Mathematical Methods for Physics is such a great course. Took the course, and an Advanced version in undergraduate when I was still attempting to be a physicist. The course is at its core a bunch of very important Mathematical concepts that a Physicist or adjacent will need that you won't quite ever come across naturally in mathematics courses unless you go into graduate studies. A "Here is the problem, here is the math and here is why" is just very useful. Legendre Polynomials, Bessel Functions, Lagrangian, Fourier Functions. Just tons of interesting and really useful math that when you hit a problem in their space you have a super power.
My professor for the courses always said it was about putting more mathematical tools in your tool box. Which is a perspective I've since carried.
All models are wrong, some models are useful. Nearly always at least one model will be useful enough. Most if not all of these models come with interesting properties that allow for analytical problem solving.
We're in something of an autodidacticism boom. I have yet to see anyone, in years, actually become proficient at college-level math/physics without formal instruction, just by doing self-study with tutorials like pdfs and videos.
There will always be major knowledge gaps or roadblocks/impasses of understanding, like notation or clarification needed. This is not to say it cannot be done but I have to see anyone do it.
The key ingredient in a formal course is the formal examination. Be it graded exercises or a written or oral exam. It forces studying at a level where you can reproduce and explain the key concepts and results and apply them to something new.
In theory one can do that also with self-study. Most people don't, they just watch some youtube video or read a Wikipedia page, and then they think they have understood something. But the deeper understanding that comes from applying this new knowledge is missing. That step is forced when you take a university course that has some form of examination in-built. Doing it yourself is possible but non-obvious, hard, perhaps even unpleasant, and it's rare people do it. Some do though, and their understanding isn't inferior to a university graduate.
Just to add some more motivation: in a typical physics undergraduate curriculum, you will spend roughly as much time doing homework as attending lectures. If you skip the exercises, you are quite literally skipping half of the education.
It's very easy to let your brain fool yourself in understanding maths - after staring at a statement for a while, you start to think you understand it, chances are you don't.
it took me 15 minutes once to understand one equation. It was the derivation of the Taylor series for the tangent function. It was a very old textbook for calculus (from the beginning of the 20th century) and it applied techniques which are not teached nowadays, so it is possible to understand mathematical expressions after staring at it ;)
Unfortunately it's really hard to put the same kind of accountability on yourself compared to that of a professor gatekeeping an accredited degree. Personally, arbitrary goals don't really motivate me as the acknowledgment of its arbitrariness overrides any pursuit of reward.
It’s entirely possible that creating a brain capable of controlling itself is more costly (in an evolutionary sense, measured by the number of generations needed to achieve this goal) than equipping a brain with the ability to check itself by communicating with others.
Nevertheless, some brains lack even that ability, gravitating instead toward echo chambers where everyone shares the same views, so no mutual checks are possible.
I have been self-studing alot. In fact, my first experience was the professor asked me to solve one of his final exam questions (for which he had provided his lecture notes for the course) within a few days after I asked for a internship.
The way I do it is to create exercise where I am required to fill as much of details as possible in the must explicit way. Sometimes, this is in the form of lecturing my friends for several hours. Or reproducing some computational results from computational physics (writing software and getting results correctly requires alot of details getting right). Similarly, when I studied such notes (for some of which I did not have accompanying course), I tried to fill as much of details as possible in the notes and do the exercises.
The ultimate state you want to reach is one where you can derive some results topic with enough details from scratch (without looking at the notes), like the way one can remember movie plot or something.
If you can take this process of understanding (and getting details right) very seriously + feynman technique, these can be helpful. But, you need to be really serious about learning and applying them (e.g. reproducing some results)
This is a strange take. Why or how would you ever see someone do it? It is very unlikely you would be aware of them.
I do concur most ppl I’ve seen lacking formal education in physics - no matter how smart - do have some gaps in their knowledge. There seems to be something of a tacit knowledge transfer involved - but I have no clue what it is. Maybe physicists radiate physics particles or something. I’m kidding. But I have no clear mechanistic model what makes physicists physicists. It’s not smarts (some required I guess, but does not suffice).
Maybe it’s just going through a specific set of rote calculations and lab work needed.
> This is a strange take. Why or how would you ever see someone do it? It is very unlikely you would be aware of them.
