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Ask HN: How do I learn math/physics in my thirties?
441 points by mosconaut 4 months ago | hide | past | web | favorite | 154 comments
I'm in my early thirties and I feel I've not really made any significant effort in learning math/physics beyond the usual curriculum at school. I realize I didn't have the need for it and didn't have the right exposure (environment/friends) that would have inculcated in me these things. And perhaps I was lazy as well all these years to go that extra mile.

I have (had) a fairly good grasp of calculus and trigonometry and did a fairly good job working on a number of problems in high school. But over the past 12-13 years, I've really not had any need to flex my math muscles other than a problem here or there at work. Otherwise it's the same old enterprise software development.

I follow a bunch of folks on the internet and idolize them for their multifaceted personalities - be it math, programming/problem solving, physics, music etc. And these people had a natural flair for math/physics which was nurtured by their environment which made them participate in IOI/ACPC etc. in high school and undergrad which unfortunately I didn't get a taste of. I can totally see that these are the folks who have high IQs and they can easily learn a new domain in a few months if they were put in one.

Instead of ruing missed opportunities, I want to take it under my stride in my thirties to learn math/physics so as to become better at it. I might not have made an effort till now, but I hopefully have another 40 years to flex my muscles. I believe I'm a little wiser than how I was a few years back, so I'm turning to the community for help.

How do I get started? I'm looking to (re)learn the following - calculus, linear algebra, constraint solving, optimization problems, graph theory, discrete math and slowly gain knowledge and expertise to appreciate theoretical physics, astrophysics, string theory etc.

There is one sure way, and it’s a test of your fortitude. You find a a college textbook with the answers to the even-numbered problems in the back. You sit down in a warm or hot room, and solve them. If the textbook is in its 4th printing or so, the answers are correct. On a few, you’ll have to work for hours. Now here is a very, very, important point. All the learning occurs on the problems you struggle with. In the blind alleys. A lot of learning in physics comprises paring down your misconceptions until the correct methodology, often surprisingly simple, appears. Then, you understand how to apply the basic laws to the problem at hand, which is what physics is. I’ll emphasize the point by stating it’s converse. A problem you can solve easily and quickly yields zero knowledge.

I would recommend two outstanding textbooks. Halliday and Resnick, early editions , printed in the late 60s and 70s. If you can do all the odd problems in this two volume set, you are an educated person, regardless of your greater aspirations. Edward Purcell’s Berkeley Physics Series Second Volume on Electricity and Magnetism. Might be the best undergraduate physics textbook ever written. Did you know that magnetism arises from electrostatics and relativistic length contraction? It’s right there. You should also get yourself a copy of Feynman’s Lectures on Physics. Warning. Read it for intuition, motivation, the story of Mr. Bader, and entertainment. It’s at much too advanced a point of view to help you solve nuts and bolts physics exercises, which is what you must do. One final warning. Every one of us sits at a desk with a powerful internet-connected computer. Don’t do this. Even get a calculator to avoid this. Of course, when you are stumped you’ll want to see how a topic has been treated by others. Do it in another room.

I agree with this suggestion. It took me a year to slowly absorb the entire book of Statistics [0] including solving all exercises. It's just like walking to school but there is no external supervision. I made a rule to complete one chapter every evening including exercises and sticked to it.

[0]: https://www.amazon.com/Statistics-4th-David-Freedman/dp/0393...

Your story implies there are 365 chapters.

It implies he completed 365 chapters but says nothing about repetition of the chapters.

> sit down in a warm or hot room

While I agree with everything else, I'd have to vehemently disagree with this. Studies [1] have shown that warm temperatures severely diminish our performance on complex mental tasks.

As some examples [2]:

> Sales for scratch tickets, which require buyers to choose between many different options, fell by $594 with every 1° Fahrenheit increase in temperature. Sales for lotto tickets, which require fewer decisions on the part of the buyer, were not affected.

> participants were asked to proofread an article while they were in either a warm (77°) or a cool (67°) room. Participants in warm rooms performed significantly worse than those in cool rooms, failing to identify almost half of the spelling and grammatical errors (those in cool rooms, on the hand, only missed a quarter of the mistakes).

[1] https://www.bauer.uh.edu/vpatrick/docs/Influence%20of%20Warm... [2] https://www.scientificamerican.com/article/warm-weather-make...

> You sit down in a warm or hot room

What is wrong with airconditioning?

I am not sure about OP's reasoning, but I personally find it a bit 'motivating' to study in a slightly not-so-comfortable environment. I mean, it gives me sense that I am actually determined and am working hard. It also reminds me of my college days when even finding an air-conditioned room anywhere was just not possible.

Could also give you the feeling of being uncomfortable. Then when you are struggling working through a problem you get so frustrated. And think "If only it weren't so damn hot in here." Then all you can think about is the heat, and you are so lost it cannot be returned. So then you give up for the day, and really haven't accomplished anything.

Exactly, your mileage may vary, but my mindset has to be completely free from distractions to be productive.

The library on my uni when I was in Math undergrad did not have AC at the beggining but was the only place where I could do any work, it was extremely difficult and I am sure impacted my progress.

I find it impossible to think or stay focused in a hot or even warm environment. People are different I guess.

I'm the same. During winter months, when I needed to cram a lot, I would open the windows wide, and sit with my jacket on. The cold would help me not fall asleep.

A hot room sounds horrible, but the memories of college days does make sense to me. My college was freezing cold, and my search would be for a room where you didn't need to wear 2 sweaters to be comfortable.

But yeah, the idea of studying in a really cold room "makes sense" to me, and this might be why.

I think that's just building the idea that it's going to be a painful and uncomfortable process

In cold parts of the world, warm has connotations of comfort, not cold...

Is this part of the process? Visit a cold part of the world, set yourself up with a physics textbook in front of a fireplace...

Actually, that sounds quite nice.

my deduction: if you done it in a warm or hot room, you surely have enough will to do it.

Seconded. I really believe there are no shortcuts to doing lots of problems. If you can afford it, getting a physics grad student to discuss problems that stumped you every now and then might also have quite good ROI, talking to physicists might also help convey some of the physics mindset(?).

Reading this made me nostalgic for my days as a physics undergrad.

I'll second this idea having survived a Physics BS doing just this. I'd also strongly recommend a series of books called Schaum's Outlines, they vary in quality but cover many advanced topics and have hundreds of solved problems in them.

Schaum's Calculus was invaluable to refresh my memory of some of the details of "Calc 2" so I could be sure of passing a waiver exam (most schools would have waived it automatically on account of my AP credits but my school limited me to how many I could waive that way...) and get on with Calc 3. The book covered some Calc 3 too so continued being useful. I have a few others in the series, very handy.

"The reader who has read the book but cannot do the exercises has learned nothing." -- J.J. Sakurai

(Incidentally, I tried reading Sakurai's Modern Quantum Mechanics on my own once and was immediately curb stomped. Lots of prep work required for that one...)

This is also what I did, going straight to the exercises except I used Calculus I by Apostol which covers some Linear Algebra. Perfect book if you need to redo math skills you've forgotten though plenty of times I had to Wikipedia, Khan Academy, and math.stackexchange in the beginning.

There's also this free book, no answers though you could stackexchange if really stuck. I finished most of Apostol before starting it https://infinitedescent.xyz/

Defs agree with op. I learned the more advanced maths I use daily in my thirties. It took about 3 years of exactly ops method. In my case, I found it motivating to take exams because it gives you a bit of skin in the game; forces you to prioritise your study at some point.

A final thing: it's really worth doing. If you long for maths; it's likely it'll conceptually take you places you won't go without it. Do it!


Is it that the text you’re referring to? We used the 5th edition in my physics course this year. It was a tough textbook to learn from but I feel like I learned a ton.

I almost never went to class in university (Waterloo Engineering) and this is how I did it. The best is not letting them explain the concept to you first. Try to invent the math as you go along by covering the explanatory pages with pieces of paper and reading only one line at a time.

It will stick with you forever.

I second this, but, you will need some help initially. Follow the examples a few times - first with help, then without. Once you build your intuition, you will then be in a position to "invent" the maths as you go along.

> Halliday and Resnick, early editions , printed in the late 60s and 70s

Any particular reason to recommend the old editions over the latter ones?

I actually had to look into this recently.

The recent ones are less "textbook." The older ones are FILLED with information with graphics here and there but it's mostly text. The recent ones are very graphical so I would assume it has less total information. With that said, it's possible that there are techniques for learning that were not considered in the older texts.

It is possible to look at samples online for you to compare if you want to see the difference. I do recommend getting the book if you decide to use it but that's just a personal preference.

Textbooks have generally gotten less information dense over time.

> If the textbook is in its 4th printing or so, the answers are correct

It's terrifying that it takes 4 printings before the answers should be considered trustworthy...

Do you write bug free code?

Is publishing a book the same thing as writing code?

No, writing a math textbook involves (perhaps) thousands of things that might be wrong, none of which will have any impact on one another.

Publishing a perfect book is difficult on par with writing code. Hell, Knuth is incredibly popular and crowdsources his error-checking, and TAOCP is still in its third edition.

Very similar in some ways. There are a vast number of interconnected details that have the potential to be wrong and far fewer automated ways to catch any errors. Your "users" inevitably catch a lot of them at "runtime".

Yes. You're programming a person instead of a computer but that's the only difference.

I'm writing a book called "A Programmer's Introduction to Mathematics". Would you like to see a draft? Shoot me an email at mathintersectprogramming@gmail.com

It's an introduction to mathematics from a programmer's standpoint, with a big focus on taste and that second level of intuition beyond rote manipulation and memorization.

