Thanks for watching! If you have any resources you'd like to recommend, feel free to comment them down below. Also, I have a math newsletter where I collate resources to learn topics in math / machine learning and deliver them to your inbox. If you'd like to sign up for the newsletter, fill out this form: forms.gle/Rt1f5StAj3yZtakE6
math gives this feeling of certitude and determinism that equations embodies, learning a new concept and solving a problem also has a unique feeling to it, totally understand it.
I would have liked to see this video like 25 years ago. All this material free online. What a wonderful time is today. I want to thank you in the name of humanity.
Books - Axler, Linear Algebra (also has videos) - Abbot, Understand Analysis (su lecture series) - Uni Toronto, Point Set - Wegert, Visual Complex Function Complex Analysis - Lang, Complex Analysis (well illustrated)* - Wesley Uni, Video Playlist Analysis of a Complex Kind - Gross, Abstract Algebra (video) - Herstein - Galois, Same channel video and Tom Lanster - Boothby, Introduction Diff. Manifolds and R. Geo. - A. Hatcher, Algebraic Topology (peer albin)
This makes me very happy. I studied comp sci at uni but have been trying to learn some higher math on my own lately. The specific resource recommendations are extremely valuable to me. I consider this a Christmas gift!
@@ashwinjain5566 Depends on the program and ur research area. What I find most useful is Advanced Linear Algebra, Differential Geometry, Topology, Tensor Calculus, Advanced Probablity Theory with a bit of Measure theory, Continuous + Convex Optimization
I appreciate how you didn't overload us with resources but gave it straight forward. Much much more appreciated than the channels that throw tons of books or resources.
DUDE you are an incredible content creator! Not only finding all the resources that best help newbies into classic maths, but also providing insight into their way of teaching!
As a theoretical physics graduate student, I can strongly recommend the textbook "Differential Geometry, Gauge theories, and Gravity" by Gockeler and Schucker to understand differential geometry with just a physics undergraduate background, where the book introduces but also applies this subject to numerous areas of modern physics - I really recommend it! Also, if you're a physics student and don't really know where to start, I would strongly recommend "Modern Mathematical Physics" By Szekeres which basically introduces from the ground up and assuming no prerequisite all the parts of maths you need to do physics at an advanced level. Hope it helps :)
Nakahara's _Geometry, Topology, and Physics_ is an excellent text as well. What it lacks in readability, it more than compensates for in completeness and rigor, explaining everything from vector spaces to cohomology to differential forms. Wholeheartedly recommend it as a referrence text for the physicist or physics-adjacent mathematician.
I'm extremely overjoyed to have found this video. I've been wanting to self-study pure maths (with some physics as a little treat) for a year or two now, and am currently strengthening my foundational understanding of the basics, but was bumbling through the undergrad stuff with the help of the MIT courses and Stack Overflow. I'm incredibly grateful that you included videos alongside the textbooks, since I tend to learn better with a combination of both (but mostly the former). This video is like finding an oasis in the midst of a desert, thank you so much!
Great video! As someone who got an undergraduate degree in math, many of the resources here are excellent resources that I made good use of. I look forward to checking out the others to fill in the gaps on some of my weaker subjects.
Hey. I just want to say you've really inspired me to get back into self studying mathematics. I had done a lot of self study in the past, when I had much less mathematical maturity, and got discouraged, in part because I kept skipping prerequisites and the like. Thank you for this and other videos.
Thank you for the video. I wish TH-cam had a "love" button. Please don't remove the previous version of this video from the channel. It was so informative and helpful. Just add "2020" or something in the title. I love that video too! Like this comment if you want that video back.
I think he already removed the previous version (people in the comments were insufferable, I think that s why). I m really glad he made a new version though
I will also put my brick in this wall. I studied mathematics a couple of years a go and I really loved all the mathematics books from Schaum's Outlines series. They have books on Group Theory, Linear and Abstract Algebra, Real and Complex Calculus and many more etc. Usually those books first present the theory and provide examples. Then you have solved problems where you see how a given problem can be tackled and finally the problems for the reader to solve (with answers at the back). I remember a lot of the content was aligned to my university courses and I could learn a lot from them. Plus they're very good for studying on your own without a teacher/mentor/professor. They are basically so good that now (a few years after graduation) I'm still collecting them since I want them in my personal library for future reference or just to have them just in case (they are really hard to come by in Poland). And one tip in advance. Look for the older versions of the books or first/second editions in some cases. The reason is because slowly some good parts or even entire sections are being cut out. I don't really understand the reason behind this but anyway - you have my opinion right here. Stays safe Comrades!
> The reason is because slowly some good parts or even entire sections are being cut out. I don't really understand the reason behind this... But you do understand, don't you?
Calculus by Michael Spivak (Third or Fourth edition) is unexpectedly a really good introduction to not just real analysis, but also some number theory and a little abstract algebra at the end of the textbook. It doesn't require any prerequisites, as stated by the author (I think). It also justifies most theorems that are brought up, has proofs for them, and most importantly, an ungodly amount of examples (like 60 per chapter, making up most of the book).
@@Miguel_Noether I have without realising what analysis was, no wonder it was so goddamn difficult. I just wanted to learn basic differentiation and integration, but ended up learning how to prove some statements from real analysis.
As someone who's doing a degree in mechanical engineering but has a very strong passion for pure math I very much appreciate this video, as well as yours and everyone else's free and amazing videos on many different math topics all across youtube. Keep up the great quality!
@@mastershooter64 You'll maybe encounter some day people with more than one interest. It's my case, too, I'm a student in microengineering and I have a strong passion in pure math
@@mastershooter64 it was hard to find a good and affordable university, I didn't have faith in my math knowledge enough to enroll into a pure math class, my passion for physics is not any lesser, and last but not least, it is significantly easier to find an enjoyable and well payed job with an engineering degree from everything that I've seen.
I would say engineering is an offshoot of mathematics or it is essentially mathematics with a different name. Many famous equations/relations in Mec.Eng are derived by mathematicians instead of engineers themselves. For example Prandtl , Blasius, Navier ,Stokes etc.
@@hussainsallar3355 Not to mention that at that time those people were really not one or the other but both, mathematicians and engineers, and sometimes also worked in other fields as well. Scientists in general (including many mathematicians and engineers), were really just called "Natural Philosophers" until the middle to end of the 19th century. That is actually one of the things that really attracts me to maths, it is not only very interesting on its own, but it is also a quite universally powerful instrument for the study of any other science. That is why I intend to pursue a masters in Applied Mathematics after I finish my bachelors in mechanical engineering.
