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Thanks for sharing!

While I'm sure that Ince does have much to offer, I'm a fan of readability. In my opinion, T&P is much easier to read than Ince. Ince is much older and has a focus on generality. I would even argue that Ince has a different target audience. What are your thoughts?

HAHAHA. Send it then be like, "Oh, sorry, I meant to send it to a friend." 😆 (gonna be explicit here and say that I'm just joking). But in all seriousness, hang in there!

@professorquarter this is a great suggestion!! Thank you for sharing!

1) Mostly problems don't require a complex solution. If you follow advice you've heard, you will get great performance boost over your classmates that don't. It looks like you want some magic pill to solve problem, not a solution. 2) Author is asked to give his opinion on problem and he gives his opinion on problem. Moreover, he/his mates probably has struggled a lot while studying. So he might know some tips that a newbie doesn't know yet. That's why it is good to ask mathematic pro for advise too. It is always "practice to get good" with some glitter. What are you waiting for?

This is a very generalized solution for learning proofs-based mathematics. This is NOT a video on how to study complex analysis or whatever. You honestly ought to be doing as he is discussing and also using more concrete worked out examples. ChatGPT is great for this and he suggests using it in the video. My preferred way to use chat is to have it poke holes in my work as its understanding of even fairly basic mathematics collapses when anything which is only implicitly involved comes into play.

Thank you! Yes, indeed, it will cover the most commonly covered topics in your usual first course in linear algebra. The first 8 chapters are going to be what you would want to read for this. Strang is an interesting case because while it will cover most of the usual content, it does not cover it in the usual way. It is very informal. Which is fine if that's what you want. Glad you could get it for free! Great stuff in there. Thank you!

@@MathematicalToolbox Thank you for your reply!!!

You mentioned "The idea is to slowly let go of the various types of scaffolding you're provided." You didn't mention, letting go of one's 'geometric crutches.' Which is another... IMO... non-sense phrase... promoted by the likes of Gauss... et al. There are few mathematicians, in history, to match Gauss. But far too many people, in history like him.

Great input! I agree!

In Europe (I recall that's where you're from), authors typically don't offer solutions?

Can't 'sell' a textbook (published for the university math course market) without... at least... some problems in it... that have no 'answers'. Many profs won't use it. (They can't assign homework problems from it.)

Discord groups, zoom clubs, reddit, etc. are always an option.

@@stevenkies802 There are a couple of 'well known' forums... and if you are willing to prostate yourself before the 'gods' that inhabit them... and admit you really are an idiot... they sometime will throw a few bread crumbs to you. I'm not saying there are not some very good mathematicians posting on those forums: there are. Just the same, and even though they know, and understand, mathematical definitions... they, don't have a poor understanding of what the term 'idiot' means, in the vernacular.

Glad to help! What subjects/fields of math interest you the most?

Thank you for your input! Much appreciated!

Don't give up on requesting mathematicians (mathematics professors) to articulate how they 'do' mathematics... and expect to evaluate your understanding. IMO, many can't, and the majority don't want to. There is a very long tradition of 'hiding' the details, and engendering elitism... at least from Fermat and Gauss onward. If you really understand, and can do, high school mathematics... especially algebra... and you know how real numbers work... you should be able to... at least mostly understand the mathematical topics... and 'do' the 'average' associated homework problems. If you can't... get 'better' textbooks (by authors that want to 'explain' mathematics)... or... get better professors... or get a better university. Also... be aware that you are part of a system that pushes as many students out of mathematics as possible, yet doesn't mind taking the money, to do it.

Hopefully between this video, and my original comment reply I answered all of your concerns!

Thank you for the nice words! I apologize for my lack of interest in algebra! I am lacking a sufficient background in algebra and geometry, so I try to stick to what I know. I've got several books that I've used lightly, that may be of use to you if you need a book recommendation or two. I will get to algebra and geometry at some point, just not sure when. Hopefully you enjoy the content enough to stick around :D

@@MathematicalToolbox no problem man, yeah you have a really chill presentation style and speak on topics that no one else does. Keep it up

I'm happy to help! My formal background in mathematics consists of a minor in applied mathematics. After university, I've delved into many subjects (at various levels of understanding): integral equations, stochastic calculus, analysis, measure theory, functional analysis, etc.

Thank you!

Indeed! Thanks again!

I have never taken this exam, and I have never attempted to take the exam. But I know from personal experience that doing problems is the way to go. I would get myself a workbook and work through the problems several times until I have them down. If you can't afford one, then I'm sure you can find one online. Perhaps you could ask Professor Joydeep Dutta here on RUclips.