Includes chapters on sets, graphs, calculus, linear algebra, and more! Each chapter has an application (a working Python implementation) of the ideas in the chapter. The applications range from physics to economics to machine learning and cryptography. One chapter even implements a Tensorflow-like neural network.

There's also a mailing list: https://jeremykun.com/2016/04/25/book-mailing-list/

I can recommend anything Jeremy does, sight unseen. Take him up on his offer.

Is this offer just for the OP, or for everyone? Disregard my email if it was meant only for the OP.

Whats the progress / ETA for this? Sounds like the perfect thing for me right now

Watch the 3Blue1Brown YouTube channel: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

These videos are frankly better explanation of college-level math concepts than most college classes.

Also, now you probably care much more about the intuitions of mathematics over the raw mechanics of it. Once again, this channel perfectly exemplifies this concept.

These videos are posted any time linear algebra is mentioned. I find it almost comical at this point. What good is this intuition? I struggle to understand the value that these videos bring, but I'm not saying there is no value. I'm just lost, and kind of jealous because I need a deep understanding of linear algebra for work.

Those are good videos, and I also endorse them. However, you will not truly know what is going on until you solve some real problems with them.

One interesting option that computer programmers have to understand linear algebra that most people do not is that you could go do some stuff with computer graphics. Grab three.js, and then once you've followed some tutorial somewhere to get a triangle on the screen, start doing things, but do them manually, including implementing matrix multiplication yourself. Modern graphics has moved so far up the stack nowadays that you probably shouldn't say that you "know 3D graphics" after that exercise, because you'll know 3D graphics circa 1995. But you will have a much better intuition for linear algebra, and those videos will either make sense, or be trivially obvious to you.

(One of the reasons why the linear algebra videos can be so helpful is that it has historically been very easy to take an entire class on the topic and just grind numbers, without ever getting to that level of intuition. Differential equations, if you took physics that did not use them, can have a very similar problem, where you just grind through problems for a semester with no motivation.)

It's not just linear algebra, 3blue1brown also has an entire series on undergraduate calculus, and series on Statistics, Linear Algebra II and Group Theory are in the works. Plus a large number of excellent videos on miscellaneous math topics.

I think I understand what you're saying -- one needs more than just cool videos and cool intuition. You need to do exercises. This is a point made multiple times in those very videos.

But, the intuition provided in those videos is absolutely excellent. As an example, look at the explanation of change-of-basis in the linear algebra I series.

Ah, the videos come with pointers to exercises? I'm quite excited about https://www.edukera.com/ and possibilities for interactive learning through automated theorem proving. Thanks!

Actually, there are not many pointers to exercises. I believe they may be planning to add some written materials, so maybe in the future, but not currently.

The videos suggest pausing and trying to figure the next bit out yourself and a couple of the videos do end with a suggestion to prove something yourself.

For me personally, understanding why it is done on a deeper level than is commonly taught helps me consolidate the concept more comprehensively and permanently.

I think what stands out about those videos is that people who have no prior higher math education still get a shadow of intuition of what's actually going on, and people who already 'grokked' the concepts still got an alternative, simpler view on those concepts, resulting in at least a view 'a-ha!' moments for almost everyone.

Did you actually watch the videos?

Ha no. I can't watch videos, I read really fast and find videos painfully slow.

Grant (the 3Blue1Brown guy) has an uncanny ability to explain difficult concepts and the fundamental intuition behind them. In many of his videos, he explains topics from the perspective of a person inventing that topic (such as in his first Calculus video [1]). I can't recommend his videos enough.

[1] https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...

It works for a lot of people, I'm just looking for something else, exercise-based computer-checked proofs to teach mathematics, something like that. Want it bad.

I do x2 speed. But not all videos are slow.

I second this. I was refreshing some of the linear algebra I learned in college recently and his videos gave far more insight into what linear algebra is actually about than I was taught in college. If everyone studying linear algebra in school watched his videos before taking the course they would have a much easier time learning it. His calculus series is of similarly high quality and I would imagine his other videos are too.

It is important to really pause the videos at some points and do the "exercises". Doing the exercises is always a very important part of learning (I believe this holds in any field). Do them with pen and paper, not just in your head.

I'm a web developer. In my 20's I used to love picking things up just for the of it. These days I'm more of a fan of JIT learning. Push the edges of my map as I go. I'm still constantly learning, but it's more iterative. Building on conquered territory and shifting the borders as needed (always outward, but the focus on which parts of the border to push changes.) Previously I was more like a crazed monkey and never holding any ground. I still feel the importance of occasionally sneaking outside my borders and going deep into enemy territory, but those are constrained efforts. Invade, gather booty, sift for intelligence value and then decide if it's worth a more serious invasion.

Maybe figure out an actual destination and then devise a plan to get there. Deep diving into math and physics just for the sake of learning etc seems to be cargo-culting. Lawyers also sound smart until you realize they write like they do intentionally to keep people from figuring them out.

> I follow a bunch of folks on the internet and idolize them for their multifaceted personalities - be it math, programming/problem solving, physics, music etc. And these people had a natural flair for math/physics which was nurtured by their environment which made them participate in IOI/ACPC etc.

Sounds like they are good story-tellers along with whatever else they do. Have you tried putting anything out for others to consume? If you want to be like these people, then it would be good to start with writing / shipping things. If you have been doing that already, then post some links. ;)

Anyone who tells you can "learn" math and pyhsics by just watching videos is lying to you. There is no substitute for actually doing lots of problems.

Pick a book, pick a pace to work through it, and spend a few months going through it. Do the exercises in the back of each chapter, work through the solutions, and ask around if you still can't figure it out. Persistance and routine are key here.

As for books, I like Stewart's Calculus, Lay's Linear Algebra, and Hammack's Book of Proof.

For physics, I don't know what your background is. Giancoli is a popular undergrad freshman year book, where as griffith's electrodynamics is a bit more advanced.

I second this but starting to watch actual undergrad lecture series is beneficial because sometimes the book your working through May just be missing that one piece of the puzzle you need to start getting enough of a grip to solve a problem.

Also with physics, you need some tricks that are usually done in derivations or calculations (cause not everything can just be solved) and those do not necessarily appear in a book - but they do in lectures.

Edit: the book gives you the fundamentals though on which you work. also, they will teach you the necessary abstraction - the first thing standing in my way of a degree in physics was my intuition and need to picture stuff. working a lot with differential geometry now, I've got some of that visualization back but with linear algebra even and quantum mechanics it can stand in your way.


For the programmers in the house, it would be like claiming you can code python if you’ve never coded before but watched some videos of expert teachers explaining the structure of python programming.

I often see people here on HN saying classroom lectures are more important than homework. But in my experience the real intuitive learning (at least for math and physics) doesn’t come until you’ve spent some time banging your head trying to figure things out solving problems (ie, applying the theory).

One text (Wangsness E&M maybe) had a great student quote, roughly “I understand the principles but I can’t do the problems.”

Giancoli's book doesn't use calculus. Halliday & Resnick (or one of its later updates from Crane) is a better bet in this regard.

People say you have to "do" math to learn it. Usually they make it sound like you need to do the exercises in the books. I think that doing just that can be boring and demotivating.

I would suggest finding projects that can motivate you and help you exercise your math. Some suggestions of mathy things I regularly work on for fun:

1. Make a video game. If it's a 3d game, you'll have to do your matrices, dot products, trigonometry, etc.

2. shadertoy.com - This is a community site where people just program cool looking graphics for fun. All the code is open, so you can learn from it. Similar to game programming but without the mathless overhead. :)

3. Machine learning projects - I love writing various machine learning things, but the project that has been a great ML playground has been my self driving toy car. It gives me plenty of opportunities to explore many aspects of machine learning and that helps drive my math knowledge. My car repo is here: https://github.com/otaviogood/carputer but a much easier project is donkeycar.com. ML will touch on linear algebra, calculus, probabilities/statistics, etc.

The most important thing for learning is to be inspired and have fun with what you're learning. :)

I can't speak to all of the things you want to learn, but I've learned some of them on my own. For calculus and linear algebra I'd go with Khan Academy, especially since it seems like all you need is a refresher for calculus. Graph theory and discrete math I did with MIT EdX courses. Their discrete course is pretty nice and I found it very easy to follow along with.

Constraint solving and optimization problems aren't things I self studied, but you can find a variety of resources to help with those based on how you learn best. For me, I did them by taking a class and relying heavily on my textbooks.

+1 to Khan Academy. Explanations are super clear. Their website allows you to work through practice problems too, which I think is the most important thing.

For physics, I believe you fall in Leonard Susskind's target audience. You can get his book, or even better, watch his large amount of lectures: https://www.youtube.com/watch?v=iJfw6lDlTuA

Susskind is an eminence - he was Feynman's buddy back in the day. And he's entertaining as hell.

Here's also Gerard 't Hooft's (Nobel laureate) list of concepts and books to master. If you finish that -in several years- you will be a qualified theoretical physicist. Whereas Susskind will give you more of an overview. http://www.goodtheorist.science/

Get a real pen and paper, get a real physical book, sit and solve problems with pen and paper for hours every day for a few months. Then you will pass the exams.

Exactly. Every time I tutor someone in math, I tell them to use up at least a sheet of paper for every interesting question. When they do, their skills improve quickly. Saving paper is a false economy when it comes to math.

