s A and B Coefficient Theory.-I also formatted them in Latex for anyone who’s interested in reading along

Electromagnetism II 8.07

Electromagnetism 2 Core Topics

Electromagnetism II Problems

Short Description: This was the second course in undergraduate Electromagnetism, it reviewed and expanded on a lot of the concepts seen in 8.02 Introductory Electromagnetism, and took them a lot further. In studying this course I mostly used the notes along with the required textbook Electromagnetism by Griffiths- which was absolutely amazing, and towards the end, I found a fantastic graduate electromagnetism course online by the University of New Mexico -which I went through most of the lectures. In terms of problems, the course had problem sets but with no solutions, fortunately, there were two exams, I went through the problems and posted them above. Besides this, I also went through several problems from past qualifying exams on MIT’s site.

Classical Mechanics III 8.309

Classical Mechanics III Core Topics

Advanced Classical Mechanics III Self-Notes

Short Description: This course reviewed several of the concepts in Analytical Mechanics and took many of them further. It covered extended topics such as Hamilton-Jacobi Theory, Chaos Theory, and Fluid Mechanics, which although somewhat non-standard quite interesting. For this course I mostly used the lecture notes along with several online resources, unfortunately, the problems it came with had no solutions, but very fortunately I went through some of the past classical mechanics 3 courses on MIT’s Dspace and found one that had very insightful problems and solutions- the only downside being that they didn’t have any problems involving fluid mechanics nor chaos theory. Furthermore, besides this, I went on to make notes explaining the concepts in latex.

Analysis on Manifolds 18.101

Analysis on Manifolds Core Topics

Analysis on Manifolds Notes

Analysis on Manifolds Problems

Short Description: The first part of the course took several concepts from real analysis and generalized them to n dimensions, while the latter half of the course dealt with certain aspects of differential geometry. This course gave me the hardest time in terms of resources, I had only the book and found it to be a tough read, more so in the latter half. This course didn’t come with problems, however, I did go through some of the problems from Munkres Textbook and compared them with solutions I found online. Later on, I discovered a course on General Relativity by Frederich Schuller, for which the first half contained mini videos consisting of solving problems involving the math used in general relativity- very similar to what the course went through, therefore I went through those and also went through several problems provided by a course Multivariable Calculus with Theory that MIT provided-which was essentially what the first part of my course covered.

Measure and Integration 18.125

Measure and Integration Core Topics

Measure Theory Notes

Measure Theory Problems

Short Description: This course covered a lot of ground, it developed many of the fundamental aspects of measure theory and towards the latter half dug further into some of the more advanced Theorems. I quite liked this course, especially the beginning foundational aspects; I happened to be very lucky in that I found Frederich Schuller’s 2 lectures on Measure theory which covered the fundamental aspects of measure theory I needed in the course, and were absolutely fantastic, to say the least, concepts that I understood instantly with his videos would have taken many hours otherwise and still probably without that great of an understanding. In terms of problems, I solved several of the problems in Rudin’s book Real and Complex Analysis, the general style of the problems was very alike to those in the real analysis except harder. This was one of the courses I felt most confident in terms of understanding the fundamental concepts, and in terms of problem-solving, I would say that at the beginning I was pretty subpar but it developed a lot faster than in other courses like functional analysis.

Abstract Algebra II 18.702

Selected Messy Notes in Algebra 2

Abstract Algebra II Problems

Short Description: This course worked around the algebraic structures of Rings, and Fields and towards the end dealt with certain aspects of Galois theory, building up to the fundamental theorem of Galois theory- yes the one associated with the unsolvability of the Quintic. In studying for this course I relied heavily on the Ring theory and Field theory lecture playlist by Elliot Nicholson on Youtube. The second time around when I went back to review, I used the Harvard Lectures along with a book called Abstract Algebra Theory and Computation. I quite liked this course, especially the final highlight of Galois theory. Unfortunately, though the course didn’t come with problems, however, I did find certain problems on MIT’s Dspace of past abstract algebra courses and did them despite not having solutions since I was decently confident in the proofs that I did. Besides this, I also did some problems from the book Abstract Algebra Theory and Computation- even though I couldn’t mark them.