When I was in college, I wanted to take more math beyond Calc III and Linear Algebra, but I didn't have the time or funding to do it. Beyond being interesting, I found it helpful with writing music and thinking about writing in different ways. I'm glad you smart professionals provide these resources for us.
Thank you so much! I'm a med student who loves math (crazy, I know). I've always regretted missing the opportunity to learn pure math at uni level. I'm excited to use these resources!!
Thank you so much for making a self study guide video again. I started down following the advice from your original, but then it disappeared... now it's back and updated :)
So I noticed the topics for the undergraduate curriculum in this newer version of the video changed slightly but either way it makes me want to bring up a somewhat list of things you would see as an undergraduate in university. First off, not everyone does calculus and differential equations in high school (generally speaking most don't): so if you're going to university, expect those. The topics of Linear Algebra, Real Analysis, and Abstract Algebra are generally the common core of every undergraduate math degree; however, most people will do more things like complex analysis and topology. A lot of American universities will have a class dedicated to learning how to write proofs and learning basic set theory and number theory. Group Theory and Galois Theory aren't usually taught on their own as courses but are rather a large chunk of a first and second semester abstract algebra course respectively. Not everyone does differential geometry or algebraic topology when they're an undergrad but a lot do. Other undergraduate courses include: Probability, Number Theory, Euclidean Geometry, Partial Differential Equations, and basically anything that says "intro to [branch of math]" is either an advanced undergraduate course or a graduate course.
Yeah differential equations is pretty uncommon in most high schools. I went to a well-funded public school, and having AP single-variable calc was honestly a privilege. I'm now in a well-funded liberal arts college, and you're on-point with the classes usually completed for the major. This video seems to assume more a magnet-school type of math education and a large university or a quite advanced undergraduate curriculum.
@@TomMS In Canada, there's an intermediary between high school and university. Most students with a stem focus use that period to take Calculus and Diff Eq, which is probably the cause of the disconnect.
Man, I got those books thanks to you, and I can say that the real analysis one is really well done! Actually, every problem that mathematic books have (not defining important things, not explaining the history of a concept, refusing to give simple example, etc.) isn't in this book so far in my reading
A minor nitpick: the study of differentiable manifolds without a distinguished 2-form (such as a metric or symplectic form) is generally referred to as differential topology, while the inclusion of such a form "elevates" the subject to geometry. So, the point where you say differential geometry ends is actually where differential geometry begins.
This is not true. Certainly not where I'm from. Differential topology is simply the study of the topology of differentiable manifolds. There is tonnes that you can call geometry without a metric or a symplectic structure. My own research for one
Perhaps a better way to think of it is that differential topology is the study of global properties of smooth manifolds while differential geometry is more about the local properties. As always there is some overlap.
@@Aetheraev I'll admit that the distinction given here isn't always exactly correct, and I hoped that my use of "generally" would cover me for situations where it fails. However, at least as far as I've seen, calculus on differentiable manifolds isn't considered geometry, which is the subject of the first six chapters of Boothby referenced in the video and by my comment. In the end, I think it comes down to semantics and it's not truly important where exactly the lines are drawn; the main purpose of my original comment was simply to correct the statement, "This part marks the end of the differential geometry portion of this book," as the Riemannian geometry that follows really has the *most* right of any material in the text to be called differential geometry.
Thank you so much! After highschool i choose to take on a psychology career even if I've really loved maths, always promising myself to self-study it but everytime getting lost, this video is a treasure
wow this is great thank you so much! I really know where to start right now! I just stopped university because stress was consuming me.. now I can just live my life and still learn some mathematics thank you so much!
Thank you so much. I'm a data scientist who has been trying to teach myself pure math for several years. I'm working on Linear Algebra Done Right now and wanted to jump to abstract algebra after I finished.
I thought this was going to be more about how to focus when doing math, how to understand concepts and solve problems and so on, but still a nice collection of resources!
A few comments from an erstwhile math student . . . +1 on Topics in Algebra by Herstein. A lovely book: rigorous proofs and accessible explanations. +1 on Algebraic Topology by Hatcher. Excellent examples and illustrations. - 1 on Intro to Differentiable Manifolds by Boothby. It's hit or miss. The pace is uneven, the exposition often too spare. It tries to cover too much in its modest length: differentiable manifolds, Riemmanian geometry, group actions, Lie groups, Lie algebras -- without giving each topic full, proper treatment. Try Lee instead (see below). NB: For differential geometry, Lee's Intro to Smooth Manifolds is a solid, modern supplement/replacement to Boothby, or suitable for self-study (perhaps if you're comfortably reading at graduate-level). For topology, Munkres' Topology book is a solid bet. For group theory, Abstract Algebra by Dummit and Foote has good breadth and depth and covers everything Herstein's does. A good starter book in group theory is Fraleigh's First Course in Abstract Algebra; suitable for self-study.
So I'm not hugely up on my Pure Mathematics as I moved toward the Applied end of the subject after graduating, but those interested in the Singularity Theory part of Differential Geometry (stuff like Cusps, Swallow Tails, and Envelopes), I would recommend "Curves and Singularities" by Bruce & Giblin. It has lots of nice pictures!
I'm very grateful for reposting these ideas. I had purchased the Understanding Analysis by Abbott book but your original video had gone offline before it arrived! My course in calculus 2 is finished and I'm looking forward to self-study on pure math in '22.
For GROUP THEORY category, my main go to book is "Contemporary Abstract Algebra" by Joseph Gallian. Current shoutout after 12 hours with it (yay Christmas) is "All the Math you missed (but need to know for graduate school)" by Thomas Garrity so far has been a fun book.
I totally agree. Gallians book is perfect as a low entry point into the subject. So many easy to understand and intuitive examples given, along with actual applications of group theory. Abstract Algebra by Gregory T Lee is also really good. It's a bit more difficult, and is drier. But is also far more concise and more rigorous. Importantly, both these books are easy to get. Unlike a lot of the older (and more expensive!) textbooks that are often recommended for Abstract Algebra and/or group theory.
I have studied both MARLOW ANDERSON and JOSEPH GALLIAN for abstract algebra. Even Though Gallian contains more content than Anderson, but I like the way anderson introduces the concepts. If you have studied number theory or just want to study number theory along abstract algebra, I think anderson will make a lot of sense than gallian. Anderson introduces concepts by ring first approach which for a beginner makes more sense than a group first approach.
