Advice for Learning Math (university)

[dbizzle]

dont post TLDR or any of that shit. If you don't want to or can't help me, just leave the thread plz.

Alright WCR, since I know there are many math efficionados here I figured where better to ask for some advice on learning math. This post is a doosy since it's important to me that I get good advice and if I just ask too generally it wont be helpful; I have to express where I'm at.

I have plenty of textbooks, but none of them are interesting enough. I actually do like math, but these things are dry, boring, and don't flow well. When I talk to people who are really into math, they often mention things like how a particular book they read when learning was really elegantly presented.

I'm taking a course right now where the prof assigns readings from major science research publications that use math extensively (it is a math course that is focused on ecology) and I find them interesting, but getting through them is often a struggle, and I find myself skipping big chunks because I know the discussion of results is somewhere in there and the math just looks too daunting for me. A lot of these articles would be written by people with decades of experience both in math and in scientific writing, so it's not unexpected that portions of their work are over my head. Regardless of that, I just don't easily switch to thinking in math and a lot of the time I'm just sort of accepting what I'm reading rather than agreeing with the math, which will prove to be a problem if I want to take my academic career in math seriously. They employ concepts I know from differential equations, linear algebra, and calculus, but weild the tools from those disciplines with such fluency that it's hard to keep up. All but 4 other students have dropped the class, which I see as a tremendous opportunity to get a foot in the door for research within the department. To do that, I would have to really step it up and show my prof that I'd be a worthy applicant. So far he has no reason to question the quality of my character, but he also has no reason to take my academic progress seriously because I'm not a standout student.

I can see that I'm gradually getting better at concepts that I've already learned just from exposure to them, but since it's not particularly interesting I can't study enough to master it. I think at this point I'm looking for math resources that are engaging enough for me to really motivate my studies. Interesting books at an appropriate level (or with tons of relevance) could be the key. I have the time to study hard if the books (or online resources?) can keep me engaged.

I need to improve on:

calculus: i took first year calc and was fine, but despite not having any other calculus prerequisites this course uses a lot of partial derivatives, more complicated implicit differentiation, and proofs that are hard to see the consequences of by their assumptions etc.... I'm also not particularly used to having to get my hands dirty as I've been tested mostly on concepts with simple, easy to solve examples. As my math focus becomes more applied that approach to learning is failing me.

linear algebra: im taking this course now but I really find it extremely dry and boring and am struggling. there are lots of really lame proofs on how matrix algebra works, vector space, and so on. I'm way behind but I'm thinking I can't catch up using course materials because they are just SO damn poorly written. this is an area of concern for me; I don't want to have to hire an expensive tutor to bring me up to speed which is what I'll have to do if I don't find a good study method by mid-april. The ecology course I mentioned uses linear algebra concepts (so far lots of stuff on Jacobian Matrices, and how to interpret them in finding stability and equilibria of systems of differential equations)

In general I want to be more comfortable at reading and writing proofs also. I've really approached this area incorrectly because like most undergrads, I don't like dealing with proofs and haven't put in the effort to improve. Proofs are where the elegance of mathematics can be found, but I haven't found it. Hopefully someone can point me in the right direction. Also, I'd love something thats Kumon style that would make me more proficient. I think I learn really well from drilling concepts. A resource for higher level math drills would be nice. I guess people are expected to be more independant and form their own study systems, but its worth a shot. I'd really appreciate hearing about what some of the best math books people can recommend are, and how I can master what I need to know. I'm actually pretty quick at picking up math concepts if they are presented to me properly.

[GregGore]

It's sad that you don't have much time but if it's possible you should really learn math proper from the bottom up:

Start at the theory of sets and basic logic which leads to the natural numbers (and proofs on them with induction), then the rational numbers. I'm confident that good lecture that matches your description also starts with these topics.

Basic Linear Algebra is also a topic that should be pretty sweet with a good book. You should at least understand when equations are solvable or unique solvable, how to solve them by the gauß jordan algorithm and what that stuff has to do with matrices and determinants. vectors and vectorspaces are a really easy topic too once you got the picture in your brain.

Then it comes to analysis: The approach to the real and complex numbers, integration and differentiation by series, progressions and power series is pretty tough if it should be formal correct but a good lecture with good illustrations could help a lot. Take the time to really understand the characteristics of the sin-, cos-, sinh-, cosh- and e-funtion and the inverse ones. There are surprisingly many people on universities that don't even know that differentiation gives the slope of the function graph, avoid such gaps in education. Don't resile from grapping paper, pencil and a calculator and investigate some things on your own.

Once you have tightened and completely understood all those basics concepts you'll be more confident at learning and executing higher stuff.

Unfortunately I don't know any (english) book that could help you but I'm sure Louis.Wain, Amazon etc do.

Btw: I did the same self studies on my own after my military service before the university started : )

[shilock]

The course you are taking sounds challenging. It takes a certain level of 'mathematical maturity' and practice to read papers comfortably. In my experience reading math 'well' requires you to be relentless: read and re-read until you understand. Understanding means that you have not only convinced yourself that the argument follows from things you already knew were true, but that you have an intuitive feeling for what is going on. Having a reservoir of commonly used arguments helps greatly with this, as does having a solid foundation in the subjects your have mentioned. Of course these things can can only be obtained by reading. I hope I have avoided being circular.

Next, my 'great' text book recommendations.

1) If you are struggling with calculus I recommend reading several chapters from Spivak's book: the chapter on limits, the chapter on least upper bounds, and the chapter on integration, which includes the fundamental theorem of calculus.

These are aspects of calculus that are commonly misunderstood in my opinion. Spivak is clear, rigorous, and entertaining to read. Also his excercises are very good for building intuition and learning to write correct mathematics.


2) For vector calculus, I recommend "div, curl, grad and all that". It is a book by a physicist that explains all of the fundamental concepts in vector calculus clearly and using physical intuition. The author skimps on technical details in some of the proofs, but when he does doing so is harmless because they are just stumbling blocks.

3) For linear algebra, I recommend Freidberg, Insel, Spence. Forget all of the tedious unmotivated proofs in matrix algebra. This book rigorously defines vector spaces, proves all of the basic facts about them, and then introduces linear transformations as functions. Matrix algebra is a consequence of properties of linear transformations, which matrices represent. The book deals with linear equations, diagonalization, inner product spaces, and canonical forms in a clear and engaging way.

4) Finally, for introductory real analysis, I recommend "Real Mathematical Analysis" by Charles Pugh. The book has almost no prerequisites, and is full of interesting examples, and excercises. Probably the best real analysis book I have come across.

5) It also sounds like you need to learn about ODE's. The only book I like is by Arnold, but it is very geometric and not very friendly for a first pass. Pugh contains the basics.

In my opinion, all of these books are "10/10". Other people will recommend different books over these. I like books with lots of pictures and examples, and am wary of books that don't do things this way.

[dbizzle]

Thanks both of you for your great tips. It's not the first time I've been recommended Spivak, but I had forgotten. I'm definitely checking some of those resources shilock. Very helpful.