@@MathematicalToolbox Thanks for your recommendation.

Short answer: kind of, but I strongly recommend elementary (calculus based) probability. Long answer: It really depends on where you want to go next and what you want your mastery of the material to be as you are reading the book. Could you read this book and then go into something like Solin and Sarkka? More than likely. But I personally found it difficult to go from this book into something like Øksendal that does require some measure theory. You could always try it, though. Especially if your probability is good. I do think it would be possible to use Calin, then Klebaner (or Shreve), and then move on to more specialized books. Klebaner's book is a rigorous introduction that assumes knowledge of linear algebra, calculus, and probability. Shreve assumes calculus and probability. Calin has another book on Stochastic Geometry that does utilize his stochastic calculus book as the primary prerequisite. If that's the route you want to go next. There's also the book by Zhang and Karniadakis on SPDEs (which I've done a video on) that is fairly self-contained, but you need PDEs, Linear Algebra, probability and even some analysis. If you're unsure if you'll like the subject, then you can use the Calin book for a good introduction. If you are more serious, then you'll definitely want to learn probability, analysis, and measure theory first. Hope that helps.

@@MathematicalToolbox Thank you very much, that was a great answer!

Thank you. I've been doing well. Yourself? As far as finding solutions to problems when there is no solutions manual, you can check out my last video (in case you haven't already). I talk about all the different things you can try when stuck. As far as forgetting stuff or not having learned stuff in a certain order, who cares. Just learn it again or learn it now. It's probably because you were not exposed to it, didn't learn it properly the first time, or didn't put in the effort to memorize it, or you just don't use that math often. It's really not a big deal. I have to go back and verify my understanding of vector spaces, linear independence, etc. pretty often tbh. I would be concerned with a pattern of negativity coming from a tutor like this. It could be just an isolated incident, though. You could say something like, "It's okay, that's why you're here. Explain it to me." I would have taken it as an opportunity to offer you more tutor time, hahaha. I still use solutions all the time, but I study for fun. You, as a university student, might want your understanding to be a little bit better than mine, though. If you are going to use solutions heavily, I suggest you use them in a healthy way. E.g. don't get stuck on problem 1 and go looking for a solution after 15 minutes. Chances are you need to go back and verify your understanding of the definitions, theorems, examples, etc. There have been several times when I was struggling with an exercise, just to find a solution, and it was pretty much an exact step by step as one of the examples! I hope that helps. I'll probably do a video on this. Thank you and good luck in your studies. Feel free to reach out whenever!

Wait, which of the two? The "expert mode" stuff or the way I go about it?

@@MathematicalToolbox The way you go about it. If itried to do every problem in each section, I would still probably be doing high school stuff.

Twenty years ago, when i was an undergrad I can across the book Advanced Calculus a Differntial Forms Approach by Edwards. It looked really cool and i could kind of get it, i had just finished calc 3 and linear algebra when i found the book. I came across the book for 30 on ebay two weeks ago and grabbed it. I made a plan, just like you mentioned. I'm doing multivariable calc out of Adams and Essex, Linear Algebra out of Strang, proofs out of Chartrand, and Real Analysis out of Abbot. Then I'm going to go through Edwards. I hope to be on Edwards in 12 to 15 months.

@@walter274 Sounds like a great journey. Best wishes. Cheerful Calculations! 🧮

I haven't read Pugh, but i heard it was good. I have a friend that is connected to Penn, so i was skimming Penn's webpage and can across the Econ department, and Pugh is the Analysis book they want their students to be comfortable with.

But aren't internet points and sounding smart what it's all about? 🙄 I'm being facetious of course. I do wonder what drives some textbook choices. Ideally, we would choose the best for our audiences. That probably doesn't always happen though; unfortunately. I don't always like the books and materials used in other sections of courses I have taught at the undergrad level, but I have kept them. Thank you for sharing your insight and experience.

@hubomba I've never heard of Pugh before. Thank you for sharing! Thank you for your input about Rudin. I have not heard/seen anyone say anything bad about it. Very enjoyable to read, too! Felt like I was reading a review. 😁

I don't know Stochastic Calculus, so take what i'm saying with a pebble of salt, but i would think some probability would be useful. I have a bizzare recomendation for that. Don't get Ross's intro to probabiity book. Get his Probability and Statistics books for Engineers, and his Probability Models book. Together they cover the same maerial as the intro book, and you can find solutions manuals online. Or so i've heard :) The excersises are often duplicated. You can get both used for like 10 each. That said there are many good probability books out there, not just ones by Ross.