I don't know why this is downvoted, but the process of writing and working problems on paper, at least for me, helps cement the knowledge.

This is exactly right. I elaborate on the method in my response.

While I agree with you, and love aj7's post, I'm going to push back slightly on the pen and paper.

I used to do all my work (solutions to problems, notes) using pen and (plain! not lined) paper. However I realized a couple of years ago that becoming fluent in LaTeX was a better option for me. The reason is that, with the proof neatly typeset, and the ability to re-work and edit repeatedly without making a mess, I found that I think more precisely and systematically. I still do scratch work on paper, but writing up a clean copy as I go is very beneficial.

In addition to those reasons, the other hugely important one is that my notes are now in git, I can grep them, and they don't add to the pile of objects that must be dealt with when moving to a new home.

For best results you need to make a nice LaTeX set up. I use the Skim PDF reader so that it autorefreshes on file save, and set up a Makefile and make it so the PDF is recompiled on every file save. But whatever works for you, I'm sure there are easier setups.

There's a lot to be said for using computer tools. If you're writing proofs, why not do it formally? [0] If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)? If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)

The downside of course is that computers are very capable distraction vehicles, you need a bit of discipline to sit at one and study / do this sort of work at the same time for prolonged periods. Pulling out the ethernet cable can help but may not be sufficient depending on one's level of discipline and access to offline distractions.

A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles. Certain modern enhancements are worth a qualified mention though.

[0] https://lamport.azurewebsites.net/pubs/proof.pdf

> If you're writing proofs, why not do it formally?

Because that requires learning a formal proof-verification language. I'm certainly interested in that, but it is a distraction from learning undergraduate mathematics.

> If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)?

> If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)

I'm confused; my post was advocating using software, so I'm unclear why you're suggesting I use software.

> A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles.

What is that, a flat contradiction of my post?

Very strange, maybe you meant to reply to a different post?

My post was mainly adding agreement to yours with more specifics, "you" used is the "generic you".

> it is a distraction from learning undergraduate mathematics

Arguably so is LaTeX. But it's desirable that students (or just people learning the same material, later) spend some of their undergraduate time learning new things, right? And not just because it's new, but hopefully because it's better. Learning new/different things is just a small step further beyond learning old things with new/different assistants. And maybe some things will have to be cut out, like 17th century prose-proofs (edit: and even just moving to structured proofs without full formal tools is an improvement...), or square roots by hand (http://www.theodoregray.com/BrainRot/)

One thing that I can add, is that the process of neatly recording something really helps cement the process. My professor for dynamics and mechanics of materials required homework to include diagrams of the problem, neatly drawn, on unlined paper. Often I would find that each problem would take three sheets of paper (I'm a horrible draftsman), but I am horribly glad after the fact that I invested all that time.

It is painful, but I don't think there is any easy way of actually learning without just sitting down and doing problems. Have you considered auditing a course at a community college? Very few people (myself included) are motivated enough to work enough problems without the threat of assigned homework. You need to do enough problems on a topic that you are no longer struggling, then do 4-6 more. Those last problems are, IMO, the most important, they actually cement the concepts in long term memory.

As far as books, I can recommend Schaums Outlines for good examples of worked-through example problems.

Edit:fixed typos

To quickly review a broad range of math up until 1st or 2nd year of university, I really recommend Khan Academy https://www.khanacademy.org/math . I am currently using it to brush up my math skills for machine learning.

Before going to Khan Academy, I started reading a rigorous math textbook, but my motivation didn't last long. You really need high motivation to complete a rigorous textbook, but Khan Academy is different and I am finally able to continuously improve my math skills.

The best thing I like about Khan Academy is the large amount and instant feedback of exercises that you don't get from regular textbooks. I really wished that Khan Academy was there when I was a kid.

To get deep knowledge of math, I think that rigorous textbooks are the way to go, but before those and to prepare for them, I really recommend Khan Academy.

Being in your thirties has little to do with learning. How you learn is much more important than your age.

If you learn best in a classroom, you may have a local college that teaches math in the evenings. (I got my Master's in Statistics that way.)

If you learn best in small chunks, Khan Academy has differential and integral calculus and linear algebra, to start you out.

If you learn best from books... there are hundreds of great textbooks.

Best wishes to you. Keep up a lifetime of learning!

Very true.

My learning actually accelerated in my 30s because knowledge pays compound interest -- the more knowledge you have, the faster it is to acquire new knowledge. Assuming one has continued to pursue learning, someone in their 30s would have built up a significant enough semantic tree to pin new knowledge to.

Most people find it hard to learn in their 30s because they lack the energy, environment (+kids, +spouse, etc.) or internal drive that provides them the impetus. Others find it hard to learn because of bad habits and a poor foundation (their semantic tree wasn't that well built up in their youth). But their actual abilities (even memory) haven't actually degraded all that much.

And of course, there are some who find it hard because they have reached the limits of their cognitive abilities (un-PC as it sounds, this is a real thing). You have to know if this is the case. Most of the time it is not.

I would start by building up a good foundation. Learn the basics well but don't get hung up on understanding every little detail.

Chunk your learning and use your little victories to drive you (brain hack: humans are a sucker for little victories). Use the Feynman method (learn by teaching).

Drill yourself with exercises rather than trying to understand everything -- math is one of those things where it is easier to learn hands-on by working on problems BEFORE understanding the definitions fully... understanding comes later (the patterns will emerge once your semantic tree is solid). It's a process of cognitive dissonance where you actively wrestle with problems rather than passively work through them.

People who try to understand math by reading alone (or by watching videos) tend to fail in real life -- they tend to be able to recite definitions but their ability to execute on their knowledge is weak.

This is a standard rookie mistake, and the reason why so many American kids are weaker at math compared to their Asian counterparts. Drilling--even if mindless at frst--really does help, especially when you're starting out on a new subject. It helps you develop muscle memory which in turn gives you confidence to move to the next level.

I could say I'm in the same boat. Always wanted to learn such things, but never found the motivation to do so in an effective way.

I picked up various books and different learning strategies along the years but couldn't move forward cause I could not see any practical use for what I was trying to learn.

Fast forward a few years and now I'm learning both physics and mathematics.

What changed is I started working with 3D development for the furniture industry and a while later I got interested in woodworking.

Started doing some woodworking projects and had to learn some basic geometry and trigonometry to calculate cuts.

Now I'm interested in mechanical machines and electrical machines. To be able to build my own machines I have to learn some physics and other branches of mathematics and that's what I have been doing for the past months.

I probably cannot work with formal physics or mathematics but I was able to learn a lot of the concepts behind the formulas and calculations and I believe that is much more important, at least, at first.

The bottom line is you need to find something that motivates you and make you want to learn. That's how it worked for me.

Honestly, despite all the crap universities get, taking an undergraduate degree with a double major in physics and maths is an awesome way to do this. You'll meet people who are similarly passionate, be naturally competitive with them which is a motivating force not to be underestimated, and you'll meet a diverse set of teachers who each will have some awesome insights into these fields and you'll get to see first-hand how they think about solving problems.

Physics, and to a lesser extent maths[1], are topics where the top 1% of ability are actually concentrated at universities. My advice would be find a cheap university nearby and start enrolling in courses. If you're bright, motivated and take ownership of your own learning, the faculty will love interacting with you. If you're doing it to learn, don't sweat about the prestige of the place. There are people everywhere who will be much better than you at this stuff, and in some ways it's extremely motivating if you feel like with some hard work you can surpass some of your teachers, and it's extremely motivating when the best teachers recognize you as having more potential than the average student. You're never gonna feel either of these things at MIT.

[1] The problem with maths in academia is that it's massively biased toward proofs of mathematics and not use of mathematics. I've met very few PhD mathematicians who are even as close to as good at applying appropriate mathematics to problems then someone with a PhD in physics who consider themselves >50% theorist. PhD mathematicians are wonderfully knowledgeable if you say "tell me about this field of mathematics" and it's a field they know. But there is a certain extent to which they like to work by building things on a frictionless ice world, and get uncomfortable if asked to build something on the rough ground of the real world.


I'm 30 and trying to relearn the math courses I did in college (Computer Science degree) and more. I am currently using Standford & MIT's open couseware. I feel like I am moving slower than I would if I were in a course but able to grasp the material better at this rate... I made good grades in my math courses but like you, I didn't have to use them in software engineering that much. I would like to get into a field that requires a stronger grasp of mathematics but also has a need for programming and computation (maybe machine learning or computational biology). I feel like I'm getting tired of being a software engineer (defense contractor) at a small company and looking for something higher level

Calculus (with a pdf version of the text book): https://ocw.mit.edu/resources/res-18-001-calculus-online-tex...

Linear Algebra (text book link: https://www.amazon.com/exec/obidos/ASIN/0980232716/ref=as_at...) https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

Optimization course & book link (Stanford) https://web.stanford.edu/~boyd/cvxbook/

Statistics: http://greenteapress.com/wp/think-stats-2e/

I'm taking you at your word and assuming you truly want to reach the cutting edge of knowledge and learn things like QFT, Gauge Theory, String Theory, etc. alongside the math needed for it.

This is a long-term project, so I'd recommend by starting a bit with "learning about learning". There is a great, fairly short Coursera course called "learning how to learn".

Things not covered in the above course: - Your learning ability is not actually much lower in your 30s than it was in your 20s. You have a relatively benign rate of learning decline, until your late 50s / early 60s, when it drops quite a bit. You can still learn a lot, but it's meaningfully harder. (learning rate != thinking rate / creation rate!!)