Quantum Theory Lectures by Frederic Schuller and Mathematical Aspects of Quantum Mechanics by Roland Speicher

Quantum-Theory-Self-Made-Notes

Short Description: Originally I intended to work through the Quantum Theory 1 course on MIT open courseware, but when I found Frederich Schuller’s Quantum Theory course I decided to switch directions, I very much liked his course, he is, without doubt, one of the best physics and math teachers I’ve learned from. This course was a formal introduction to Quantum Theory, one geared mainly towards mathematicians as it spent a lot of time formalizing the mathematics yet still leaving room for many of the fundamental physical systems. This course had a tone much different from that of perhaps 99 percent of all other quantum theory courses; the difference came in that they proceed to do things in a mathematically clean way, leaving no stone unturned in that regard. Roland Speicher’s course went even further in mathematics, in the sense that he developed the theory of various types of operators more than Frederich Schuller’s course, but I found Frederich Schuller’s course gave me more insight.

Elementary Number Theory 18.781

Elementary Number Theory Core Topics

Elementary Number Theory Problems

Short Description: This course dealt with many of the fundamental aspects of number theory- as one may expect from the title. For this course I heavily used Michael Penn’s Number theory lectures on Youtube- they are truly amazing. Fortunately, this course contained problem sets and exams with solutions which helped a lot as oftentimes the course used the ideas and concepts in unexpected ways, and only through problems was I able to grasp that. I liked this course a lot, as one of the things that really got me interested in math happened to be numbers and that’s what this course is about, using certain aspects and theorems to ultimately better understand numbers. On the other hand, the one thing I didn’t like as much in Elementary Number theory was that the motivation behind many concepts is often lacking, but nevertheless, I would highly recommend it.

Introduction to Computer Science and Programming in Python 6.000

Introduction to Computer Science and Programming in Python Core Topics

Introduction to Computer Science and Programming Problems

Short Description: This course dealt with the fundamental features of algorithms and programming, discussing the basic semantics and data structures of python, but more importantly it sought to give the learner a basic algorithmic toolbox. This course is probably one of the most essential courses I’ve taken thus far, in fact, one of the lecturers even made a hypothesis that in the future learning programming would be on the same footing as reading and writing. I went through this course using the MIT-open courseware lectures, I also had gone through the edx lectures a little bit earlier when I was planning to use it as part of my high school transcript at the time. Although I learned a lot of interesting things this was surely one of the hardest courses I took despite it being an introduction, often with algorithms there are a lot of things you have to keep track of and precisely this is what left me dizzy and confused, especially in the the beginning stages I often found myself having to constantly repeat the lectures. Besides this the course also had problems and exams with solutions as well as lecture videos from previous years and recitation videos which were extremely useful

Statistical Mechanics of Particles 8.333

Statistical Mechanics Selected Notes- Large

Statistical Mechanics Problems

Short Description: This course quite literally exhausted most of what one learns in Statistical Mechanics aside from certain concepts like that of the Ising Model and Mean field theory. I would say that this was by far the MIT course with the most resources in the sense that it had literally nearly tons of exams with solutions- if you count also past exams from previous years, as well as a full set of lectures. I was first planning to do Statistical Physics 2 and I kept it that way- and went through that course, only towards the last few months did I decide to switch gears completely as I saw how many resources this course had. Besides concepts from kinetic theory and a few other approximation schemes, there was a lot of overlap, with the difference being that this course took things further.

Introduction to Mathematica and Latex:

Short-Description: One of the things I regretted not doing in the challenge was to have left out the need to learn to use many of the basic programs and applications that are heavily used in the sciences and mathematics, so I decided to go back and add it onto the list of to do’s for the challenge. In terms of learning, I thought that for things of this sort it would be best to struggle through and take a very hands-on approach rather than watching lectures teaching you how to use them, I did however use a few online lectures relevent to Mathematica and Latex.