Such a great video! I've just finished my MSc in Statistics but I'm now looking to spend time self-studying the pure maths modules I had to miss to focus on stats... So this video couldn't come at a better time!
This video collects lots of useful online resources and recommended textbooks. The content is clear and well-structured and I appreciate your work to illuminate a way to learn pure mathematics!
"Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation.
Im in first year of electronics engineery and i just can feel how many holes there are in my pure math education, so im trying to improve in order to be a better engineer tomorrow day, pluss its a cool hobbie...!
Pure math gives you a comprehensive background about the origins of mathematical models and tools used in engineering: Its formulation and its scope (sometimes in engineering we ignore this information, resulting in a wrong interpretation and bad application of some math concepts)
Thank youuu. I am interested in maths but I have a very different life so I can't just go and study it. But self learning guide is what I needed so far because that's what I was doing. Found some books and was checking youtube to learn it but this video will be so helpful. Thank you so much❤❤
Yeah, I think it present linear algebra in very understandable manner and also in the correct way, Axler is good but his determinant phobia is not great
In the United States, Real Analysis is not studied until 3rd year undergraduate. And Complex Analysis is not usually studied until 4th year. And Algebraic Topology and Differential Geometry are usually studied in the 1st 2 years of grad school. 0:130:13
I agree we need more visualizations. Just seeing the checkerboard for complex functions lead me to realize you can use similar checkerboards to illustrate Stokes’s theorem.
I had several attempts to gather math and none of them was a success due to eventual focus loose. I'm gonna give it another try. Thanks for sharing this.
Excellent books suggestions. Some I've enjoyed and some I've never seen. Also, the Wesleyan complex analysis lectures are on Coursera, so can be watched with quizzes and exercises there. Richard E Bouchard's lecture courses on TH-cam are well worth a watch too. Group theory, Galois theory, complex analysis and algebraic topology are there, along with number theory and other areas.
@@Phoenix-he1mm I would learn them in this order: linear algebra-->real analysis-->complex analysis-->group theory-->point-set topology-->Galois Theory-->algebraic topology-->differential geometry
Great work compiling all of this together! I do think there are some issues with order and what material is presented. First (and most glaringly) a genuine abstract algebra treatment is not given. Groups are certainly important, but rings are equally if not significantly more important. While modules are typically covered in a graduate course, they certainly can be discussed (at least free modules or modules over PIDs) in an undergrad class. Galois Theory is almost certainly graduate level material, and you need a good amount of field theory to even begin discussing it seriously. My first Algebra book was Aluffi's "Algebra Chapter 0" but that's quite fast and uses a lot of category theory, Artin's Algebra book would be a good undergrad level book to work through. I also think that point-set topology need not be a standalone class. Basic point-set topology should be covered in Real Analysis (open/closed sets, compactness, closure, limit points, etc.) , and advanced point-set topology should be introduced on an as-needed basis in an algebraic and/or differential topology class. If one were to do Real Analysis through Spivak (which I recommend) then this would happen organically. Overall though I think you did a great job. The book recommendations were on point and the rough order was solid.
This is amazing! We need experts from every field to make a video like this. You could learn everything that you would in college for free! I know that online free courses are out there but it's often hard to tell which ones are good or bad and the order you need to take them.
I've watched all of your videos thus far and I just have one request for you: PRODUCE MORE! :D I really like your style and neck for explaining things. PLplspls, you are great!
Agreed, Group Theory by Harvard Prof Benedict Gross online lecture ("Abstract Algebra") is the best. I viewed most of the videos (almost 30 series) in 2012, then I attended Prof Gross' NUS public seminar in 2013, during the tea break I thanked him in person for his online free "Abstract Algebra" lecture which benefited me to understand at ease his lines of thought in that public seminar, where he introduced his indian PhD student Bhargava's thesis research work (which won for Bhargava the Field Medal 2014).
Thanks a lot bro. It is quite rare to find good "manuals" and useful advice for self study. And it takes a lot of time to get into a book and trying to understand the concept the book works and how the author tried to deliver his knowledge. Thanks a lot for this list. Especially for the Playlist!
I used to be one of the better math/science people I knew. at least among my peers. However, since graduating high school, all my friends are studying math and science while I study outdoor leadership which is an arts degree and has no required math courses which made me sad. However, outside of just going and getting another degree, I didn't have a strong grasp for how best to learn these things. Thank-you so much
I recommend complex analysis by saff and snider. I’m a freshman in high school and personally I find it engaging and not too difficult and it seems comprehensive enough
I'm a CS undergraduate student and wanting to explore the fields of mathematics and understand very well the math foundations of my field. But your videos are making me want to dive deeper than i should haha.
MIT ocw aggregates a lot of resources and problem sets, with lecture notes and practice exams for essentially all of undergrad and grad school. Probably one of the most comprehensive resources in existence.
as a physics student I really recommend the Wu Ki Tung's book for group theory. It's pretty easy to follow, with you a lot of emphasis in examples and uses in physics.
I find "Elementary Linear Algebra" by Howard Anton to be a great book with plenty of illustrations and problems to solve. I used it much more than the book we used in class and it builds up the topic from scratch.
Books may help a lot when you have good basics. I'm tutoring math-related stuff, and often the case is when people don't have any clue what is proof and what is not. Asking "why this is true?" resulting into answer describing what we do and argument that because we do this, thus it is true. Completely unrelated things. Also, I often hear in reply "but this is true!", and can't deny that what we trying to prove is true, but sometimes hard even tell difference between just saying it's true and real proving. So, just wanna tell, that not anyone can study pure math by these books without any help.
As too understand proofs and do some of them your self you have to have some knowledge of logic.Both some knowledge of formal logic and how proofs are done in mathematics is needed for a study of mathematics.For formal logic I would recommend "An introduction to formal logic" by the author Peter Smith (Cambridge University Press).For how the proofs are done in mathematics I would recommend "A logical introduction to proof" by the author Daniel W. Cunningham (Springer).
surprised Dummit and Foote (Abstract Algebra) was not mentioned at all in this video. Extremely useful resource for Group theory, Galois theory, and much more.