You'll likely be able to write good papers into your late 60s, and perhaps 70s. There are exceptions and people who do significant work even later, but that's more unlikely.

When I started learning stuff again in my late 20s, I felt frustrated because I'd take a couple courses over a year, and by the time the year's over, I'd forget the first one. We all know what this feels like - we've forgotten most of what we've learned in college that we don't use in our profession.

When I was a teenager, I had great recall for things I've learned only once or twice. I didn't realize this was unusual and thought I got very bad at learning. In fact, the vast majority of people, including high IQ people, will need spaced repetition and study to retain things for a long period.

I'd recommend the following "schedule" to absorb things into y memory permanently. (By "learn" I mean read, do problems, write summaries.. it's a wide range):

Days to repeat: 0 (initial learning), 1, 6, 15, 37, 93, 234, 586, 1464, 3662, 9155

This would suggest interleaving classes instead of learning things sequentially for optimal time management. It's also a bigger time investment than people usually think of upfront, but pays dividends later on as the material builds-up like a cathedral of knowledge.

Like other commenters I'll also repeat: Do problems, problems, problems. The struggle is where the learning happens.

On the other hand, I wouldn't worry too much about super-high IQ etc. I don't think it's a strict requirement to have an extraordinary IQ to learn grad school physics and math.

Great post. Rather thought provoking. Though I dropped out of college in my late teens, I started taking classes again 17 years later later am doing much better than I did before. As a sysadmin, I was always reading all sorts of subjects and pursuing different hobbies that further expanded my knowledge. The one subject I have been having issues is with Math, but that is due to lack of effort and stretching myself too thin.

That is a fascinating number series. Is it taught in the Coursera course you mentioned?

Hi, I calculated it basing it on the super-memo algorithm. That algorithm is more sophisticated and geared towards cards / smaller pieces of knowledge; but I think it works equally well as "re-study" / "re-learn" reminders for larger chunks of material. The overall idea is to capture https://en.wikipedia.org/wiki/Forgetting_curve fairly well with the repetition frequency.

I took a Masters of Mathematics with the Open University in my thirties.

The (my) short answer is grind. Get a good textbook on subject of interest, start reading, start scribbling, start answering the questions. That's how I did it. Three or four days a week, two to six hours a day, grind grind grind GRIND GRIND GRIND GRIND. It's geology; time and pressure.

Now and then, when really stuck, finding someone who can illuminate a point for you is worth it, but that helped me a lot, lot less than one might think.

Anticipate not understanding large chunks of it. Anticipate pressing on anyway. Anticipate not being able to answer many questions. Anticipate having to read three or four different treatments of the same thing in order to get a real understanding. Anticipate that some of it you will never understand. Anticipate that watching youtube is not a substitute. Anticipate that the sheer information density of well-written text means you might spend an hour on a single page. Anticipate questions taking you six hours to solve, leaving your table and floor strewn with the history of your consciousness. If you're prepared for all that, and it's a price you're willing to pay, there is no reason to not simply start now. Pick up the first good textbook, start grinding now. Time and pressure. It's so easy to waste time preparing to start learning; beyond making sure it's a decent textbook and getting some pencils and paper in a quiet room, the only preparation is accepting that this is going to be a long grind, and embracing it.

I'm doing my OU Masters in Maths now, in my 40s. It is definitely hard, but I'm enjoying it. Personally I struggle to learning things thoroughly unless I'm working in the subject, or I have exams to do. My own learning workflow is to flip through a chapter to get an overview, then read/re-read it thoroughly, then go through the exercises on the chapters quickly looking at the answers. Read the chapter again, then try doing the exercises without help. I've found YouTube pretty good for getting the intuition behind some ideas.

Read some books, practice exercises, and find an area of interest.

Start with some liberal-arts introduction to a particular topic of interest and delve in.

I often find myself recommending Introduction to Graph Theory [0]. It is primarily aimed at liberal arts people who are math curious but may have been damaged or put off by the typical pedagogy of western mathematics. It will start you off by introducing some basic material and have you writing proofs in a simplistic style early on. I find the idea of convincing yourself it works is a better approach to teaching than to simply memorize formulas.

Another thing to ask yourself is, what will I gain from this? Mathematics requires a sustained focus and long-term practice. Part of it is rote memorization. It helps to maintain your motivation if you have a reason, a driving reason, to continue this practice. Even if it's simply a love of mathematics itself.

For me it was graphics at first... and today it's formal proofs and type theory.

Mathematics is beautiful. I'm glad we have it.

[0] https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...

Update: I also recommend keeping a journal of your progress. It will be helpful to revisit later when you begin to forget older topics and will help you to create a system for keeping your knowledge fresh as you progress to more advanced topics.

Here's how I would do it, my 2 cents:

1) Find a good source of information --- typically, this is either very good lectures (like on youtube), a good textbook, or good lecture notes.

2) Do problems. There is a fairly large gap between those that just watch the lectures and those that have sat down and try to go through each and every step of the logic, and that's what everyone here (on HN) is pointing out when they similarly mention doing problems.

2b) Have solutions to those problems. I make this a separate point because it's important to spend quality time on a problem yourself before looking at the solutions. At the end of the day, if you read the problems and then the solution right away, that's much closer to reading the textbook itself instead of the more rigorous learning one goes through when trying things themselves.

If you were to ask me what textbooks or lectures I recommend, I think that's a more personal question than many here might guess. What topics are you most interested in? Are you really just solely interested in a solid background? How patient are you when doing problems?

Regardless, I'll give my two cents for textbooks anyway. In no particular order:

1) Griffiths E&M: https://www.amazon.com/Introduction-Electrodynamics-David-J-...

2) Axler, Linear Algebra Done Right: https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Ma...

Good luck!

First, you can't go back to your twenties and you shouldn't try.

When you are still in a school environment, there's an environment that for doing problems for problem's sake. For all you have trained so far, be able to solve problems is how you were measured and feels like a life's purpose.

By thirty you probably get the hint that life is not about solving fake problems, and most of the knowledge you learn at school is useless and pointless. If you try the suggestion to sit down and do exercises, I doubt you would be able to keep at it long enough for any gain.

But at thirty, you have the luxury of not worry about midterms and finals and you probably can afford multiple books. So first, find your love of physics. Second, collect old text books. Third, read them. Fourth, criticize them. Fourth, throw away the book that you've digested or is bad. When you can throw away all the books (the knowledges are all online anyway), you are learned.

I'm learning Japanese at 35 with the goal of becoming business-fluent in five years. I decided to be very systematic about it.

Anki is a very good memorization tool, I would use it aggressively.

I study for one hour every day before work. Math probably requires more time to grind through hard problems.

I would hire a tutor or find a partner.

A few things I'd recommend:

write down why you're doing what you're doing.

write down the gap in your abilities you'd like to fill so you can track your progress

don't compare yourself to others, compare yourself to yourself.

It is very possible to develop engineering math chops late in life (I did it!) but outside of a fulltime education context it requires organization and sacrifice.

edit: on the bright side, I'm so pleased with my current language learning pace that I intend to double down on it and make it a lifelong commitment. It really feels good to learn something new and interesting. I'm eyeballing self-learning an EE degree next, so I'm curious how it goes for you

Oh Anki, how I love it. 2199 cards and counting

Math is something that needs to be learned by rigorous practice

I find it helpful to first learn the theory via 3blue1brown


I'm currently working through the bookofproof math problems found here


It goes into how to actually write mathmatical proofs / discrete math which I believe is extremely important to any math branches and computer-science in general.

I was lucky to have a great foundational math background in highschool, my calc 1/2 teacher was considered one of the best in my state. I practiced at least 30 problems every night in that class for 2 years

If you are located in Los Angeles, I would recommend looking into the Michael Miller math class series taught thru UCLA extension. This is an 11 session 7-10pm math graduate level math course.

I've taken courses on Algebraic Number Theory, Lie Groups and Lie Algebras, and Measure Theory. It's hard, but definitely doable. It is definitely more on the abstract math side, which I enjoy a lot. It is very different than the practical engineering-focused math I learned in school.

One of the students has an introduction into what you can expect. http://boffosocko.com/2015/09/22/dr-michael-miller-math-clas...

The epitome of what you want is to find a mentor, a chalkboard, and 3-4 hour chunks of time you can dedicate to learning. Repeat about 2x a week for a year, and do independent study with a book on one side of the table and a notepad on the other between classes.

I enjoyed reading the No Bullshit Guide to Math and Physics: https://gumroad.com/l/noBSmath It's a brief overview but covers a lot of topics. The author's writing style and the book's layout worked well for me.

The same person wrote a book for linear algebra as well but I have not read it: https://gumroad.com/l/noBSLA

Hi atxhx, thx for the plug!

Here is a link to an extended PDF previews of the books: https://minireference.com/static/excerpts/noBSguide_v5_previ... and https://minireference.com/static/excerpts/noBSguide2LA_previ...

May I ask what your goal is? Do you want to be able to say read papers? Do research? It's easier to offer suggestions from there. Are you more interested in math or physics? I think at your stage, having a problem that interests you and then seeking out knowledge to understand that problem might be a good way to go...What is your current area of work? For example, if you are a programmer, then collaborating with a physicist or mathematician on a problem could be a worthwhile exchange.

For motivation, if there is a nearby university, start attending the relevant departments' colloquia. They are generally open to all, and will start to get you up to date on what's new across all of math/physics.

My favorite undergraduate students when I was TA'ing were all students who had returned to school after spending time in the world. They knew why they were there, knew that the material was worth learning, and asked lots of questions.

