Whereas Susskind will give you more of an overview. 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. 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. Clubber on May 15, 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! 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. 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? 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. 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. Because that requires learning a formal proof-verification language. I'm certainly interested in that, but it is a distraction from learning undergraduate mathematics.
I'm confused; my post was advocating using software, so I'm unclear why you're suggesting I use software. 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". 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.
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 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 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. 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 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.
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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. 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 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.
Tomminn on May 16, 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.
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. 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. Hi, I'm 30 and trying to relearn the math courses I did in college Computer Science degree and more. 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'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.
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 can still learn a lot, but it's meaningfully harder. 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 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. MockObject on May 16, 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. Taste, you say? EliRivers on May 16, 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. 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. 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 .
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 Mathematics is beautiful. I'm glad we have it. 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. 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.
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. 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. It is very possible to develop engineering math chops late in life I did it! 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. Kagerjay on May 16, I practiced at least 30 problems every night in that class for 2 years. IvyMike on May 15, This is an 11 session pm math graduate level math course. 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. The epitome of what you want is to find a mentor, a chalkboard, and 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. The author's writing style and the book's layout worked well for me.
Hi atxhx, thx for the plug! 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. ISL on May 15, For motivation, if there is a nearby university, start attending the relevant departments' colloquia.
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. Good speakers and even non-physicists can follow the whole talk regardless of subject matter.
But good speakers are rare. So you might learn about super cool research happening, which is great. 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.
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. Also, I should mention, one big lesson learned Maths build on each other. The actual rules of calculus are simple, easy to grasp and dare I say it, intuitive 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. 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. MikkoFinell on May 16, Covered for linear algebra? Yes those videos have some nice visuals but the material is just scratching the surface. Raphmedia on May 15, 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.
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!
The founder, Alex Coward, is an ex-Berkeley math professor. 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.
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. TangoTrotFox on May 15, A phenomenal resource for physics is the text for the Feynman lectures.
The one thing those lectures are desperately missing is a similarly well organized and presented problem set. Was a physics student. 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. 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. You understand a topic only if you can solve the problems. Not all topics are equally important.
Focus on the important parts. Online courses, college material especially problem sets! You will likely have questions. Cultivating some relationship that allows you to ask questions is invaluable. It is really difficult to skip any step along the way. Often, what turns people off is that they do not get things quickly e. If you find yourself thinking hours about seemingly simple problems, do not despair! That is normal 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.
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. Learning often happens in non-linear ways. If you hit an insurmountable roadblock, just keep going.
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Buy a book and take a course. 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. 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. Think very carefully where you want to spend your motivation and discipline. Understanding the "Why" important. 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. 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. See the website for how that works. It's a lot of fun! 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. BigChiefSmokem on May 15, 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.
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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. 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? Youtube, EDX , brilliant. 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.
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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. Music: 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 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. 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.
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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. 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.
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 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.
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. 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 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 , the currently accepted approach to advanced probability, stochastic processes, and statistics. That's a start. RickJWagner on May 16, YouTube is my preferred method of learning. It's easy and can quickly expose you to a variety of teaching styles. 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. MrFantastic on May 15, 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. But there is one other motive which is as strong as any of these -- the search for beauty. Mathematics is an art, and as such affords the pleasures which all the arts afford. The book can also act as a self-study vehicle for advanced high school students and laymen.
Professor Kline begins with an overview, tracing the development of mathematics to the ancient Greeks, and following its evolution through the Middle Ages and the Renaissance to the present day. At the same time, these concepts are linked to pure and applied science, engineering, philosophy, the social sciences or even the arts.
In one section, Professor Kline discusses non-Euclidean geometry, ranking it with evolution as one of the "two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century. Mathematics for the Nonmathematician exemplifies Morris Kline's rare ability to simplify complex subjects for the nonspecialist.
About the Author Morris Kline: Mathematics for the Masses Morris Kline had a strong and forceful personality which he brought both to his position as Professor at New York University from until his retirement in , and to his role as the driving force behind Dover's mathematics reprint program for even longer, from the s until just a few years before his death.