You would see someone be proficient at math or physics and then get surprised they had no degree, just like anything else. You can find such programmers everywhere, even algorithms are very easy to self study compared to math and physics, but people proficient in math or physics without a corresponding degree is almost non existent.
> Maybe it’s just going through a specific set of rote calculations and lab work needed.
I think the biggest problem is error correction. Everyone who learns math and physics accumulates errors as they learn, and if nobody is there to correct the wrong assumptions they will just continue to accumulate. To do that you need an instructor grading your tests, or at least almost everyone needs that since they can't grade their own answers fairly, their incorrect understanding leads to incorrect grading of their own answers.
This is why programming and algorithms is so easy to learn in comparison, since there its easy to grade yourself. If the program works you did it right, otherwise not. For algorithms there are plenty of sites you can practice implementing algorithms for problems and they tell you if you did it right. There are no such sites for math or physics, so there unless you are a genius at self correcting you need an instructor to just get proficient.
Edit: In short, if you can write code that works the first time you run it without testing etc, and you can do that over and over and never be wrong, then you can self learn math and physics without an instructor.
But if you need to execute your code and test to get it right, then you need an instructor to learn math and physics correctly as well.
Posts by Susan Fowler "so you want to self-study math/physics" have gone hugely viral here and I have yet to see anyone in the comments or later report that they actually became really good after following her tutorial. Same for any other tutorial or guide or self-study overall. maybe one blogger whose name eludes me but extremely rare.
This is a strange take.
How is it strange? What you meant to say is you disagree.
> Posts by Susan Fowler "so you want to self-study math/physics" have gone hugely viral here and I have yet to see anyone in the comments or later report that they actually became really good after following her tutorial. Same for any other tutorial or guide or self-study overall. maybe one blogger whose name eludes me but extremely rare.
Well, when you learn something, do you remember everything that contributed to your learning it, and go back and acknowledge it? I try very hard to acknowledge those from whose instruction I have benefited, and I doubt I come anywhere near that ideal.
Come to that, one measure of real expertise is knowing you're never done learning, so, even for someone who does acknowledge every single source along the way, the more they know the more presumptuous it'll seem to say they are done learning it, so they probably won't.
In general there are 8+ billion people on the planet, of which some number are trying to autodidact through MSc physics equivalent. It’s possible you are in a circumstances where you engage with lot of autodidacts, but I personally have no model how you could do that (unless you are a tutor which ofc you may be!).
I’m also guessing most autodidacts arent observable through the internet.
It begins when you realise all the physics examples are toy models, and reality is basically unmodellable. That physics is not mathematics, and the rules of inference in physics cases are the rules of reality -- not of mathematics. That what guarentees your modelling works is observation, that "hand waving" is something one does to force one's thinking to be empirical.
I suspect people who learn physics via the textbooks end up learning some paralllel reality which is closed-form and models engineered trinkets.
Physics isnt mathematics, and it isnt engineering. I think this is communicated in how its performed, in the emotional tensor of its practice. A good physicist has a healthy contempt of both.
The key thing is doing the exercise problems; most textbooks have them at the end of each chapter. Yes, it takes time. Yes, you will struggle, and it's hard work going through them. But it's really the only way to make sure you really understand the subject at a sufficient level vs. you believing you understand it because you watched some videos.
People who write game engines easily fall in this category. Especially when they start thinking about fluids, fibers, cloth, fire, etc. it's high level math and physics under 60fps constraints
Granted, this is anecdotal, but I definitely found that there were some things that I could learn on my own, and others that I needed to learn in a classroom. The latter were in fact math and physics. I was considered to have good aptitude in those areas, yet I still needed that structure for whatever reason.
On the other hand, I easily taught myself programming and electronics, after an "intro" course in both subjects.
One problem is that both classroom and self learning of physics are almost entirely confined to theory, but physics is an experimental discipline. In fact most laypeople are unaware that experimental physicists outnumber theoreticians. In grad school, there was a noticeable distinction, that the students who went into theory were scary smart. With a few exceptions, the experimental students including myself were considered to be the lesser intellects. I barely survived the coursework, and owe my degree to my work in the lab.
In physics, we do have tests. Mother nature provides them.
