Tips for independently studying mathematics

[@simonreiff3889]

I've been a self-taught amateur in math for over 20 years now (meaning, no instructors or professors whom I could ask questions). My very first subject that I taught myself after college was multivariable calculus. I read the Schaum Outline for Honors Calculus cover to cover and did literally every problem (over 1000 problems!). I Ioved that book, typos and all, and strongly recommend it for anyone from pre-calc to multivariable and they even have a couple of intro chapters on real analysis, because the presentation was so logical and appealingly laid out, but also because there were so many examples and solved problems, plus solutions for about half the exercises, and hints even for unsolved problems. It obviously was nowhere near 1 chapter per day since I made myself do all the problems, but the book started easily enough and I was still going strong after Chapter 2 which gave me motivation to finish the entire book and find all the typos and convince myself that they were indeed typos and not my own errors. I subsequently taught myself number theory, linear and abstract algebra, real and complex analysis (multiple times at several levels of difficulty), mostly from online texts and KZbin math courses, and I completely agree--independent study goes incredibly slowly at first, but eventually you get faster as you recognize more things in whatever new text you pick up or course you watch. My approach to deadlines and how many pages or chapters to read per unit of time is different. I always take my time and set only general goals, never hard deadlines, for getting through a particular text or subject. Although completely impractical for formal academic study, I strongly recommend doing the same for any other self-taught amateurs and can promise that the no-deadline approach really does work as long as you set goals like an hour of study per day or are otherwise motivated to keep pushing yourself regularly to learn the material. I do try hard not to get completely stuck on one single problem whenever possible and will cheat by looking up the answer after a bit, but at the same time, I also might take weeks on a single very hard problem (while trying to solve lots of other problems near it in the book in the meantime and learning from other sources) if I can't solve it or find a solution online that I understand--the inability to solve a problem, especially if it isn't particularly computational, usually indicates to me that I don't really understand the material as well as I thought. I have repeatedly discovered that it is extremely useful to assume that all problems and exercises in any given text are interconnected and required reading, and that the proof technique for one problem will be used to show other things later. This does stymie me at times and isn't invariably true, admittedly. However, being not just independent but also not having many friends into math in my social circle and just one who is stronger than me or at about my level mathematically, I depend almost entirely on myself to find errors in my reasoning, so I have to take a lot of time--maybe too much!--to be really confident that I actually have something right and haven't missed a negative sign or something obvious even when I feel like I have successfully solved a problem. On the other hand, once I have taken that time, I really do feel confident that I know the material and can prove it by solving others like it. My proof style is probably too verbose, but because I lack regular feedback from instructors, I need to be sure my proofs are right and cannot afford to wave my hand at any details. I guess this can get philosophical as to just how deeply anything really must be proven. Once I unstuck myself from a hard problem, though, things go very quickly after that and usually for quite some time afterwards because I took all the effort to learn the surrounding details thoroughly. I will say that graduate level texts are WAY harder to use for self-learning and usually require a ton of supplemental reading, KZbin videos, etc. Even Stein & Shakarshi, my favorite measure theory book which I am studying now for analysis, is a bear. I love your suggestion of looking at the table of contents for familiar terrain along with the first few pages of the book. If you're already feeling like it makes no sense at that point, it isn't likely to be a good book to read for now. Of course, I thought Lang's Algebra, on that basis, was a good choice, and now I'm multiple chapters in and feel like I married that book and have to see it all the way through. Maybe by the time I'm 80 I will finish it! But I figure nobody reads Lang cover to cover and that if I somehow can, I probably will be capable of doing independent research in algebra, so I'll just take my time and keep going for the next decade or however long it takes me to get to the end. Anyway, all of this is just a longwinded way of thanking you for these videos and for motivating amateurs like me to continue studying math while showing us how you do it. Congrats again for passing your quals and good luck in all your research!

[@Colonies_Dev]

didchu try making mathematical art by creating shaders or textures via code? you can use a lot of awesome techniques like boolean logic for example to combine or intersect black/white images~ this means u can essentially combine procedural noise or other patterns to create more complex graphics.. u can then also use linear algebra to for example transform the UV coordinates~ UV being a kind of xy vector which helps to map pixel colors into triangles on a mesh. i think computer-programming and maybe videogames/simulation tools/graphics can be an awesome way to put your knowledge into practice and get new perspectives on it all. since i used to be mostly theoretical, try to train my mathbrain, but now its more natural since i encounter problems that weren't created artificially but instead must be solved to create something.