It depends on what you mean by "average level." You can use the book mentioned in the video. The blue book by Professor Calin. No probability is required, just calculus. The prerequisites from probability are given in the second chapter. I would consider having some probability and learning stochastic calculus using a book that assumes probability to be an average level. This is because the subject usually requires measure theoretic probability. Measure theoretic probability would be at the advanced level, I would say. You can use something like Blitzstein and Hwang, or Carol Ash's Probability Tutoring Book, or Lefebvre for probability. There are dozens of really good probability books. Then, you can use something like Klebaner, Durrett, Shreve, etc. to learn the subject. So, the shortest path at an average level could be Calculus I -> Calculus II & Linear Algebra -> Probability -> Stochastic Calculus

Doing alright, thank you tor asking! Yes! The book Problems and Solutions in Mathematical Finance Volume 1 by Chin, Nel, and Olafsson is a collection of theorems and their proofs. Unfortunately, I am not aware of any book that is a collection of concrete problems and their solutions. Apparently, there is a way to utilize Ito's formula to solve almost any SDE. I have only heard of this and have not put in the time to actually learn the method. Send me an email, and I'll forward you the work a subscriber had sent me on an SDE a while back. Thank you!

I haven't published them here on RUclips yet. The memberships are pretty much ready to go, but I want to upload all the content I've already created before people give me their hard earned money. I have not gotten around to this yet. So sorry! I am planning on uploading the same content on Patreon and RUclips. Currently, I have a book video on integral transforms and more than a few short discussions on books. I've got $1, $5, and $10 tiers on there. Thank you so much for considering supporting the channel. patreon.com/MathematicalToolbox?Link

@@MathematicalToolbox Thank you!

I am a bit of a luddite, so I am not sure, but from my perspective, I would recommend Strang. Imo Axler is not a good first book on the subject. I tried using it when getting back into math and thought it was lacking. It had been a few years since I had last seen linear algebra, but I felt I needed to refresh with the usual lower level undergrad books. You can check out Dimitri Bianco here on RUclips. He's a Quant, so he might have a better answer for you. Also, you might want to check out the book on deep learning architectures by Professor Calin.

@@MathematicalToolbox Thank you! Also, do you think Hammock's book of proof is sufficient going into Tao's book on Real Analysis?

Yes! Hammack's book is my go-to on the subject. Tao's book is very good, too, and is one of my recommendations for a serious analysis book for a serious student. Tao also has a short refresher on intro to proof on his first analysis book. By the way, you might want to also consider some of the following analysis books as they are some of the easier ones on the subject (in no particular order): Binmore Zorn Ross Speight Howie Alcock Cummings Bartle and Sherbert Ash These are all at roughly the same level, with the exception of Ash (I did a video on his analysis book). It's a more serious book (intended to prepare the reader for measure theory, functional analysis, etc.), but it is very readable, has an intro to proofs, and has full solutions. It's also low cost! Hope this helps! Feel free to keep asking away or send me an email! ruclips.net/video/FEI-dFYQYes/видео.htmlsi=Y8c5tTkoDUmvECiP

@@MathematicalToolbox Thank you so much!

In general, I like the writing style of this book much more. Axler is fine as far as a regular textbook goes, but this one is better for me. It looks like it has much of the same content coverage. This subject has been pretty difficult, and quite honestly, I wouldn't want to use anything else.

Thank you so much for this feedback. I had started to use Shreve while on my own journey years ago and put it down in favor of Calin. To be quite honest, I had forgotten about it. This is my mistake. I'm glad to hear that you found your way to it and that it has been helpful to you. Thank you again for sharing your experiences. I should definitely revisit this book. I remember it having some of the best and most intuitive explanations, as you say. I apologize for not mentioning it to you in our previous conversation. By the way, if you do a Google search online, you might be able to find a solutions manual!

Oh yeah, for sure. You definitely still need to know something about the topic. It doesn't do well at teaching new concepts, but it's still a useful tool.

@@MathematicalToolbox yes, those are "expert systems", so they assist an expert, but you must know what you're doing. LLMs help tremendously in programming, they can generate fully functioning code from specifications given as human readable text.

I think there's a lot to unpack in your comment, and I think it deserves its own video, to be quite honest. I 100% agree with you. A lot of members of the math community insist that we do things the way that they do them and that if you can't or won't, then you're dumb or wasting your time. As you say, this is toxic. Furthermore, this mentality is not wanted or acceptable here. Many of us here in *this* community are learning math for fun and trying to learn math the same way a PhD. student would learn math will only lead to frustration. I realized fairly early on that I could not do math as you say on 'expert mode'. This is not because I'm some 200IQ mathematician, but because I had (historically) and a low tolerance for suffering, and that low tolerance would lead me to quit a lot of things. Yes, you can achieve a decent enough understanding without having to go on expert mode, despite what some would have you believe. I usually only pick out 5-10 exercises in any given problem set, and I *always* verify my answer. As much as I talk about books on this channel, I still don't think I've done a good enough job of explaining how finding the right book for you can be life changing. It might not take 2-3 years to get to where I am. I would say learning analysis (or topology) is the best thing anyone interested in higher mathematics could do. There are a lot of really good, readable books on both of those subjects. Thank you so much for sharing!