Go get it -- start small, don't stop.

This will expose you to cutting edge research going on. But I disagree entirely on this approach to learning fundamentals of math and physics.

The colloquia are usually very subject specific. When I did my PhD in condensed matter physics, depending on the speaker, sometimes it could be 10 minutes into the lecture when it delves into narrow field-specific material I don’t understand (eg, a speaker talking about particle physics or astrophysics). Good speakers and even non-physicists can follow the whole talk regardless of subject matter. But good speakers are rare. And colloquia are typically for benefit of the department (students + faculty), so will quickly gloss over fundamentals into the real meat.

So you might learn about super cool research happening, which is great. But you aren’t likely to learn key fundamentals.

I went back to school in my late 20s for this. Ultimately what everyone says is true, you learn the stuff by doing problems and at the end of the day lectures are of marginal use and really the learning happens when it's you and the textbook(s).

I personally went back to school because it was a way of putting pressure on myself. There's a lot of boring drills before it gets particularly interesting. The biggest value of school/coursework for me was the fact that it was a declaration of "I am going to do this now, and there are external motivators (grades, exams) to ensure that I stay on track."

That said, if I had it to do over again, for the money I spent, I wonder if hiring a graduate students/postdocs or even professors as tutors would have been better. I suppose that even then, when the material gets boring or time consuming it's easy to walk away, whereas if you invest time and money into a class you basically are in good shape to shame yourself into succeeding. (plus being in the academic environment helps to get a better sense of the broader landscape of material)

Maybe you could find a buddy to work with, like people do with the gym or whatever to shame each other into staying on task.

Also, I should mention, one big lesson learned... Maths build on each other. Don't be embarrassed to start with a pre-calculus text/course.

You probably don't remember much from high school, and most low-div calculus problems are some simple calculus rules combined with a bunch of high school algebra/arithmetic manipulation that if you don't have it all ready at your fingertips, you _will_struggle. (this means memorizing all those trig identities again) The hardest part of low-div calculus is this stuff... The actual rules of calculus are simple, easy to grasp and dare I say it, intuitive.... if you can do the pre-calc.

What works for me is purchasing and reading textbooks (look for online college syllabuses for good ones). Probably the best way to read maths texts is to work the problems, but what I do is read it through once or twice. Then switch to a different text on the same subject. Things will slowly start to click, although of course you don't understand something until you can explain it (i.e. write a blog post about it) to someone else.

> purchasing and reading textbooks

Plus, you can get really, REALLY good deals on used college textbooks (some of which are still in pristine - as in never even been opened - condition).

https://www.3blue1brown.com has you covered for linear algebra. Now as far as the math needed for physics, I highly suggest Roger Penrose's 'Road to Reality' (https://www.amazon.com/Road-Reality-Complete-Guide-Universe/...). As the reviews say it's not an easy read but what it does provide you with is all the mathematics you're going to need to learn to understand today's physics. The book provides a high-level overview of the mathematics - which is technically complete but so concise that it's difficult to learn from. So use that to take a deeper dive into a mathematical subject. What the book is really providing is a roadmap: you need to understand these concepts from these mathematical disciplines to understand this area of physics and then proceeds with the high-level description of those concepts. Take the deep dive as needed and you'll be amply rewarded.

Covered for linear algebra? Yes those videos have some nice visuals but the material is just scratching the surface.

If you want to be serious about math you should get a feel for what mathematics means to mathematicians.

A common view is that mathematicians prove theorems. While true, a skill that is not emphasized enough is learning (by heart) and understanding definitions. Learning by heart here also means something slightly different than simply being able to recite definitions and theorems. You have to be able to compare the objects that you define and get a feel for how a definition is really a manipulation of a basic intuition.

A great book to start with is Rudin's Principles of mathematical analysis. You will get a much deeper appreciation of what calculus is. If you want any chance of understanding the mathematical tools used in theoretical physics (operators, Hilbert spaces, Fourier decompositions to get solutions to differential equations etc.) you will have to understand analysis.

You might also find https://physicstravelguide.com/ useful. It's an expository wiki that sorts physics and related math explanations/books etc. by the required level of sophistication.

This means, especially as a beginner when you are stuck you can easily find an explanation that you can understand.

I would recommend that you start with physics and only learn math on a "just-in-time learning" basis. Mathematics is infinitely large and it's too easy to get lost. Physics, in comparison, is relatively constrained. When you learn the basics of modern physics you will automatically get a good understanding of lots of important math topics plus you will always automatically know why they are important.

I'm in a similar situation myself. My plan is to buy a high school textbook and work my way through it, chapter by chapter. I assume that the selection of topics in such a curriculum is reasonable and if the presentation is deficient I'll supplement with YouTube etc until I understand.

Same here. I've been eyeing few books on Amazon myself. Most come with answer sheet at the back to check your work.

It's all about doing it like we did in high school. Pen, graph paper, and maybe a calculator. Drilling down the practice will help with theory.

You want personal sessions with a mathematics professor to help plan your curriculum and direct your learning! Here's where you can make that happen:

Check out https://edeeu.education/. The founder, Alex Coward, is an ex-Berkeley math professor. To apply to have him as your director of studies, fill out the form at the bottom of https://edeeu.education/alexander-coward. This is affordable (currently $330/month).

I have the same aims as you, also learning math/physics in my 30s, and it's been a huge boost to move from purely self-directed online learning to having sessions with a math professor and a planned curriculum.

I did the same as you, but in my 40’s. It’s better to start with books that are “too easy” than those that are “too hard” (for your level, whatever that is). I started with Ken stroud’s ‘engineering math’, then did calculus 1,2,3 and linear algebra at community college (online, with proctored exams), then used chartrand’s “mathematical proofs” to learn proofs, which you must know in order to learn, understand and enjoy upper div math (and Physics). I’m now doing an MS in math and stats and loving it, but you need to pace yourself in a way that you can understand and enjoy the math and physics you are doing - and as many people have commented, that means doing lots of problems . Feel free to message me if you want to chat

Take refresher classes at community college. This is what I did. I did all the calculus and linear algebra classes on offer. For me this was very valuable. My goal was to be able to read mathematics in research papers.

A phenomenal resource for physics is the text for the Feynman lectures. And the great part - they're all online, excellently organized, and free: http://www.feynmanlectures.caltech.edu/ The lectures alone will not provide mastery, but they will provide a very solid foundation for understanding the breadth of most of all of physics. The one thing those lectures are desperately missing is a similarly well organized and presented problem set.

Was a physics student. IMO the best book you can drill questions from is Boas' Mathematical Methods in the Physical Sciences. It offers fairly succinct yet comprehensive overviews of various fields of math. I still have my copy sitting at home. It is considered an essential textbook for any physics student.


I am not sure where you got the impression that it's essential for any physics student, but I have never used it and don't think it's mattered much in my upbringing as a physicist.

I thought I would also add my two cents, though there have been many excellent responses already. I am about to defend my PhD in Physics at MIT.

First of all - great idea! It is never too late to learn math and physics! In fact, with hard work and commitment, anybody can muster them to a high level.

(1) Reading =/= understanding in math and physics. You understand a topic only if you can solve the problems.

(2) Work through the solved problems you encounter in textbooks carefully.

(3) Most people around me have never read any physics textbook cover to cover. E.g. reading Halliday, Resnick & Walker completely might take you years! Not all topics are equally important. Focus on the important parts.

(4) You need guidance on what is important and what is not. Online courses, college material (especially problem sets!), teaching webpages could be a helpful guide. Checkout MIT OCW, once you are ready.

(5) Finding someone to talk to is really useful. You will likely have questions. Cultivating some relationship that allows you to ask questions is invaluable.

(4) College courses in math and physics have a very definitive order. It is really difficult to skip any step along the way. E.g. to understand special relativity, you must first understand classical physics and electrodynamics.

(5) Be prepared that the timescales in physics are long. Often, what turns people off is that they do not get things quickly (e.g. in 15-30 minutes). If you find yourself thinking hours about seemingly simple problems, do not despair! That is normal in physics.

(6) You have to 'soak in' physics. It takes time. Initially, you might feel like you do not make a lot of progress, but the more you know, the quicker it will get. Give yourself time and be patient and persistent.

(7) Often, just writing things down helps a lot with making things stick. It is a way of developing 'muscle memory'. So try and take notes while reading. Copying out solved problems from textbooks is also a good technique.

(8) Counterintuitive: If you get completely stuck, move on! Learning often happens in non-linear ways. If you hit an insurmountable roadblock, just keep going. When you return in a few days/weeks, things will almost certainly be clearer.

Buy a book and take a course. It won’t be enjoyable but being examined on what you learn will provide focus. Find a small group of people doing the same or similar learning, so you can discuss the different problems each of you will have. Persevere and plan carefully. Otherwise you’ll waste time and energy. Don’t get sidetracked by great presentations and new tech from internet-based resources; remember what the objective is...

Why? What purpose do you have for textbook physics? Think of what you're going to do with that knowledge. Are you going to become a physics teacher? Do you just want to impress "a bunch of folks on the internet"?

The time in your life for grinding on textbook knowledge is over. That was age 0 to {insert age at end of college/academic career} non-inclusive.

Think very carefully where you want to spend your motivation and discipline.

Understanding the "Why" important. However, it seems clear that he/she has already answered that question, and now they are trying to figure out the "how". Knowledge is power no matter where it comes from - a textbook, the internet, a master or simply studying the natural world. Seems odd to discourage someone from expanding their understanding of the world.