[@Srfs777]

Jesus pal you ever consider a degree in writing? Could take off a 747 off that runway of an essay. All in all, very motivating text indeed.

[@AutoDisheep]

thank you for sharing your experience.

[@mastershooter64]

This is unrelated but you're the most American sounding math student I've ever heard

[@PhDVlog777]

🇺🇸🇺🇸🇺🇸

[@SweatyMathMajor]

​@@PhDVlog777 Merica!

[@josefernandez5652]

you are my favorite yapper, love your videos and i dont even study anything math related

[@Epilogue_04]

I can relate in this, long story short, i use to hate math up to high school due to bad experiences, i ended up trying physics as career i had the luck of having good professors in the first semesters that hooked me, and i had both confusion while reading and while listening the professor. But ive always liked to try to do stuff by myself, so i started reading the books they used and all the stuff started to seem more clear to me, off course i say this easy but i took like 1 or 2 years to get comfortable with it

[@josephstalin5003]

Books those things expensive any way to download the suckers online for free I just started college

[@Epilogue_04]

@@josephstalin5003 yep when I said books I mean like 95% pdfs 😂

[@mastershooter64]

@@josephstalin5003 if you actually look for the books you want online, you can find the PDFs 99.98% of the time

[@peamutbubber]

Guts is literally me

[@maviselK]

@@josephstalin5003 you do clearly remunerate your user name.

[@mathsmachinelearning1885]

I have been struggling with math notes and I think that thanks to this video I might know why. I have been trying to take the perfect math notes with all the small details in them, but, as you have said, the key is not to write a lot, the actual key is to write as least as possible. I am going to take notes with this mindset for a while and see what happens.Thanks a lot for your videos, you are definitely my favourite KZbinr.

[@TheCyberneticVortex]

It's always very helpful to hear other people's methods, so thank you for making this! I'm about to begin my first research semester, studying Conformal Field Theory, so I'll be using the CFT book by Francesco. It's a 900 page monster, and my first real exposure to learning from a textbook (textbooks weren't ever necessary for my undergrad in Australia) so it'll be slow going, but this will help a lot with the struggle.

[@chovuse]

I am self-studying Applied Mathematics for Engineers, Computer Science & Data Science (AI, Machine Learning, Deep Learning, etc). I had to stop math in middle school, so I have a LONG way to go ! After some internet research here is my Math progression to reach this goal. > High School Math (O+A-Level) : 1 - Arithmetics, Algebra I, II & III, Geometry & (Analytic Geometry), Trigonometry - ( Equations, Functions, Logarithms, Summations, Series, etc. ) > College Math : 2 - Discrete Mathematics - ( Logic, Number Theory, Proofs, Sequences, Sets, Relations, Counting & Discrete Probability, Graphs, etc ) 3 - Linear Algebra - ( Systems of Linear Equations, Matrices, Determinants, Vector Spaces, LinearTransformations, etc ) 4 - Calculus I & II - ( Limits, Single-Variable ( Differentials (Derivatives) & Integrals ), ODE, etc) 5 - Calculus III - ( Vectors, Multi-Variable ( Partial Derivatives & Multi Integrals ), 2nd ODE, etc ) 6 - Differential Equations - ( ODE & PDE ) 7 - Probability & Statistics ( Random Variables, Distributions) & ( Inference, Regression) Hope this is correct. Note taking will help a lot. Its a Long way to Tipperary ! Its a Long way to Go ! 😬

[@harisserdarevic4913]

I would say that your focus on the principles of algebra are going to be so fundamental to your study that you should really focus hard on that. Understanding algebraic manipulation of equations will be crucial for calculus. Proofs and sets can be learned at any time right after you get the skills of algebra down. Once you have the solid base of algebra and proofs, then the fun begins

[@chovuse]

@@harisserdarevic4913 Many thanks !

[@peamutbubber]

About to start a 4 year masters degree in mathematics 😳 glad I have this channel for wise words

[@surajpowar4915]

Nice video mate. Could you please make a video on how you do homework as well? It would be interesting to know how much time you dedicate and other stuff.