I am currently coasting through a mathematics undergraduate degree after having taught myself how to do proofs using a proof assistant. The last barrier to autodidacticism in mathematics is overcome with them, because one no longer needs direct contact with the mathematical community via academia to learn the techniques used (at the undergraduate level.)
I don't think the math libraries for proof assistants are good enough to support undergrad math very well, but they may be occasionally useful in specific places. I still think you will dominantly be doing informal proofs to make progress with any velocity.
The issue is a lack of credential means you are putting in a lot of effort to teach yourself something hard, with little evidence to show you understand the material. If universities simply offered a for credit exam where learners had to self teach I reckon we would see a lot more success stories.
The first two years of my degree where I learned most of my university-level math/physics were online-only. There are likely millions of students who learned entirely from videos and PDFs.
I don't think the OP here was imposing anything.. It seems more like a fitting reference for those who have performed physical calculations before and are looking to refresh.
Reading the numerous lecture notes on mathematical methods used in physics that are available online is like reading stories about love: it never gets old, and each time you learn something new!
Tangential: Has anybody here read Mathematical Methods and Physical Insights: An Integrated Approach by Alec J. Schramm.? Would love to hear your opinion on this.
Most likely these were lecture notes that were transcribed by students. The figures are just pictures of the professor's drawings.
True, they're not hard to make on a computer, but who is spending time to do that? And is the result more clear? More educational? Or simply more beautiful?
Mathematical Methods for Physics is such a great course. Took the course, and an Advanced version in undergraduate when I was still attempting to be a physicist. The course is at its core a bunch of very important Mathematical concepts that a Physicist or adjacent will need that you won't quite ever come across naturally in mathematics courses unless you go into graduate studies. A "Here is the problem, here is the math and here is why" is just very useful. Legendre Polynomials, Bessel Functions, Lagrangian, Fourier Functions. Just tons of interesting and really useful math that when you hit a problem in their space you have a super power.
My professor for the courses always said it was about putting more mathematical tools in your tool box. Which is a perspective I've since carried.
All models are wrong, some models are useful. Nearly always at least one model will be useful enough. Most if not all of these models come with interesting properties that allow for analytical problem solving.
We're in something of an autodidacticism boom. I have yet to see anyone, in years, actually become proficient at college-level math/physics without formal instruction, just by doing self-study with tutorials like pdfs and videos.
There will always be major knowledge gaps or roadblocks/impasses of understanding, like notation or clarification needed. This is not to say it cannot be done but I have to see anyone do it.
The key ingredient in a formal course is the formal examination. Be it graded exercises or a written or oral exam. It forces studying at a level where you can reproduce and explain the key concepts and results and apply them to something new.
In theory one can do that also with self-study. Most people don't, they just watch some youtube video or read a Wikipedia page, and then they think they have understood something. But the deeper understanding that comes from applying this new knowledge is missing. That step is forced when you take a university course that has some form of examination in-built. Doing it yourself is possible but non-obvious, hard, perhaps even unpleasant, and it's rare people do it. Some do though, and their understanding isn't inferior to a university graduate.
Just to add some more motivation: in a typical physics undergraduate curriculum, you will spend roughly as much time doing homework as attending lectures. If you skip the exercises, you are quite literally skipping half of the education.
It's very easy to let your brain fool yourself in understanding maths - after staring at a statement for a while, you start to think you understand it, chances are you don't.
it took me 15 minutes once to understand one equation. It was the derivation of the Taylor series for the tangent function. It was a very old textbook for calculus (from the beginning of the 20th century) and it applied techniques which are not teached nowadays, so it is possible to understand mathematical expressions after staring at it ;)
If there's a method to feel the pressure of classroom test or the requirement of finishing a project would really help home DIY learning.
Unfortunately it's really hard to put the same kind of accountability on yourself compared to that of a professor gatekeeping an accredited degree. Personally, arbitrary goals don't really motivate me as the acknowledgment of its arbitrariness overrides any pursuit of reward.
It’s entirely possible that creating a brain capable of controlling itself is more costly (in an evolutionary sense, measured by the number of generations needed to achieve this goal) than equipping a brain with the ability to check itself by communicating with others.
Nevertheless, some brains lack even that ability, gravitating instead toward echo chambers where everyone shares the same views, so no mutual checks are possible.