This video is a nice christmas gift to us, the viewers... Thank you for sharing this with the world; I've been engaged with the channel since "The derivative isn't what you think it is" and this channel truly has a unique style. Your way of explaining topics make them feel approachable and nice. ¡Merry christmas, cardinal of the natural numbers!
woah, i dont have any words to explain. thank you so much. not for the book, but the path you showed. Here at university we are reading just to pass the subject, but deep down we can have fun doing math problems. I miss when i was young and loved to do maths.
To understand easy not to remember but to understand its conceptual and adapt to match your thought to correct answer.Key:Given problem, using tool, answer, testing answer feedback.
Hi Aleph0, I have just started my journey towards self studying Mathematics, and I understand that Applied Mathematics is a different study track from what you have outlined here? Could you please make a similar video on how to self study Applied Mathematics (like Probability, Game Theory, etc.)?
it depends since applied math is pretty broad. but you'd still have to learn a lot of the same core subjects (calculus, linear algebra, discrete math, abstract algebra, real analysis), but if you're self studying, you might just skip what you don't need. if your interest is computer science, then you'd want to learn logic, recursion theory (so like incompleteness and undecidability), maybe cryptography, etc if your interest is statistics, then you'd maybe want to learn probability, measure theory (which helps you abstract probability), game theory, maybe functional analysis, etc if your interest is physics, there's multiple routes you can take (quantum, chaos, fluids, etc), but you might learn more on ODEs and PDEs, and computational methods would be useful. but again, it's going to depend on your interest
For real analysis, i found Tao's (subsittuted by Zorich for missing topics) to be a really great self-study resource on analysis. Same for Shilov's linear algebra, i think Axler's is great book for an alternative look at lingebra, but his phobia of determinants was not helpful for a first take on the subject to me. Although real analysis books usually have a decent-ish introduction to logic that just about suffices for writing all the proofs, i found Rautenberg's introduction to logic to be a great resource especially coming from a high-school that never really bothered with rigorous or even semi-rigorous takes on maths. Can't really speak further on maths because, i was focused on applications thereof in physics - highly recommend Frederic Schuller's lectures for anyone interested in physics.
I graduated from college with a double in pure math and econ in may. Abstract algebra convinced me that no matter how much I loved analysis, a math grad program wasn’t for me. Been doing some industry stuff and I’ve been surprised by how much I miss learning this stuff. Maybe I’ll go back one day, but I think this self study stuff sounds like a good middle ground
Thank you very much. The problem for students today is not having enough resources, it's having too many resources - which, if you think about it, is just like not having any resources at all. When the signal that you are searching for is hidden in a cacaphony of incomprehensible noise, you might as well be deaf. Your insight is extremely valuable.
Stillwell's _Naive Lie Theory_ is an excellent introduction to fundamental ideas in continuous groups. It restricts to matrix/classical groups so as to avoid the complication of the term "smooth manifold," explaining the vast majority of useful continuous symmetries in less than 200 pages, with extensive bibliographic work and historical exposition. It's part of Undergraduate Texts in Mathematics, same as Axler's wonderful linalg text. That whole textbook series is fabulously written, edited, and printed; I imagine if they have a book on the topic it compares favorably to most others.
Thanks for watching! If you have any resources you'd like to recommend, feel free to comment them down below.
Also, I have a math newsletter where I collate resources to learn topics in math / machine learning and deliver them to your inbox. If you'd like to sign up for the newsletter, fill out this form: forms.gle/Rt1f5StAj3yZtakE6
I used the organic chemistry teacher to get me started in math, because I have a GED to take. I actually like math now
replies
Analysis of a Complex Kind playlist:
th-cam.com/video/CVpMpZpd-5s/w-d-xo.html
As someone who is depressed, this is the most joyful video I have watched in a long time. Thank you for posting this.
... what
I don't mean to offend you, but this is not a wholesome video or something... It's a math learning guide, how does that cheer you up
Lmfao
Bot
math gives this feeling of certitude and determinism that equations embodies, learning a new concept and solving a problem also has a unique feeling to it, totally understand it.
@@Sciencedoneright welcome to aesthetics. In case you haven't noticed, it's quite subjective.
I would have liked to see this video like 25 years ago. All this material free online. What a wonderful time is today. I want to thank you in the name of humanity.
That’s true
Books
- Axler, Linear Algebra (also has videos)
- Abbot, Understand Analysis (su lecture series)
- Uni Toronto, Point Set
- Wegert, Visual Complex Function Complex Analysis
- Lang, Complex Analysis (well illustrated)*
- Wesley Uni, Video Playlist Analysis of a Complex Kind
- Gross, Abstract Algebra (video)
- Herstein
- Galois, Same channel video and Tom Lanster
- Boothby, Introduction Diff. Manifolds and R. Geo.
- A. Hatcher, Algebraic Topology (peer albin)
This makes me very happy. I studied comp sci at uni but have been trying to learn some higher math on my own lately. The specific resource recommendations are extremely valuable to me. I consider this a Christmas gift!
same boat. i need a lot of math in my research that my CS degree simply didn't prepare me for. These kind of resources are amazing!
Plz help those students who can't go to school in this corona pendamic one click for them
i am pursuing an undergrad cs degree. what sort of math should i be learning myself that my degree wont have in its syllabus?
@@ashwinjain5566 Depends on the program and ur research area. What I find most useful is Advanced Linear Algebra, Differential Geometry, Topology, Tensor Calculus, Advanced Probablity Theory with a bit of Measure theory, Continuous + Convex Optimization
man, i want the same for this year.
I appreciate how you didn't overload us with resources but gave it straight forward. Much much more appreciated than the channels that throw tons of books or resources.
This video is left as an exercise for the reader.
DUDE you are an incredible content creator! Not only finding all the resources that best help newbies into classic maths, but also providing insight into their way of teaching!
Should keep me busy for the next 5 years
lmaoooo
quite literally me
As a theoretical physics graduate student, I can strongly recommend the textbook "Differential Geometry, Gauge theories, and Gravity" by Gockeler and Schucker to understand differential geometry with just a physics undergraduate background, where the book introduces but also applies this subject to numerous areas of modern physics - I really recommend it!
Also, if you're a physics student and don't really know where to start, I would strongly recommend "Modern Mathematical Physics" By Szekeres which basically introduces from the ground up and assuming no prerequisite all the parts of maths you need to do physics at an advanced level. Hope it helps :)
Nakahara's _Geometry, Topology, and Physics_ is an excellent text as well. What it lacks in readability, it more than compensates for in completeness and rigor, explaining everything from vector spaces to cohomology to differential forms. Wholeheartedly recommend it as a referrence text for the physicist or physics-adjacent mathematician.
perfect timing
Are you also graduated from MIT,and perhaps your name is gordon?