[@nolanalexander8696]

Now you passed the exams, the desk are more tidy and we get fun sketch for each videos😊 Thats a good drawing of Homer btw.

[@PhDVlog777]

Thank you 😊

[@daved1113]

The number one tip is that you have to enjoy it. If it's just an unpleasant grind no matter what then it means you should switch to something you like. Studying a language, programming, or whatever. You don't have to force yourself to do math.

[@jay_william17]

Thanks for the vid. Just started grad school for CS&E and these AI math courses have been a wakeup call to actually read the textbooks

[@choijamts9095]

I dunno what is actually good about watching your videos honestly, but we are on the same boat as for the struggling part of graduate life. And your videos do not really have fancy thumbnails, graphics, edits, just one casual chill guy (atleast to me, further I don't know xd) talking about maths and how he overcame the obstacles that faced earlier. So, the thing is obstacles that you are talking are almost exactly the same ones, even when I am just an structural engineering kind of guy. It's kind of weird to say, but after hours of "researching" I find your videos "soothening" me (I am not a native speaker so, yeah,)

[@researchchannel1767]

Your videos can help bro definitely help me motivated again to do math and i became the president for the math club in my school

[@researchchannel1767]

By the way Struggling Grad Student, khan academy math videos are kinda outdated (no offense) but do you also watch OCT? (The organic chemistry tutor) they are different from my perspective and im confused who should i watch lol...

[@SavvySaxy]

So I should practice. That sounds so simple but I haven’t been doing it. I just got overwhelmed by books and stopped. I took AP Calc AB and now I’m wanting to pick up where I left off in James Stewart’s “Calculus” but I just get so overwhelmed and learn nothing because I don’t know how to do it then I give up. So it might be better if I try to be more consistent about it and just do it in smaller bursts for now so I don’t get overwhelmed. Thanks :)

[@fabianbaier8049]

I'm structuring my notes quite similar to yours, but usually I leave out proofs and I also try to mark the main conditions for important theorems

[@natanprzybylko7227]

Gonna watch these videos instead of independently studying math

[@Halderchess]

What about problem solving? How do you do them? Even if you read through the whole book its really hard to solve those problems.I suck at them.

[@PhDVlog777]

It depends, if I am in a class, then I focus on the hw assignments. If I am on my own, then I at least try them or just go to stack exchange. There are a ton of solved problems out there, but I at least want to understand the solution and make it my own.

[@rexraptor1094]

I've been waiting for this video

[@PurpleMiniMoose]

Aww man I was curious how research in mathematics is done 😞

[@Efesus67]

Ooh, that would be interesting.

[@matthewbyrne6344]

chris staecker has a really good video on that iirc, he talks all the way through the process of writing a research paper

[@samuel5153]

I really want to take math a lot more seriously now. I already graduated grade 12, soon will go to college (still can’t decide what to major in, which explains why am still not enrolled). Recently I’ve thinking about Math, watching math videos even though I can’t seem to follow through. As for my math knowledge, I suck at everything after pre-algebra or when they decided to add numbers in. I didn’t really pay attention in class, so I’m wondering what course of action should I take, how and where should I start? I’m really lost right now (in life too lol)

[@jennifertate4397]

Good vid. Thanks.

[@iancraig5415]

18:35 the John Cazale of mathematics

[@briangronberg6507]

What’s a good way of approaching difficult (at least to me!) computational courses like PDEs? Is there a book you like?

[@oraz.]

When you're self learning from a textbook to you ever create an abridged "curriculum" to get through the most important parts?

[@siddharthjain2127]

I'm assistant professor of mathematics and i always really love maths book

[@sanjursan]

I notice you have Kolmo on the top of your pile. Great book! Try his FA book also. He covers some of same stuff from a slightly different angle. I really like Kolmo.

[@PhDVlog777]

Kolmogorov was my favorite book for a while, and I still like it, but I’ve found other books that are more my style.

[@sanjursan]

Maybe check free pdf of Shlomo Sternberg "Theory of Functions of a Real Variable." (See eg., page 15 def of closed set!!)

[@blackcat5771]

I'm starting an engineering degree in mathematics, if anyone has gone through it, what is it like?

[@Sejdr]

You improve your learning retention by writing by hand instead of useing a keyboard. I always write by hand since I like the tactile feeling of it.