I have been self-studing alot. In fact, my first experience was the professor asked me to solve one of his final exam questions (for which he had provided his lecture notes for the course) within a few days after I asked for a internship.
The way I do it is to create exercise where I am required to fill as much of details as possible in the must explicit way. Sometimes, this is in the form of lecturing my friends for several hours. Or reproducing some computational results from computational physics (writing software and getting results correctly requires alot of details getting right). Similarly, when I studied such notes (for some of which I did not have accompanying course), I tried to fill as much of details as possible in the notes and do the exercises.
The ultimate state you want to reach is one where you can derive some results topic with enough details from scratch (without looking at the notes), like the way one can remember movie plot or something.
If you can take this process of understanding (and getting details right) very seriously + feynman technique, these can be helpful. But, you need to be really serious about learning and applying them (e.g. reproducing some results)
”but I have to see anyone do it”
This is a strange take. Why or how would you ever see someone do it? It is very unlikely you would be aware of them.
I do concur most ppl I’ve seen lacking formal education in physics - no matter how smart - do have some gaps in their knowledge. There seems to be something of a tacit knowledge transfer involved - but I have no clue what it is. Maybe physicists radiate physics particles or something. I’m kidding. But I have no clear mechanistic model what makes physicists physicists. It’s not smarts (some required I guess, but does not suffice).
Maybe it’s just going through a specific set of rote calculations and lab work needed.
> This is a strange take. Why or how would you ever see someone do it? It is very unlikely you would be aware of them.
You would see someone be proficient at math or physics and then get surprised they had no degree, just like anything else. You can find such programmers everywhere, even algorithms are very easy to self study compared to math and physics, but people proficient in math or physics without a corresponding degree is almost non existent.
> Maybe it’s just going through a specific set of rote calculations and lab work needed.
I think the biggest problem is error correction. Everyone who learns math and physics accumulates errors as they learn, and if nobody is there to correct the wrong assumptions they will just continue to accumulate. To do that you need an instructor grading your tests, or at least almost everyone needs that since they can't grade their own answers fairly, their incorrect understanding leads to incorrect grading of their own answers.
This is why programming and algorithms is so easy to learn in comparison, since there its easy to grade yourself. If the program works you did it right, otherwise not. For algorithms there are plenty of sites you can practice implementing algorithms for problems and they tell you if you did it right. There are no such sites for math or physics, so there unless you are a genius at self correcting you need an instructor to just get proficient.
Edit: In short, if you can write code that works the first time you run it without testing etc, and you can do that over and over and never be wrong, then you can self learn math and physics without an instructor.
But if you need to execute your code and test to get it right, then you need an instructor to learn math and physics correctly as well.
Posts by Susan Fowler "so you want to self-study math/physics" have gone hugely viral here and I have yet to see anyone in the comments or later report that they actually became really good after following her tutorial. Same for any other tutorial or guide or self-study overall. maybe one blogger whose name eludes me but extremely rare.
This is a strange take.
How is it strange? What you meant to say is you disagree.
> Posts by Susan Fowler "so you want to self-study math/physics" have gone hugely viral here and I have yet to see anyone in the comments or later report that they actually became really good after following her tutorial. Same for any other tutorial or guide or self-study overall. maybe one blogger whose name eludes me but extremely rare.
Well, when you learn something, do you remember everything that contributed to your learning it, and go back and acknowledge it? I try very hard to acknowledge those from whose instruction I have benefited, and I doubt I come anywhere near that ideal.
Come to that, one measure of real expertise is knowing you're never done learning, so, even for someone who does acknowledge every single source along the way, the more they know the more presumptuous it'll seem to say they are done learning it, so they probably won't.
> What you meant to say is you disagree.
No, I meant what I wrote.
In general there are 8+ billion people on the planet, of which some number are trying to autodidact through MSc physics equivalent. It’s possible you are in a circumstances where you engage with lot of autodidacts, but I personally have no model how you could do that (unless you are a tutor which ofc you may be!).
I’m also guessing most autodidacts arent observable through the internet.
I may be totally wrong of course!
It begins when you realise all the physics examples are toy models, and reality is basically unmodellable. That physics is not mathematics, and the rules of inference in physics cases are the rules of reality -- not of mathematics. That what guarentees your modelling works is observation, that "hand waving" is something one does to force one's thinking to be empirical.