I just finished fourth year quantum. I was confused by the linear algebra book that didn't talk about determinants.
Wow thank you! Excited to start Szekeres
I'm extremely overjoyed to have found this video. I've been wanting to self-study pure maths (with some physics as a little treat) for a year or two now, and am currently strengthening my foundational understanding of the basics, but was bumbling through the undergrad stuff with the help of the MIT courses and Stack Overflow. I'm incredibly grateful that you included videos alongside the textbooks, since I tend to learn better with a combination of both (but mostly the former). This video is like finding an oasis in the midst of a desert, thank you so much!
Great video! As someone who got an undergraduate degree in math, many of the resources here are excellent resources that I made good use of. I look forward to checking out the others to fill in the gaps on some of my weaker subjects.
Where did u graduate & what r u doing currently?
Malayali aanalle. Naatil evideya?
As some one in an engineering major we study just as much math and its PAINFUL!!
@@dontreadmyusername6787 you're not studying as much maths as a maths undergrad
@@sandraaiden8587 maybe but there's a lot of math in electrical engineering
Hey. I just want to say you've really inspired me to get back into self studying mathematics. I had done a lot of self study in the past, when I had much less mathematical maturity, and got discouraged, in part because I kept skipping prerequisites and the like. Thank you for this and other videos.
this is my favorite math channel by far, your voice is soothing and your explanations are always easy to follow.
I love you. Self teaching math is stupid hard for no reason.
Thank you for the video. I wish TH-cam had a "love" button. Please don't remove the previous version of this video from the channel. It was so informative and helpful. Just add "2020" or something in the title. I love that video too! Like this comment if you want that video back.
I think he already removed the previous version (people in the comments were insufferable, I think that s why). I m really glad he made a new version though
@@everlastingideas8625 what was in the previous version???
I will also put my brick in this wall. I studied mathematics a couple of years a go and I really loved all the mathematics books from Schaum's Outlines series. They have books on Group Theory, Linear and Abstract Algebra, Real and Complex Calculus and many more etc. Usually those books first present the theory and provide examples. Then you have solved problems where you see how a given problem can be tackled and finally the problems for the reader to solve (with answers at the back). I remember a lot of the content was aligned to my university courses and I could learn a lot from them. Plus they're very good for studying on your own without a teacher/mentor/professor. They are basically so good that now (a few years after graduation) I'm still collecting them since I want them in my personal library for future reference or just to have them just in case (they are really hard to come by in Poland). And one tip in advance. Look for the older versions of the books or first/second editions in some cases. The reason is because slowly some good parts or even entire sections are being cut out. I don't really understand the reason behind this but anyway - you have my opinion right here. Stays safe Comrades!
Your mommy
> The reason is because slowly some good parts or even entire sections are being cut out. I don't really understand the reason behind this...
But you do understand, don't you?
Calculus by Michael Spivak (Third or Fourth edition) is unexpectedly a really good introduction to not just real analysis, but also some number theory and a little abstract algebra at the end of the textbook. It doesn't require any prerequisites, as stated by the author (I think). It also justifies most theorems that are brought up, has proofs for them, and most importantly, an ungodly amount of examples (like 60 per chapter, making up most of the book).
For everyone else please don't use Spivak as first calculus book, I beg you
@@Miguel_Noether why? :)
@@maiiaskrypnyk5234 It's very challenging. Even Caltech only uses Apostol.
@@Miguel_Noether I have without realising what analysis was, no wonder it was so goddamn difficult. I just wanted to learn basic differentiation and integration, but ended up learning how to prove some statements from real analysis.
@@Miguel_Noether I came looking for copper, but instead found gold lol
As someone who's doing a degree in mechanical engineering but has a very strong passion for pure math I very much appreciate this video, as well as yours and everyone else's free and amazing videos on many different math topics all across youtube.
Keep up the great quality!
lol why didn't you go into math? engineering is like the antonym of math
@@mastershooter64 You'll maybe encounter some day people with more than one interest. It's my case, too, I'm a student in microengineering and I have a strong passion in pure math
@@mastershooter64 it was hard to find a good and affordable university, I didn't have faith in my math knowledge enough to enroll into a pure math class, my passion for physics is not any lesser, and last but not least, it is significantly easier to find an enjoyable and well payed job with an engineering degree from everything that I've seen.
I would say engineering is an offshoot of mathematics or it is essentially mathematics with a different name. Many famous equations/relations in Mec.Eng are derived by mathematicians instead of engineers themselves. For example Prandtl , Blasius, Navier ,Stokes etc.
@@hussainsallar3355 Not to mention that at that time those people were really not one or the other but both, mathematicians and engineers, and sometimes also worked in other fields as well. Scientists in general (including many mathematicians and engineers), were really just called "Natural Philosophers" until the middle to end of the 19th century.
That is actually one of the things that really attracts me to maths, it is not only very interesting on its own, but it is also a quite universally powerful instrument for the study of any other science. That is why I intend to pursue a masters in Applied Mathematics after I finish my bachelors in mechanical engineering.
When I was in college, I wanted to take more math beyond Calc III and Linear Algebra, but I didn't have the time or funding to do it. Beyond being interesting, I found it helpful with writing music and thinking about writing in different ways. I'm glad you smart professionals provide these resources for us.
Thank you so much! I'm a med student who loves math (crazy, I know). I've always regretted missing the opportunity to learn pure math at uni level. I'm excited to use these resources!!
You’re in the wrong course
@alfredhitchcock45 what do you mean?
@@rijakhalid9011 you should have taken bs math or engineering
@@alfredhitchcock45 too late lol I'm already in 4th year of med school
Thank you so much for making a self study guide video again. I started down following the advice from your original, but then it disappeared... now it's back and updated :)
So I noticed the topics for the undergraduate curriculum in this newer version of the video changed slightly but either way it makes me want to bring up a somewhat list of things you would see as an undergraduate in university.