I fell in love with physics by reading this guy: Paul G. Hewitt https://www.amazon.com/Conceptual-Physics-High-School-Progra...

I've posted a full undergraduate curriculum based on free resources here:


I'm also accepting students. See the website for how that works. It's a lot of fun!

This is a bit of an evergreen so just searching HN will net you piles of threads with lots of advice and references to resources.

Pick a related topic that makes use of what you want to learn, and learn what you want to learn as side effect from practice in the related topic. Another poster already mentioned linear algebra for simple computer graphics.

From there you could branch out into more dynamic stuff, like realtime 3D rendering or particle simulations, where you'd need calculus.

Getting the right material to study is only small part of it. The real difficult part is finding all the time it takes and also finding company that has similar interest and is willing to invest time seriously with you. I struggled with the latter part. It is very frustrating to not be able to ask someone if what I am doing is right or not.

You might find my site interesting. It attempts to cover basic algebra in a more formal, proof oriented style. I designed it with adults revisiting mathematics and wanting to move on to higher mathematics in mind.


If you don't have the patience for doing random math problems and just want to understand things abstractly, try this channel:


But how do I learn math/physics in 2018?

Jokes aside, try https://www.coursera.org/ or https://www.khanacademy.org/

The Feynman Lectures on Physics. I found 50th anniversary hard-bound edition at the Los Alamos book store. Very readable. Feynman explains things like no other. The audio recordings are out there, though video should have been made of these. A real loss for humanity.

There are recordings of the lectures on YouTube. I'm not sure if they're complete or not, but there's a good bit there. I just wish Feynman had presented his Lectures on Computation similarly. The book is great, though usually hard to find.

I learned Calculus myself through a book called "Calculus - an intuitive and physical approach", by Morris Kline. It is cheap and very didatic.

I never really studied physics, but I found the first books from the "Feynman lectures on Physics" to be very good.

I forgot to mention that the Feynman lectures on Physics are available online for free: http://www.feynmanlectures.caltech.edu/

Pick a problem that interests you and involves lots of physics and math. This worked for me.

I've had the same thoughts about (re-)learning some math. I ran across this a while back, anyone know if these are good?


You should go here: https://betterexplained.com/ I found this site explains basic math concepts intuitively and very easy to understand.

Youtube, EDX , brilliant.org and Khan academy are all good resources. I'm taking hybrid online Math classes at my local community college; trying to get through all the Math requirements.

I am doing this now. I just took a Discrete Math class and I am taking a calc refresher in the Fall. I have been reading several text books as well for practice and reinforcement.

Start with simple books to warm up those grey cells. For maths, I recommend Mathematical Circles: Full of fun discrete math problems.

I would suggest getting text books with loads of homework problems with solutions and actually sit down to work through the problems.


Get a piano, look up the basics of how to read music, find the keys on the piano, see my post on music theory and the Bach cello piece, get a recording of some relatively simple music you do like, get the sheet music, and note by note learn to play it. After 3-4 such pieces, get an hour of piano instruction and continue on.

Violin: Much the same except need more help at the start. From my music theory post, learn how to tune a violin. Get a good shoulder rest -- the most popular is, IIRC, from Sweden and is excellent. Look at images of violinists and see what rests they are using. Get Ivan Galamian's book on violin. Start in the key of A major and then branch out to E major and D major. Get some good advice on how to hold the violin and the bow; look at pictures of Heifetz, etc. Learn some scales and some simple pieces, get some lessons, and continue.

Math: High school 1st and 2nd year algebra, plane geometry (with proofs), trigonometry, and hopefully also solid geometry. Standard analytic geometry and calculus of one variable.

For calculus of several variables and vector analysis, I strongly recommend

Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/} Addison-Wesley, Reading, Massachusetts, 1957.\ \

Get a used copy -- I did. Actually, it's not "modern" and instead is close to what you will see and need in applications in physics and engineering. There, relax any desire for really careful proofs; really careful proofs with high generality are too hard, and the generality is nearly never even relevant in applications so far. Maybe do the material again if want to do quantum gravity at the center of black holes or some such; otherwise, just stay with what Apostol has. For exterior algebra of differential forms, try hard enough to be successful ignoring that stuff unless you later insist on high end approaches to differential geometry and relativity theory.

Linear algebra, done at least twice and more likely several times. Start with a really easy book that starts with just Gauss elimination for systems of linear equations -- actually a huge fraction of the whole subject builds on just that, and that is close to dirt simple once you see it.

Continue with an intermediate text. I used E. Nearing, student of Artin at Princeton. Nearing was good but had a bit too much, and his appendix on linear programming was curious but otherwise awful -- linear programming can be made dirt simple, mostly just Gauss elimination tweaked a little.

Mostly you want linear algebra over just the real or complex numbers, but nearly all the subject can also be done over any algebraic field -- Nearing does this. Actually, might laugh at linear algebra done over finite fields, but the laughter is not really justified: E.g., algebraic coding theory, e.g., R. Hamming, used finite fields. But if you just stay with the real and complex numbers, likely you will be fine and can go back to Nearing or some such later if wish.

So, concentrate on eigen values and eigen vectors, the standard inner product, orthogonality, the Gram-Schmidt process, orthogonal, unitary, symmetric, and Hermitian matrices. The mountain peak is the polar decomposition and then singular value decomposition, etc. Start to make the connections with convexity and the normal equations in multi-variate statistics, principle components, factor analysis, data compression, etc.

Then, of course, go for P. Halmos, Finite Dimensional Vector Spaces, grand stuff, written as an introduction to Hilbert space theory at the knee of von Neumann. Used in Harvard's Math 55. Commonly given to physics students as their source on Hilbert space for quantum mechanics. Likely save the chapter on multi-linear algebra for later!

For more, get into numerical methods and applications. You can do linear programming, non-linear programming, group representation theory, multi-variate Newton iteration, differential geometry. Do look at W. Fleming, Functions of Several Variables and there the inverse and implicit function theorems and their applications to Lagrange multipliers and the eigenvalues of symmetric or Hermitian matrices. The inverse and implicit function theorems are just local, non-linear versions of what you will see with total clarity at the end of applying Gauss elimination in the linear case.


Work through a famous text of freshman physics and then one or more of the relatively elementary books on E&M and Maxwell's equations. Don't get stuck: Physics people commonly do math in really obscure ways; mostly they are thinking intuitively; generally you can just set aside after a first reading what they write, lean back, think a little about what they likely really do mean, derive a little, and THEN actually understand. E.g., in changing the coordinates of the gradient of a function, that's not what they are doing! Instead they are getting the gradient of a surface, NOT the function, as the change the coordinates of the surface. They are thinking about the surface, not the function of the surface in rectangular coordinates.

For more than that, you will have to start to specialize. Currently a biggie is a lot in probability theory. There the crown jewels are the classic limit theorems, that is, when faced with a lot of randomness, can make the randomness go away and also say a lot about it.

For modern probability, that is based on the 1900 or so approach to the integral of calculus, the approach due to H. Lebesgue and called measure theory. In the simple cases, it's just the same, gives the same numerical values for, the integral of freshman calculus but otherwise is much more powerful and general. One result of the generality is that it gives, via A. Kolomogorov in 1933, the currently accepted approach to advanced probability, stochastic processes, and statistics.

That's a start.

YouTube is my preferred method of learning. It's easy and can quickly expose you to a variety of teaching styles.

get spivaks calculus. Took me months to get though chapter 1 :D, but gave me through understanding of how to think about maths and how to prove stuff and that proof are the real fun of math. If you can't prove it, you don't understand it.

I have no idea why someone would have down-voted you - Spivak is brilliant. It's not really about calculus, it's really about Real Analysis, and it's excellent.

Highly recommended.

I think your goal to "learn math/physics" is too broad. Steven Hawkins was still trying to learn about physics before his death.

Define what "understanding physics" means to you and then figure out how to get to your goal.

There are two approaches that work well. The first is to embark on the standard, formative curriculum. The second is to start with a handful of problems that interest you and go pick up stuff piecemeal on the way to solving those.

Approach 1. Decide if you want to learn physics or applied mathematics. They're not the same. The math you describe is more on the operations research side of applied mathematics, and not terribly relevant to physics. You say your goals are physics. In that case you're in luck. The curriculum is utterly standard and consists of four passes through the material.

The first pass is one or two years long and is roughly what's in Halliday and Resnick or Tipler's physics books: Newtonian mechanics, some wave motion, some thermodynamics and statistical mechanics, electromagnetism, and a little "modern physics" (special relativity and a bit of quantum theory). Meanwhile you study calculus of a single variable, multivariable and vector calculus, and a little bit of ordinary differential equations, and do a year of laboratories.

The second pass is a semester of classical mechanics covering Lagrangian and Hamiltonian mechanics, a semester of statistical mechanics and thermodynamics, a year of electromagnetism, and a year of quantum mechanics, paired with a year of mathematical methods (linear algebra, special functions, curvilinear coordinates, a little tensor calculus, some linear partial differential equations, and a lot of Fourier analysis) and a year of more advanced laboratories. Here ends the undergraduate curriculum. At this point astrophysicists tend to separate off and start learning the knowledge for that domain instead of the second semesters of electromagnetism and quantum mechanics. Their labs are also different.

The third pass is the first couple years of graduate school, and goes through the same subjects again in more depth. No labs this time. A mathematical methods course only if a student needs more help. Quantum field theory for those going that direction. General relativity for those going another. Advanced statistical mechanics or other special topics for those going into condensed matter.