I suspect people who learn physics via the textbooks end up learning some paralllel reality which is closed-form and models engineered trinkets.
Physics isnt mathematics, and it isnt engineering. I think this is communicated in how its performed, in the emotional tensor of its practice. A good physicist has a healthy contempt of both.
> I think this is communicated in how its performed, in the emotional tensor of its practice.
A typo for "tenor," I presume, but a funny one!
The key thing is doing the exercise problems; most textbooks have them at the end of each chapter. Yes, it takes time. Yes, you will struggle, and it's hard work going through them. But it's really the only way to make sure you really understand the subject at a sufficient level vs. you believing you understand it because you watched some videos.
People who write game engines easily fall in this category. Especially when they start thinking about fluids, fibers, cloth, fire, etc. it's high level math and physics under 60fps constraints
Granted, this is anecdotal, but I definitely found that there were some things that I could learn on my own, and others that I needed to learn in a classroom. The latter were in fact math and physics. I was considered to have good aptitude in those areas, yet I still needed that structure for whatever reason.
On the other hand, I easily taught myself programming and electronics, after an "intro" course in both subjects.
One problem is that both classroom and self learning of physics are almost entirely confined to theory, but physics is an experimental discipline. In fact most laypeople are unaware that experimental physicists outnumber theoreticians. In grad school, there was a noticeable distinction, that the students who went into theory were scary smart. With a few exceptions, the experimental students including myself were considered to be the lesser intellects. I barely survived the coursework, and owe my degree to my work in the lab.
In physics, we do have tests. Mother nature provides them.
Yes, one can learn from online available sources. I had Calculus I in school and then I found Prof. Gilbert Strang, MIT OpenCourseWare. [0]
openstax.org has a lot of good books. [1]
[0] https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D [1] https://openstax.org/subjects/math
edit: i am an autodidact who loves to learn.
Math Academy has solved this.
I am currently coasting through a mathematics undergraduate degree after having taught myself how to do proofs using a proof assistant. The last barrier to autodidacticism in mathematics is overcome with them, because one no longer needs direct contact with the mathematical community via academia to learn the techniques used (at the undergraduate level.)
I don't think the math libraries for proof assistants are good enough to support undergrad math very well, but they may be occasionally useful in specific places. I still think you will dominantly be doing informal proofs to make progress with any velocity.
Which proof assistant did you use?
The issue is a lack of credential means you are putting in a lot of effort to teach yourself something hard, with little evidence to show you understand the material. If universities simply offered a for credit exam where learners had to self teach I reckon we would see a lot more success stories.
The first two years of my degree where I learned most of my university-level math/physics were online-only. There are likely millions of students who learned entirely from videos and PDFs.
I don't think the OP here was imposing anything.. It seems more like a fitting reference for those who have performed physical calculations before and are looking to refresh.
I am not implying that the OP is imposing anything. i am just asking a related question.
Where is your question?
Reading the numerous lecture notes on mathematical methods used in physics that are available online is like reading stories about love: it never gets old, and each time you learn something new!
Mathematics is one of our love stories?
Yes! All the endings are happy and the unfinished stories are utterly fascinating. And there are already so many stories with happy endings!
Mathematics is our best bildungsroman
> Yes! All the endings are happy and the unfinished stories are utterly fascinating.
I'm not sure the practice of mathematical research supports that all the endings are happy, but definitely all the journeys are interesting.
Happy as in Halmos Tombstone
Ronald L. Graham, Donald E. Knuth, Oren Patashnik: oncrete Mathematics
https://archive.org/details/concrete-mathematics/page/n11/mo...
Nice, but not "for Physics" per se.
Nice! This will make a good companion to Arfken & Weber
https://archive.org/details/Mathematical_Methods_for_Physici...
Tangential: Has anybody here read Mathematical Methods and Physical Insights: An Integrated Approach by Alec J. Schramm.? Would love to hear your opinion on this.
Big no on the badly hand drawn figures. Those graphics are not hard to make on a computer.
Most likely these were lecture notes that were transcribed by students. The figures are just pictures of the professor's drawings.
True, they're not hard to make on a computer, but who is spending time to do that? And is the result more clear? More educational? Or simply more beautiful?