First off, not everyone does calculus and differential equations in high school (generally speaking most don't): so if you're going to university, expect those. The topics of Linear Algebra, Real Analysis, and Abstract Algebra are generally the common core of every undergraduate math degree; however, most people will do more things like complex analysis and topology. A lot of American universities will have a class dedicated to learning how to write proofs and learning basic set theory and number theory. Group Theory and Galois Theory aren't usually taught on their own as courses but are rather a large chunk of a first and second semester abstract algebra course respectively. Not everyone does differential geometry or algebraic topology when they're an undergrad but a lot do. Other undergraduate courses include: Probability, Number Theory, Euclidean Geometry, Partial Differential Equations, and basically anything that says "intro to [branch of math]" is either an advanced undergraduate course or a graduate course.
I 100% can attest to this. I go to Florida State University and that's exactly how it's set up here.
Yeah differential equations is pretty uncommon in most high schools. I went to a well-funded public school, and having AP single-variable calc was honestly a privilege. I'm now in a well-funded liberal arts college, and you're on-point with the classes usually completed for the major. This video seems to assume more a magnet-school type of math education and a large university or a quite advanced undergraduate curriculum.
alg top is hardly an undergrad subject
@@TomMS In Canada, there's an intermediary between high school and university. Most students with a stem focus use that period to take Calculus and Diff Eq, which is probably the cause of the disconnect.
@@wontpower Interesting. Didn't know that!
Man,
I got those books thanks to you, and I can say that the real analysis one is really well done!
Actually, every problem that mathematic books have (not defining important things, not explaining the history of a concept, refusing to give simple example, etc.) isn't in this book so far in my reading
did you find them on the internet for free?
update : all of them except one are on the internet for free
@@salwaadlouni1855 Library Genesis btw
A minor nitpick: the study of differentiable manifolds without a distinguished 2-form (such as a metric or symplectic form) is generally referred to as differential topology, while the inclusion of such a form "elevates" the subject to geometry. So, the point where you say differential geometry ends is actually where differential geometry begins.
This is not true. Certainly not where I'm from. Differential topology is simply the study of the topology of differentiable manifolds. There is tonnes that you can call geometry without a metric or a symplectic structure. My own research for one
Perhaps a better way to think of it is that differential topology is the study of global properties of smooth manifolds while differential geometry is more about the local properties. As always there is some overlap.
@@Aetheraev I'll admit that the distinction given here isn't always exactly correct, and I hoped that my use of "generally" would cover me for situations where it fails. However, at least as far as I've seen, calculus on differentiable manifolds isn't considered geometry, which is the subject of the first six chapters of Boothby referenced in the video and by my comment.
In the end, I think it comes down to semantics and it's not truly important where exactly the lines are drawn; the main purpose of my original comment was simply to correct the statement, "This part marks the end of the differential geometry portion of this book," as the Riemannian geometry that follows really has the *most* right of any material in the text to be called differential geometry.
Thank you so much! After highschool i choose to take on a psychology career even if I've really loved maths, always promising myself to self-study it but everytime getting lost, this video is a treasure
wow this is exactly what I was looking for just now, can't believe it was uploaded today! thanks for this!
wow this is great thank you so much! I really know where to start right now! I just stopped university because stress was consuming me.. now I can just live my life and still learn some mathematics thank you so much!
Thank you so much. I'm a data scientist who has been trying to teach myself pure math for several years. I'm working on Linear Algebra Done Right now and wanted to jump to abstract algebra after I finished.
Socratica's short course of abstarct algebra was pretty amazing
I thought this was going to be more about how to focus when doing math, how to understand concepts and solve problems and so on, but still a nice collection of resources!
Highly recommend the real analysis. Took a semester of real analysis and the book's been a huge help
A few comments from an erstwhile math student . . .
+1 on Topics in Algebra by Herstein. A lovely book: rigorous proofs and accessible explanations.
+1 on Algebraic Topology by Hatcher. Excellent examples and illustrations.
- 1 on Intro to Differentiable Manifolds by Boothby. It's hit or miss. The pace is uneven, the exposition often too spare. It tries to cover too much in its modest length: differentiable manifolds, Riemmanian geometry, group actions, Lie groups, Lie algebras -- without giving each topic full, proper treatment. Try Lee instead (see below).
NB:
For differential geometry, Lee's Intro to Smooth Manifolds is a solid, modern supplement/replacement to Boothby, or suitable for self-study (perhaps if you're comfortably reading at graduate-level).
For topology, Munkres' Topology book is a solid bet.
For group theory, Abstract Algebra by Dummit and Foote has good breadth and depth and covers everything Herstein's does. A good starter book in group theory is Fraleigh's First Course in Abstract Algebra; suitable for self-study.
I think this is one of the videos that got me to switch from physics to maths. Thank you!
This video is pure gold. Thank you so much for putting this resource together!
Thank you for bringing all this together, true hero! Your voice is wonderfully relaxing too!
Amazing video. I've used quite a few of those myself and I've also found a couple of new things that will come in handy this semester. Thanks
¡Gracias!
So I'm not hugely up on my Pure Mathematics as I moved toward the Applied end of the subject after graduating, but those interested in the Singularity Theory part of Differential Geometry (stuff like Cusps, Swallow Tails, and Envelopes), I would recommend "Curves and Singularities" by Bruce & Giblin. It has lots of nice pictures!
I'm very grateful for reposting these ideas. I had purchased the Understanding Analysis by Abbott book but your original video had gone offline before it arrived! My course in calculus 2 is finished and I'm looking forward to self-study on pure math in '22.
For GROUP THEORY category, my main go to book is "Contemporary Abstract Algebra" by Joseph Gallian.
Current shoutout after 12 hours with it (yay Christmas) is "All the Math you missed (but need to know for graduate school)" by Thomas Garrity so far has been a fun book.
I totally agree. Gallians book is perfect as a low entry point into the subject. So many easy to understand and intuitive examples given, along with actual applications of group theory.
Abstract Algebra by Gregory T Lee is also really good. It's a bit more difficult, and is drier. But is also far more concise and more rigorous.
Importantly, both these books are easy to get. Unlike a lot of the older (and more expensive!) textbooks that are often recommended for Abstract Algebra and/or group theory.
@@maxbesley1412 "All the Math you missed" is a blessing. Thanks for the recommendation!
I have studied both MARLOW ANDERSON and JOSEPH GALLIAN for abstract algebra. Even Though Gallian contains more content than Anderson, but I like the way anderson introduces the concepts. If you have studied number theory or just want to study number theory along abstract algebra, I think anderson will make a lot of sense than gallian. Anderson introduces concepts by ring first approach which for a beginner makes more sense than a group first approach.