The fourth path is the student by themselves, integrating it all in preparation for doctoral qualifying exams.

If this approach sounds like what you're after, start by getting a 1960's edition of Halliday and Resnick's physics (which is better than the present editions and quite cheap used), a Schaum's outline of calculus.

Approach 2. Pick a handful of problems. What actually interests you? Not what sounds fascinating or what seems prestigious. What's interesting? Cloud shapes? Bird lifecycles? What are you actually curious enough about to spend some time poking at?

I'm a bit wary of some of the suggestions here. I'm a similar position to you (at least, when it comes to maths). The difference for me is that I'm fairly certain that I'm one of the least accomplished people responding to your post. Whereas most of the others responding appear to be of the same ilk as the people you idolise. I know I'm cut from a different piece of cloth as those people (and I don't mean that resentfully - I just want to be realistic about myself). So some of the suggestions may be perfect for them. For people like me (and I'm not suggesting that you are, but your circumstances sound similar to mine), something with more of a safety net may be more realistic. I know I don't have the fortitude to sit and persevere with a text book for hours on end. Sometimes, I need some of the groundwork to be laid down for me - at least, when it comes to things like maths and scientific ideas.

I can't say I've found a successful way of learning and retaining maths knowledge over the past few years. That retaining bit is important - I can pick up something, give it a go and get it right, but if I don't do it again, I forget what I've learnt. I had some early success with the OU book series 'Countdown to Mathematics' (https://www.amazon.co.uk/Countdown-Mathematics-1-v/dp/020113...), but the problem ultimately is that it covers materials in a block format; once you've covered a topic and solved problems, you move on to the next topic. Khan Academy has the same issue and is arguably worse because, so far as I've found, it requires learners to know what they don't know, rather than structures a learning programme where each topic follows sequentially (maybe there is a way to do this but I've found the KA UX to be complicated and confusing).

I've tried more 'sophisticated' maths learning solutions that claim to account for learners' knowledge and weaknesses, but there are various shortfalls with them and none is aimed at learners older than schoolchildren. It's not so much the provision of learning materials that I'm frustrated with but the process of testing and retaining what I learn. I would dearly, dearly love to find a tool that can assess what I know, tell me what to learn next, test me on the topic BUT continues to do so as I progress, making use of spaced repetition and interleaving. It seems like the perfect use of both techniques - like Anki - but so far as I've found, nobody has done this for maths learners. Anyway, I've digressed...

I studied Mechanical Engineering but gradually ended up as a full fledged software engineer from one who wrote ME related programs. So I have sort of always been in the midst of mid level maths and physics, but, as it happens, I lost it all except for some fundamental concepts. I had a similar epiphany as yours in my early thirties and this is what I did and it helped me greatly.

First of all, you have to realize that you are learning these stuff only for the sake of learning, as an intellectual challenge, rather than making a career out of it. So, you need not follow a pattern that is made for late teen students attending university. That includes not needing to religiously solve problems in textbooks, especially the numerical ones. If you have to, solve the conceptual ones. Further, you wouldn't even need textbooks.

What you do need, however, is a perspective. Unlike university students whose primary aim is to pass the exams that matter, graduate and get a job, for which they learn by rote, what you need is to understand the reason why learning certain topics is essential. You need to learn about the topics first in order to appreciate whatever you will encounter later. For this purpose, well regarded popular science books should be your first choice. Any worthy textbook will also give a bit of perspective in the preface, early chapters or the appendix, but it can never compete with popsci books which are designed for the purpose of elucidating academically complex stuff to an otherwise educated public. Good popsci books provide enough stimulation to your mind. They prime you for further, formal exploration of the topic.

Often times what happens is that highly technical topics such as those you mentioned look attractive from the outside but get painfully dry, boring and difficult once you pick up a text book and start studying from it. What I realized is that for many topics what I really crave is a deep understanding as an educated person as opposed to deep academic knowledge. For such topics popsci books regarded as 'hard' are better than proper textbooks. These days, with the availability of opencoursewares, you can simply watch a few lectures before deciding if its worth your time delving deep into the topics with a textbook. The textbooks are often listed in the course content, and we all know where we can find the appropriate pdf versions.

These books worked for me. They are in my home library and I try to flip through some of them once a year.

For maths, books like Prime Obsession by John Derbyshire, e The story of a Number by Eli Moar, Number by Tobias Dantzig, The Unknown Quantity by John Derbyshire provide good orientation for number theory, analysis & calculus and algebra. The second book might be tough to finish as its really really deep for a thin book. You might also look at books like God Created the Integers by Hawking, The Calculus Gallery by William Dunham that curate interesting and historically important results, where original works are often reproduced.

For physics, my goto book (which I have never finished) would be The Road To Reality by Roger Penrose. This self contained 1000+ page monster builds up to advanced physics in a methodical fashion from scratch. The first half deals with all the necessary maths from elementary to advanced and the second half covers physics. It has chapters with names such as calculus on manifolds, hypercomplex numbers, the entangled quantum world, gravity's role in quantum state reduction etc which sound pretty deep and complex for a recreational learner. Beyond that, you can also read books such as The Evolution of Physics by Einstein himself, Thirty Years that Shook Physics by George Gamow to get a historical perspective on the development of advanced physics. For a midway academic treatment, you can read Feynman's lecture on Physics volumes.

For discrete maths, books like The Algorithm Design by Skiena and Algorithm Design by Kleinberg and Tardos should serve you well. They don't push you into a maze of mathematics like Knuth books do, or a maze of source code like CLRS or Sedgewick books do, but show you the practical side of things with limited code and pseudocode. That said, Knuth's The Art of Computer Programming 4A Combinatorial Algorithms Part 1 will just blow your mind.

Here's how I go from dumb to less dumb:

- on Saturday, I wanted to learn how to write a Kubernetes configuration file from scratch, so I decided I'd deploy a static web page

- for the static web page, I decided it should return pictures of teapots w/ a 418 status code, and initially tried to return responses using netcat, which I got working on my local machine, but not in a container

- instead of using netcat, I decided that nginx is for people that don't like over-engineering their weekend hack projects, so clearly I needed to write my own web server and hacked out a janky Elixir server that serves up a poop emoji teapot image [1][2]

- then I started working on an overly over-engineered HTTP server, which so far only has date headers [3]

- then last night I randomly wondered how HTTP 2 works, looked for the RFC [4]

- then I remembered working on the date header, and I wondered how headers work in HTTP2, and I learned they use Huffman encoding, and so my next side tangent is to read up on Huffman encoding and add HTTP 2 header support to my HTTP server! [5]

TL;DR -- I made a poop joke and turned it into a learning opportunity

[1] https://imgur.com/a/rZcL0Rw

[2] https://github.com/amorphid/i_am_a_teapot_container

[3] https://github.com/amorphid/hottpotato-elixir

[4] https://tools.ietf.org/html/rfc7540

[5] https://tools.ietf.org/html/rfc7541

This is awesomely overengineered and not-invented-here, and I mean that in the best possible way. Well done!

First up, I am no expert, but I have traveled this road for a while so I'll share a bit. I see Susskind's theoretical minimum is plugged here. Good. That is a fantastic place to start. Start at the beginning. Skip nothing.

Leverage what you know against what you don't. You remember how to find an extremum of a function from Cal 1? The first order necessary condition for an extreme point is that the derivative of the function be zero. Remember that when you get to (deterministic) optimization and a gradient needs to be zero. Next with a constrained optimization you will see the reformulation of an objective function and constraints into one functional by using new variables (Lagrange "multipliers") so that when the gradient of this new functional is zero, not only are you somewhere in the intersection of the constraints and the objective, but you also meet the first order conditions necessary for an extremum to be found. (Second order sufficiency conditions (SOSC) are needed to show that you aren't instead at an inflection point but we are moving fast and breaking things)

Hmm, I didn't say that very well, but there is much intuition to be found in optimization problems. This will serve you well in physics.

Calculus of variations: Nail down cold how to derive the Euler Lagrange Equations for a functional. Hint, Apply the FONC (first order necessary conditions for an extremum) to the functional (now the Action) in question. See Lanczos "Calculus of Variations" (Dover Books) to sort out your initial questions and learn the smooth little trick with integration by parts. Susskind's first book comes in here.

Learn cold the integral of the Gaussian distribution and how to play around with it to find more complicated integrals. Orthogonality. Dot products and Fourier transforms have a lot in common! Fourier's trick is crucial. Oh man... have you got some fun in store here. Do not proceed to Quantum mechanics without having these tools at your disposal. Otherwise QM is just linear algebra over complex numbers together with complex amplitudes from circuits/naval architecture/ spring mass dampers in the frequency domain. (I mention all of these to raise awareness that complex amplitudes are not unique to QM) Supplement Susskind's second book with Griffiths intro QM book. Also see Griffeths E&M book for a great explanation of Fourier's trick and separation of variables in the chapter on special techniques... At least in the 1999 Third edition. (beware, it falls apart under little abuse!) Oh, and pick up the power series methods for solving tricky differential equations - don't learn it from the physicists. Learn it from "advanced differential equations materials". It's not bad in isolation - soon you will be computing recusion relations and bessel functions.

Soon it will be time to start thinking about field theory. A side quest is available if you want to get into fluid dynamics. Incidentally this is a great way to learn about uses for Green's functions... And if you dive deep, your first look at singular integral equations in field theory. But we will proceed dead ahead... to the book reviews (you will need more than one book):



Somebody else plugged 't Hooft here http://www.goodtheorist.science/ But there it is again. Now is a good time to speak of renormalization: There is an intro book out there: https://www.amazon.com/Renormalization-Methods-William-David...

