this is probably getting repetitive

but shit, I need to focus..

Idk what the hell my problem is

I’ve been having a lot of trouble with math recently. I’ve found it really frustrating to keep trying to get through Spivak/Munkres for analysis. I feel like I’m stuck. Maybe I need to reformulate my attack plans.

I tried to narrow down the books I will focus on for the rest of the semester. I have Stein and Shakarchi for complex, Dummitt and Foote for Algebra, do Carmo for DG, and Munkres/Spivak for analysis. Although I feel like narrowing to this scope has probably helped me focus a bit, I still feel like I’m kind of helplessly drifting right now. I’m going to try to get a good night’s sleep tonight, and hopefully tomorrow will be more fruitful. It’s entirely possible that I’m just getting worn out. I find myself trying to read, but I end up just getting bogged down in proofs and unable to make any significant progress toward understanding the material.

I’m even getting kind of frustrated with algebra. It’s supposed to be the place I can go to for solace from all of the analysis that I have coming up, but it seems like even looking at that frustrates me. I feel like everything I read I either know or is over my head. I just need to focus and systematically work through D&F, making sure I get everything, even if that isn’t what I find most interesting.

On a more positive note, I think that I at least finally gained some insight into what differential forms actually are, although I’m not sure if it’s an entirely mathematically sound insight. Either way, it helped me at least a bit in my continuing studies, because the notion of differential forms had been a large barrier up to this point for me.

whatever. hopefully this week will end up better than it started.

1.5 Weeks Done

So  yeah, as the title indicates, I’m 1.5 weeks into my summer adventures in math.

Got through Folland. Skimmed most of the last part of it, but whatever. Integration is a lot more intuitive in multiple dimensions than differentiation, imo. I really really don’t like differentiation.  I was also trying to just think about the ideas of the big theorems – Stokes’, Green’s, Divergence – and I’m still not sure I entirely get the idea. I guess the gist is that you get the same result whether you integrate a function over the boundary of a region or integrate the derivative (in some sense) of a function over the region. This is a pretty cool idea, but I’m still not sure I entirely understand. I’m not even sure if what I said is correct. Hopefully I’ll gain more of an understanding as the weeks go on.

I’m going out of town next week, but I’m going to try to keep up with my studying, although not as intensely.

Now I’m going to focus on Spivak’s Calculus on Manifolds and Munkres’ Analysis on Manifolds. I’m probably going to alternate between them, reading the different treatments of the subjects as I go. I hope this will finally give me an understanding of what’s going on. Although I’ve been putting a lot of time into this project, I still feel like I’m not entirely grasping the subjects. Maybe it’s because I don’t really know how to learn mathematics on my own. In the classes I took this year, we weren’t tested on our understanding of the proofs of the main theorems themselves – we were tested on our ability to apply the theorems to new situations. While this is clearly an important skill to learn, I guess it caused me to focus only on that aspect of mathematics while neglecting the more fundamental part of actually learning results and why they work. I know the results of Rudin and H&K, but I don’t know that I’d be able to restate any of the proofs, and that kind of bothers me. In the coming weeks I’m going to try to focus on actually understanding and internalizing the proofs in my readings, especially in Spivak and Munkres. I think this is what I need to work on the most.

Also this week, I got the chance to look at books for the three classes I’m signed up for next semester in at least some detail. I’m excited for Algebra. I looked through some of Dummit and Foote this week. I’m excited to get back into abstract algebra. I think that was probably my most stimulating class in high school, and it will be fun to return to that.

I also am really excited for Complex Analysis. I looked through the first 2 chapters of Stein/Shakarchi, and it seems like a beautiful subject. It seems like (at least in the portion that I looked at) the epsilon-delta ugliness of the analysis I’ve seen so far is kept to a minimum. Obviously it’s present, but it doesn’t seem as glaringly omnipresent as it does in what I’ve seen before. Also, the Cauchy integral formula (I think that’s what it was called) is a beautiful result.

Finally, I looked at do Carmo’s Differential Geometry of Curves and Surfaces for a bit on Friday. I’m also excited for this. Idk…I didn’t get quite far enough to get a good taste of the subject, but what I got through seemed pretty interesting.

Two things I’ve not been doing enough of: Arabic and Problem Solving. I need to focus more on both in the coming weeks. There’s just too much to do and not enough time. However, I do still have several months to learn, so I’m sure I’ll get to them eventually.

That’s it for now.

First Home Update: Been Had Math Bookz

Exciting parts of being home: YZ, Math Books, Nessie

I ordered an absurd number of math books from Amazon to celebrate being done with my first year. I probably ordered far more than I actually need, but whatever.

List:

Lang – Complex Analysis

Schey – Div, Grad, Curl, and All That

Tenenbaum/Pollard – Ordinary Differential Equations

Gamelin/Greene – Introduction to Topology

Edwards – Advanced Calculus of Several Variables

Kreyszig – Differential Geometry

Cartan – Elementary Theory of Analytic Functions of One or Several Complex Variables

Stein/Shakarchi – Fourier, Complex, wa Real Analysis

Spivak – Calculus on Manifolds

Munkres – Topology

Pressley – Elementary Differential Geometry

*BONUS* – Brustad/Battal/Al-Tonsi – Al-Kitaab fii Ta’allum al-‘Arabiyya II

Hungerford – Algebra

Kuhnel – Differential Geometry: Curves-Surfaces-Manifolds

do Carmo – Differential Geometry of Curves and Surfaces

Dummit/Foote – Abstract Algebra

Lang – Algebra

Munkres – Analysis on Manifolds

Rudin (RIP) – Real and Complex Analysis

This is legit probably way too many books. I guess part of me feels like by buying more books I’ll be more inclined to work through them. I don’t know…we’ll see how that goes.