@@ABHISHEKSINGH-nv1se Will add the anderson book to my birthday wishlist this year.
@@ABHISHEKSINGH-nv1se what's title
OH MY GOD YOU'RE BACK!!!! i didn't see your last 2 uploads until now!!
I really like Munkres’ “Topology” as an undergrad text for point/set topology.
Math is beautiful and a great way to look into the wonders of the universe! Bravo!
Awesome content. Please cover self study for other fields of mathematics eventually. Thanks for this.
So happy to see all these resources being shared.
Such a great video! I've just finished my MSc in Statistics but I'm now looking to spend time self-studying the pure maths modules I had to miss to focus on stats... So this video couldn't come at a better time!
From which University did you get your M. Sc?
This video collects lots of useful online resources and recommended textbooks. The content is clear and well-structured and I appreciate your work to illuminate a way to learn pure mathematics!
The one-sentence-descriptions of these subjects are just pure gold.
"Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation.
I like how learning pure math basically makes you a better electronics engineer. Thanks for providing such insightful list.
Im in first year of electronics engineery and i just can feel how many holes there are in my pure math education, so im trying to improve in order to be a better engineer tomorrow day, pluss its a cool hobbie...!
Pure math gives you a comprehensive background about the origins of mathematical models and tools used in engineering: Its formulation and its scope (sometimes in engineering we ignore this information, resulting in a wrong interpretation and bad application of some math concepts)
Thank youuu. I am interested in maths but I have a very different life so I can't just go and study it. But self learning guide is what I needed so far because that's what I was doing. Found some books and was checking youtube to learn it but this video will be so helpful. Thank you so much❤❤
The Linear Algebra book by Kenneth Hoffmann and Ray Kunze is very good. It puts the Algebra back into Linear Algebra.
Yeah, I think it present linear algebra in very understandable manner and also in the correct way, Axler is good but his determinant phobia is not great
@@ankitsuman1110 Kenneth Hoffmann's book also extends determinants to consider exterior products. It is a gold mine of information.
@@ankitsuman1110 I wonder how he can do Gauss- Jorden elimination
In the United States, Real Analysis is not studied until 3rd year undergraduate. And Complex Analysis is not usually studied until 4th year. And Algebraic Topology and Differential Geometry are usually studied in the 1st 2 years of grad school. 0:13 0:13
I agree we need more visualizations. Just seeing the checkerboard for complex functions lead me to realize you can use similar checkerboards to illustrate Stokes’s theorem.
I had several attempts to gather math and none of them was a success due to eventual focus loose. I'm gonna give it another try. Thanks for sharing this.
Excellent books suggestions. Some I've enjoyed and some I've never seen. Also, the Wesleyan complex analysis lectures are on Coursera, so can be watched with quizzes and exercises there.
Richard E Bouchard's lecture courses on TH-cam are well worth a watch too. Group theory, Galois theory, complex analysis and algebraic topology are there, along with number theory and other areas.
Currently working through Borcherd's series on Group Theory. Highly recommend.
Could you please share the link of Wesleyan complex analysis as I didn't get them
@@subikshakannan8570 It is in the description.
Do you guys recommend studying some of the mathematical concepts listed in the video concurrently or individually?
@@Phoenix-he1mm I would learn them in this order: linear algebra-->real analysis-->complex analysis-->group theory-->point-set topology-->Galois Theory-->algebraic topology-->differential geometry
I failed my maths test today and TH-cam shows me this. Great!
Great work compiling all of this together! I do think there are some issues with order and what material is presented.
First (and most glaringly) a genuine abstract algebra treatment is not given. Groups are certainly important, but rings are equally if not significantly more important. While modules are typically covered in a graduate course, they certainly can be discussed (at least free modules or modules over PIDs) in an undergrad class. Galois Theory is almost certainly graduate level material, and you need a good amount of field theory to even begin discussing it seriously. My first Algebra book was Aluffi's "Algebra Chapter 0" but that's quite fast and uses a lot of category theory, Artin's Algebra book would be a good undergrad level book to work through.
I also think that point-set topology need not be a standalone class. Basic point-set topology should be covered in Real Analysis (open/closed sets, compactness, closure, limit points, etc.) , and advanced point-set topology should be introduced on an as-needed basis in an algebraic and/or differential topology class. If one were to do Real Analysis through Spivak (which I recommend) then this would happen organically.
Overall though I think you did a great job. The book recommendations were on point and the rough order was solid.
completely agree
This is amazing! We need experts from every field to make a video like this. You could learn everything that you would in college for free! I know that online free courses are out there but it's often hard to tell which ones are good or bad and the order you need to take them.
Would you want to form a pure math study group ?
I do want to learn more math but I'm pretty busy rn. I might make a video like this about my field though. I have a PhD in mechanical engineering.
I've watched all of your videos thus far and I just have one request for you: PRODUCE MORE! :D
I really like your style and neck for explaining things. PLplspls, you are great!
I've been looking for a video like this for years now!
Agreed, Group Theory by Harvard Prof Benedict Gross online lecture ("Abstract Algebra") is the best. I viewed most of the videos (almost 30 series) in 2012, then I attended Prof Gross' NUS public seminar in 2013, during the tea break I thanked him in person for his online free "Abstract Algebra" lecture which benefited me to understand at ease his lines of thought in that public seminar, where he introduced his indian PhD student Bhargava's thesis research work (which won for Bhargava the Field Medal 2014).
Thanks a lot bro. It is quite rare to find good "manuals" and useful advice for self study. And it takes a lot of time to get into a book and trying to understand the concept the book works and how the author tried to deliver his knowledge. Thanks a lot for this list. Especially for the Playlist!
Thanks so much for this hard work for others! It's very confusing to explore the world of pure math by self-study, and your channel is very helpful.
I used to be one of the better math/science people I knew. at least among my peers. However, since graduating high school, all my friends are studying math and science while I study outdoor leadership which is an arts degree and has no required math courses which made me sad. However, outside of just going and getting another degree, I didn't have a strong grasp for how best to learn these things. Thank-you so much
I recommend complex analysis by saff and snider. I’m a freshman in high school and personally I find it engaging and not too difficult and it seems comprehensive enough
I'm a CS undergraduate student and wanting to explore the fields of mathematics and understand very well the math foundations of my field. But your videos are making me want to dive deeper than i should haha.