It's okay but I have yet to derive more utility from it than from various field theory books. Oh and studying complex singular integrals in isolation is good too. Generally have some experience with contour integration around singularities.

See Sydney Coleman on symmetry breaking. (Man in the magnet, flip the sign of a term in your potential to get a mexican hat, see that the ground state is now different etc.)

There are tons of free resources out there for learning QFT. Use many sources. Expect to get stuck with any one of them. Bounce between them to un-stick.

Next up, differential geometry, tensors, and GR.... Google is your friend. I like Schutz's "a first course in general relativity" and really like Zee's Einstein Gravity book but I am working into this now and so my recommendations are running out. Check out this video as an intro to GR: https://www.youtube.com/watch?v=foRPKAKZWx8

To be inspired/get prepared for things to come get John Baez's Gauge-Knots_Gravity book: https://www.amazon.com/GAUGE-FIELDS-KNOTS-GRAVITY-Everything... and maybe Penrose's Road to Reality - which is like cosmology if the universe consisted of all the math and physics needed to understand all this math and physics. Baez's book probably has more exercises and is more focused in general. Stuff like this will help you see where more advanced mathematics comes in. By now you will be seeing manifolds and fibre bundles. Think of parameterizing a surface instead of a curve and see how a tangent bundle describes a whole new vector space - one vector space for every point in the manifold. Yang Mills... internal symmetry... ok I'd love to talk about how these are new expressions of ideas we've seen before but at some point up there we've passed my pay grade, I have to beg off until I can learn some more!

The best route is a community college. You can get most of this through a CC. Some offer discrete math, not many offer graph theory, and you probably won't get much astro (but you can get astronomy) or string theory from them. But it is good to get academic contacts who can give you direction. Someone who personally understands your drive and work ethic will have a better ability to give you suggestions. So this is my number one suggestion.

If you want to self learn, well let's go through some books and then youtube channels.

Books: Missing Discrete and Graph Theory

(You can get previous versions to save money. The content of these is mostly the same). Mostly in order of level (math then physics)

Calculus: Stewart's Calculus[1] (this is pretty much the standard) This has calc 1,2, and 3 (multi variable)

Linear Algebra: David Lay [2]. Start sometime after calc 2 (series problems). This will start you on some optimization and constraint solving. Stress learning eigen values/vectors and least squares. I don't have a good level 2 book, but that would mean looking into coordinate transformations, QR decomposition, and some more stuff.

Differential Equations: Blanchard Differential equations [3]. You will need diff eq to gain a true appreciation for physics. You will also gain a lot of the pre-req's for optimization and constraint solving.

Physics: Halliday and Resnik[4] is one of my favorites. But this is the lower college level (3 courses: Classical, E&M, Rel/quant). If you are relearning you can skip to below (though you might struggle a little more) Req: Taken or taking Calc 1 (differentiation and integration required later)

Classical Dynamics: Thornton[5] You will learn A LOT about constraint, optimization, and simple harmonic motion (necessary!!). You will also learn about Hamiltonian Systems. (1.5 courses) Req: Diff Eq, Calc 3

Electrodynamics: Griffiths [6]. Another standard. You won't find a better book than this for E&M. (1.5 courses) Req: Diff Eq, Calc 3

Quantum Mechanics: Griffiths [7] (He's the man, seriously) (1.5 courses) Req: Diff Eq, Calc 3 (lin algebra is nice, same with a tad of group theory)

Astrophysics: BOB [8] Lovingly called the "Big Orange Book" you will see this on every astrophysicists' shelves. (2+ courses) Req: Calc 1

Particle Physics: That's right! You guessed it! Griffiths![9] Take after QM.


BlackPenRedPen[10]: Fantastic teacher. He will help you with calc and help you understand a lot of tricks that you might not see in the above books. I can't stress enough that you should watch him.

Go find MIT OCWs, I'm not going to list them.

Honorable mentions: 3Blue1Brown[11], Numberphile[12], Veritasium[13], StandUpMaths[14], SmarterEveryDay[15]. All these people talk about some neat concepts that will help you gain more interest and think about things to pursue. But they are not course channels, they are much more casual (somewhere between what you'd see on the Discovery Channel and a classroom, more towards the latter).


> I follow a bunch of folks on the internet and idolize them for their multifaceted personalities

Don't stress too much about being like those people you idolize. I guarantee that you see them as much more intelligent people than they are or think of (not dissing on them, but we tend to put these people on pedestals and this is a big contributor to Imposter Syndrome. Which WILL have, probably already does, an effect on your learning process). Don't compare yourself. You can get to most of these peoples' levels by just doing an hour or two a day for a few years.

> I can totally see that these are the folks who have high IQs and they can easily learn a new domain in a few months if they were put in one.

This is a skill. A trainable skill. Just remember that. Some people are much more proficient at it, but I be you'll see that they have much more experience. In music you sight read. Doing the same thing with math, physics, engineering, etc will result in the same increase in talent.

[1] https://smile.amazon.com/Calculus-Early-Transcendentals-Jame...

[2] https://smile.amazon.com/Linear-Algebra-Its-Applications-3rd...

[3] https://smile.amazon.com/Differential-Equations-Tools-Printe...

[4] https://smile.amazon.com/Fundamentals-Physics-David-Halliday...

[5] https://smile.amazon.com/Classical-Dynamics-Particles-System...

[6] https://smile.amazon.com/Introduction-Electrodynamics-David-...

[7] https://smile.amazon.com/Introduction-Quantum-Mechanics-Davi...

[8] https://smile.amazon.com/Introduction-Modern-Astrophysics-Br...

[9] https://smile.amazon.com/Introduction-Elementary-Particles-D...

[10] https://www.youtube.com/channel/UC_SvYP0k05UKiJ_2ndB02IA

[11] https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw


[13] https://www.youtube.com/channel/UCHnyfMqiRRG1u-2MsSQLbXA

[14] https://www.youtube.com/user/standupmaths

[15] https://www.youtube.com/channel/UC6107grRI4m0o2-emgoDnAA

I'm in a similar situation to you. I have been really enjoying the 3blue1brown videos on Youtube. He has a sequence on calculus and linear algebra and both of them are worth watching and thinking about before going through a book. I also recently started watching some of Dr. Norman Wildberger's math lectures. He's a finitist crank, but most of his class lectures are great despite this. This inspired me to get some more textbooks and try to go through them. I'm pleased if I can make it through a chapter at all on any time scale, and I think having low expectations is probably healthy.

One thing a friend of mine said, which I think has been very good advice, is to get several books on a single topic. Eventually every author will lose you and you'll get stuck; having alternate discussions will help you get through it. This is easy to do and inexpensive if you pick up some Dover math books, but I've been making heavy use of the local academic library. The math books you want to read are not in high demand at the library! You can get three or four and see if any of them are good enough to warrant a purchase later on, because you probably won't get through them in a month or whatever.

I find that for many topics there is a really good text. Calculus by Spivak is a great example, it straddles the line between calculus-in-college and analysis. Every topic seems to have a few really good books like this one, and there are often books that will take a totally different approach, like H Jerome Keisler's nonstandard calculus book using infinitesimals.

I used to see it as a real problem that I was learning math outside class, but more and more I see it as a benefit, because you can pick up the stuff you want at the resolution you want and benefit from the best books rather than whatever the publishers are bribing professors to use. Going at your own pace, you're not going to go through as much stuff as quickly, but you will actually _really_ learn it. I've spent the last three weeks or so thinking about the construction of the real numbers... in a classroom setting, you would be forced to get through this quickly to get on with the rest of the curriculum, even if you aren't interested in the rest of it.

I think both our roads are eventually going to lead us to differential geometry, and the only thing I know about that is that there appears to be a very good book on Amazon (Tapp, Differential Geometry of Curves and Surfaces), and that you may want to avoid older books that use the older notation for it. I have heard great things about the book Gravitation, but I'm totally afraid of it, not ready to go there yet. Also check out Physics from Symmetry, that book looks amazing to me but I haven't read it yet, just flipped through the contents, but it might be exactly what you're after, since it discusses the math right before applying it to specific areas of physics.



Regardless of whether you will actually put a maths degree to use on the job market—which no doubt you could, once you attain it—pursuing one can bring great rewards. I am in my thirties too (mid-thirties now) and recently took up maths again. Lucky for me, I come from the German-speaking realm, where there is a distance learning university that offers a solid BSc programme in mathematics at roughly one eighth of what someone would pay for tuition in the UK. I do not know what options you have in that regard, but if being enrolled in a programme does not seem off-putting to you, it might be worth checking out. I am mostly on my own, so I guess if I were studying entirely independently, I would not be doing much different. There are benefits though that I would not want to miss out on—I get my (bi-weekly) assignments reviewed, there are platforms where your teachers and fellow students are communicating, and, last but not least, there are exams that provide for the 'hard facts' as to whether you have been studying your stuff right. Not to mention the degree you are awarded if you succeed. To me it is beyond question that distance education is the right way for me to do this. Like you, I am working in business software development, and I simply cannot attend a brick and mortar university because I do not have the time, or would have to accept the severe cut-back in income resulting from a reduction of my work hours. (Besides, I have been to brick and mortar before, and not being very social, I do not think I am missing out on the social aspects of it at all.) So if there is any distance education option that suits your needs, it might greatly augment your self-directed learning.

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