[@chad.breece]

Shout out to Linear Algebra Done Right, great book

[@preston3291]

is textbook the best way to learn math

[@paradiseexpress3639]

It's the main way. but you should also supplement with other ways

[@Kt-hp7cv]

A lot of mathematics is self-studying.

[@шляпик-и1м]

Doing exercises and proving proofs from books is the way to go

[@MSS864]

Yes, reading the physical textbook is the best way to study math.

[@climitod8524]

do you read solutions to proofs if you get stuck? Please I need to know. Also do you try and find counter examples after you think you've proven it?

[@PhDVlog777]

If there are solutions then yes, I will read them. I think it is good to tweak the assumptions a little and try to come up with counterexamples. It gives a you a good intuition for this kind of math.

[@Youssef-Abotaleb]

Hello , i wanted to say that i enjoy your videos , but i have a question, i am now in grade 12 , after 10 months from now i will graduate, i want to become a physicist, but my parents want me to become a doctor and go to medical school , what should i do ? I dont hate biology but am much more interested in physics , but at the same time i am scared that i wont make money after graduation if i become a physicist, i am also not that good at math and this year i didnt choose math as one of the subjects but i take physics, whats your opinion, should i make my family happy by becoming a doctor even tho i am not that interested in it , or should i become a physicist which i am now passionate abt ?

[@soupy5890]

If you become something you really don't want to be for your family, you'll just become resentful and build regret, however maybe my opinion is trash, so I'd recommend doing some research about if you have any prospects anywhere as a physicist, such as job opportunity and who hires that sort of person, what marketable skills you'll have if you go through with it, and if you can study something that covers a lot of physics perhaps, but is more focused into something you'll both have a future in, like some sort of engineering or robotics perhaps.

[@purefnm7861]

hey, i’m a 3rd ur premed student and it will be very hard to get in without giving a proper reason. they try to not admit people who are applying to med schools because of their parents, explain this to them and do what u want

[@purefnm7861]

u also won’t make good money after graduation as a doctor, will take several years to reach 6 figures

[@purefnm7861]

3rd year*

[@theepicman8160]

If you develope the right skill set then you can get paid really well after you become a physicist. You can work for a company which pays you well and work on your research side by side. You don't have to worry about that, I advise you to follow your passion here. If something goes wrong, you can always pick up on mathematics and become an analyst or you can become a computer scientist.

[@eleazaralmazan4089]

I personally love the way you take your notes. It reminds me of how I would take notes as an undergrad.

[@ramyramy6896]

Working Thanx

[@cherry1leii]

why do you use ruled paper for maths?

[@alvydagr8108]

I took a break from math because I wasn’t finding it fun anymore, anything to recommend? Im taking a lower level class as a “fun way” to get back into it but now i just feel lazy 😩

[@PhDVlog777]

Maybe try some problem solving with YT guides? Mind your decisions is a great channel that does this kind of stuff.

[@alvydagr8108]

@@PhDVlog777 yeah, talking to other students I guess I lack knowledge of good/reliable YT channels like this one. This and a couple of studying channels is all I follow when it comes to classwork

[@theepicman8160]

The thumbnail is genius

[@artophile7777]

I don't see a thumbnail, it's a black screen!

[@theepicman8160]

@@artophile7777 Exactly!

[@jidrit999]

i try to do self study. i study for month and when it gets hard i loose interest. without instructor it is hard

[@PhDVlog777]

Try something on a smaller scale.

[@jidrit999]

what would that be in analysis and algebra@@PhDVlog777

[@Yo-qf4ch]

See, mit ocw

[@iisimaginary]

Is there anyway for me to contact you? Perhaps an email in your profile page? Thanks for the vlogs!

[@PhDVlog777]

I’m thinking of setting up an email. More details later

[@adityaraj6377]

10:40 dude she isn't jealous.. she is just totally into you...trust me brother...

[@theconservativeone2690]

Face reveal

[@manueldelossantos1892]

.

[@VijaySharma-nv5yd]

third

[@WazapianArmy]

Sounds like you might have a potential GF if you're impressing the math department lol but hope ur doing good now that you don't have to worry about ur qual

[@PhDVlog777]

Well, her husband and five kids might object to us going out lol

[@AdoptedPoo]

math books are pretty hard to read from imo.