I recommend Tao’s & Zorich’s book for real analysis
Also Smale’s book on differential equation is a great start to dynamical system
You put this video up again. Thank you so much!!!
MIT ocw aggregates a lot of resources and problem sets, with lecture notes and practice exams for essentially all of undergrad and grad school. Probably one of the most comprehensive resources in existence.
Thank you so much for taking the time to present and share this subject matter.
as a physics student I really recommend the Wu Ki Tung's book for group theory. It's pretty easy to follow, with you a lot of emphasis in examples and uses in physics.
Just a little info - the kind that can make a bigger difference than you'd think. Thanks for this! 👍
I find "Elementary Linear Algebra" by Howard Anton to be a great book with plenty of illustrations and problems to solve. I used it much more than the book we used in class and it builds up the topic from scratch.
Wow I love this video this will certainly help with my studies even tho I’m not a math major
Books may help a lot when you have good basics. I'm tutoring math-related stuff, and often the case is when people don't have any clue what is proof and what is not. Asking "why this is true?" resulting into answer describing what we do and argument that because we do this, thus it is true. Completely unrelated things. Also, I often hear in reply "but this is true!", and can't deny that what we trying to prove is true, but sometimes hard even tell difference between just saying it's true and real proving.
So, just wanna tell, that not anyone can study pure math by these books without any help.
As too understand proofs and do some of them your self you have to have some knowledge of logic.Both some knowledge of formal logic and how proofs are done in mathematics is needed for a study of mathematics.For formal logic I would recommend "An introduction to formal logic" by the author Peter Smith (Cambridge University Press).For how the proofs are done in mathematics I would recommend "A logical introduction to proof" by the author Daniel W. Cunningham (Springer).
surprised Dummit and Foote (Abstract Algebra) was not mentioned at all in this video. Extremely useful resource for Group theory, Galois theory, and much more.
Probably the best undergraduate book for abstract algebra.
One of the best video on youtube for Maths major.
This video is a nice christmas gift to us, the viewers... Thank you for sharing this with the world; I've been engaged with the channel since "The derivative isn't what you think it is" and this channel truly has a unique style. Your way of explaining topics make them feel approachable and nice.
¡Merry christmas, cardinal of the natural numbers!
This video is extremely informative, it spent me a morning to follow every topics one by one! Good job, man!😀
I am amazed at how I cannot understand anything from any of your videos. Guess I just have to study more.
woah, i dont have any words to explain. thank you so much. not for the book, but the path you showed. Here at university we are reading just to pass the subject, but deep down we can have fun doing math problems. I miss when i was young and loved to do maths.
I remember liking Ian Stewart’s Galois Theory for its very slow pace and many examples. I think it’s a good book for self-studying Galois Theory.
I got that book for Christmas!
@@fyggy5480 Lucky you. Math books cost a lot
To understand easy not to remember but to understand its conceptual and adapt to match your thought to correct answer.Key:Given problem, using tool, answer, testing answer feedback.
Who else when watching this video: FINALLY.
This is beautiful. Thanks for all the references to dig deeper.
Hi Aleph0, I have just started my journey towards self studying Mathematics, and I understand that Applied Mathematics is a different study track from what you have outlined here? Could you please make a similar video on how to self study Applied Mathematics (like Probability, Game Theory, etc.)?
it depends since applied math is pretty broad. but you'd still have to learn a lot of the same core subjects (calculus, linear algebra, discrete math, abstract algebra, real analysis), but if you're self studying, you might just skip what you don't need.
if your interest is computer science, then you'd want to learn logic, recursion theory (so like incompleteness and undecidability), maybe cryptography, etc
if your interest is statistics, then you'd maybe want to learn probability, measure theory (which helps you abstract probability), game theory, maybe functional analysis, etc
if your interest is physics, there's multiple routes you can take (quantum, chaos, fluids, etc), but you might learn more on ODEs and PDEs, and computational methods would be useful. but again, it's going to depend on your interest
@@The2378AlpacaMan A very late reply, but thank you so much for the detailed reply!
As someone who hates commets starting with "as someone who...", I think this is a really great post. Thank you!!!
why was this video taken down before?
thank for bringing it back
This is the most helpful video for self learners I've seen, thank you
For real analysis, i found Tao's (subsittuted by Zorich for missing topics) to be a really great self-study resource on analysis. Same for Shilov's linear algebra, i think Axler's is great book for an alternative look at lingebra, but his phobia of determinants was not helpful for a first take on the subject to me.
Although real analysis books usually have a decent-ish introduction to logic that just about suffices for writing all the proofs, i found Rautenberg's introduction to logic to be a great resource especially coming from a high-school that never really bothered with rigorous or even semi-rigorous takes on maths.
Can't really speak further on maths because, i was focused on applications thereof in physics - highly recommend Frederic Schuller's lectures for anyone interested in physics.
We share the same opinion! That's odd.
I wrote a comment a few minutes ago but cannot see it. I've exited the video 4 times but still nothing.
"Visual Differential Geometry and Forms" and "Visual Complex Analysis" both by Tristan Needham.
I graduated from college with a double in pure math and econ in may. Abstract algebra convinced me that no matter how much I loved analysis, a math grad program wasn’t for me. Been doing some industry stuff and I’ve been surprised by how much I miss learning this stuff. Maybe I’ll go back one day, but I think this self study stuff sounds like a good middle ground
Plz help those students who can't go to school in this corona pendamic one click for them
Loving the lelouch pfp
Go study statistics if you love analysis
Thank you very much.
The problem for students today is not having enough resources, it's having too many resources - which, if you think about it, is just like not having any resources at all.
When the signal that you are searching for is hidden in a cacaphony of incomprehensible noise, you might as well be deaf.
Your insight is extremely valuable.
Stillwell's _Naive Lie Theory_ is an excellent introduction to fundamental ideas in continuous groups.
It restricts to matrix/classical groups so as to avoid the complication of the term "smooth manifold," explaining the vast majority of useful continuous symmetries in less than 200 pages, with extensive bibliographic work and historical exposition.
It's part of Undergraduate Texts in Mathematics, same as Axler's wonderful linalg text. That whole textbook series is fabulously written, edited, and printed; I imagine if they have a book on the topic it compares favorably to most others.
I Found this video in the middle my mid term test and honestly it's so hepfully to me. Thanks for ur video
Thank you so much!
This is wonderful. For a very simple "pre-intro" to serious Group Theory, I love Nathan Carter's Visual Group Theory