For a long time I have been struggling with a similar approach to studying mathematics. For me the underlying reason was this feeling that, otherwise, I won’t *remember* it well later and therefore would have trouble to deal and comprehend whatever is coming next. That I would have to read and study the subject again and use my time inefficiently. Suffice it to say that *this* attitude is inefficient. I am curious about your take on it.

In general, what I have experienced is that most books will test you on the content of the main part of the text, not the exercises themselves. So if you are dedicated to your subject (you might not even need to be, to be quite honest), you can study/memorize the definitions and the proofs of theorems/propositions. Sometimes, the information in previous chapters is very heavily used in subsequent chapters, but oftentimes, it is assumed that you have a working familiarity with it. The only way to know for sure is to *try*. This is an issue I used to have and something that I saw in my wife when I was trying to share some concepts with her: the reluctance to try certain things. "Oh well, I know it won't work, so I'm not gonna try," nope. Try it anyway. There's also this feeling of almost fear. Perhaps this is due to a fear of failure. But in mathematics, I have become very good friends with the concept of failure. If I'm failing, it means I'm trying, and trying is always a good thing. I would say that attempting to memorize the definitions and proofs of theorems/propositions has had the most impact on my feeling of having mastery or command of a subject. I've done this in my studies with measure theory. Now, I need to go back and do this with linear algebra as I always forget how to work with some of those ideas. 😂 Hope that helps!

I'm not 100% sure. This is a little advanced for me tbh. I've heard of people doing things like giving internet access to GPT, but I've never delved into any of this myself.

I don't know about the free version, but I know that ChatGPT plus allows you to use "Wolfram GPT" which will do all that kind of stuff for you.

It can but it's still not 100% trustworthy, because how plugins work is the model uses the plugin and then generates text from the whatever the tool produces So the eventual output is more reliable but not perfect. I still find chatgpt very useful for study. I treat it as a smart friend that I can discuss things with, ask questions etc, but make sure I check the answer carefully for myself.

Well 3 questions actually :)

As far as I understand it, perhaps the biggest limitation of the method is boundary conditions. The method is only appropriate to use with homogeneous boundary conditions. The series does not always exist. These are nice cherry-picked initial conditions to ensure somewhat easy calculations and a nice closed form solution. Sometimes, you'll get something that looks like it does not converge, but if you were to use software, you could get a nice closed form solution. Yes, the nonlinear DE can be as complicated as you want it to be, but this will change the Adomian polynomial that you'll be using. You can have two or more nonlinear terms. You can have a sin(u), e^u, etc. I strongly suggest you check out the Wazwaz book. It's just amazing. Really great questions here, I appreciate it! Edit: One more limitation is that the ADM will only ever find *one* solution to a DE. This is problematic because nonlinear DEs often have two or more solutions. Perhaps this is the biggest limitation.

Thank you for the book recomendation. Have a nice day and keep up the good work!

Thank you, you too! BTW, I just want to make sure you saw my edit on the biggest limitation of the method on my original reply!

I did :) 👍

It definitely is! I was looking through it a couple of weeks ago and went through the first chapter. Do you plan on buying any of the Phi learning books?

@@MathematicalToolbox Right now, I'm only looking at buying two math books in the next couple of months. Advanced Calculus a differential forms approach, and strang's linear algebra and machine learning book. Stran'gs book is pretty unique, and the Edwards book I saw 20 years ago in my local Borders, and i've always wanted to take a crack at it. I will almost definitely buy something from phi in the future.

It's quite hilarious that I've never noticed the lack of series solutions in C&H. Thanks for pointing that out. I wasn't required to use them when I took ODEs so I never cared for them. Nowadays, I use the Adomian Decomposition Method anyways. Regardless, have you tried Edwards and Penney? I like it much more than B&D, although it is certainly not my first option for a book on ODEs. Hope that helps! Thank you!

@@MathematicalToolbox I haven't. I'll take a look at it, thanks.

For sure!

I would try for a different book like Edwards and Penney or Campbell and Haberman. T&P is great too but these two are modern. Good luck and feel free to reach out if you need more help!