Today was also my first official day of studying. It was hard to get back into the analysis mindset. I think I might go back through Rudin to kind of remind  myself how some of this stuff works. I just feel like it’s a different way of thinking than algebra. To take a break today, I worked through the first few sections of Dummit and Foote. It was mainly review, but I like thinking about algebra more than I do analysis. Maybe that will change, since next year will be fairly analysis/topology heavy. I’m thinking right now Algebra/Complex/Differential Geometry first semester, and Topology/Real/214, with DG/214 being optional depending on how I feel I’m doing. I want to take 214 so I can maybe take a class on Algebraic Number Theory Junior Fall. I don’t really know how Junior seminars work, but I’d like to take Algebraic Geometry Junior fall too, assuming I do well enough in algebra. idk we’ll see how it goes.

I’m having trouble focusing on proofs as well. Maybe it’s just because it’s the beginning of summer. I’ll see how it goes. I basically read the first chapter in a bunch of books on analysis, trying to remember how compact/connected/open/closed and all that stuff works. Maybe tomorrow will be more fruitful. I think I’m just going to focus on analysis again tomorrow, trying to get a grasp on this stuff again. I’ll start on Arabic again next week.

That’s it for now.

First Summer Update

Studying has been going alright.

I basically read through all of Stewart’s Calculus, skimming the first 1000 or so pages as a review, then really trying to read and understand the vector calculus/multivariable calculus stuff. I’m still a bit hazy on all of the Green’s/Stokes’/Divergence theorems, but hopefully more clarity will come later. Apart from those, I think I at least have the gist of most of what’s going on. I probably couldn’t wholly explain it to someone, but at least it’s partially sorted out in my mind.

Right now, I’m going through Marsden and Tromba doing more exercises to beef up my understanding. Hopefully understanding of those last few parts will come through this book. Also, it ends with a chapter on differential forms, which I hope will be an ok introduction for when I start into the real analysis of this stuff.

I’m also reviewing all of Rudin, probably doing some extra exercises to kind of get my bearings with analysis again before I start in on the legit multi.

I’m also excited to have little forays into other branches of mathematics when I get home. I ordered 20 books on various topics, and they’re all getting to my house around this time. I’ll probably post a picture once  I get home of my massive stack of books. I think I’m most excited to start on algebra when I get home with Dummit and Foote. I feel like it’s the most beautiful subject I’ve encountered so far, and I’m excited to get a deeper understanding of it and also to hopefully get a grasp on Galois Theory. The more I think about algebra the more I like it. Looking through Rudin again, especially chapter 7, I remembered how ugly some analysis can be. Ugly integrals with crazy exponents aren’t pretty to me. I guess  I can appreciate the beauty that others find in them, but at this point they just aren’t as interesting/beautiful as algebra seems.

On a final note, I’ve also been working through Zeitz’s The Art and Craft of Problem Solving when i don’t feel like tackling hardcore new stuff. It’s been a good place to kind of take a break. One problem I found pretty interesting (although easy) was this:

Let S be a subset of the reals that is closed under multiplication, with T and U disjoint subsets of S whose union is S such that the product of any 3 elements of T is in T and the product of any 3 elements in U is in U. Prove that at least one of T, U must be closed under multiplication.

This is A1 from a Putnam in the 90s (I think). It took me about 5 minutes to solve – made me wish I had taken the Putnam that year. However, I felt like it was still kind of cool. What appears to be a (decently) hard problem at first glance – especially with the reputation Putnam problems have of being hard – is actually about a 2 line solution. I don’t know maybe since it’s A1 I shouldn’t assume it’ll be hard, but still, I thought it was cool.

I’m also going to make more of a conscious effort to keep up with Arabic when I get home. I haven’t really been doing anything this week with it. I’m afraid I’ll forget everything. Hopefully not.

Anyway, that’s it for now. Back to studying.

Let us Meet Again as Learned Men

It’s been a good year.

This Summer Pt. 2

I’m going to try and set up some concrete goals for this summer, which will probably be modified as I get a better idea of how my work ethic holds up and how fast I end up going through these books.

Essential:

1. Learn Multivariable Calculus via Stewart and Marsden. I think this should be doable in the week after finals and perhaps the following week when I return home. I don’t want to spend too much time on it, considering it should just be a straightforward extension of single-variable calculus (for the most part).

2. Learn Multivariable Analysis. I have several books on the subject, and I will probably use a mix of them. I will be using a combination of Spivak’s Calculus on Manifolds, Munkres’ Analysis on Manifolds, and the 218 course notes. I plan to do all of the 218 problem sets, and to do a final or two from the math website to make sure I really know the material. I’d like to finish this up maybe 2/3 of the way through the summer at least, so I can get a head start in the classes I’m taking in the Fall.

3. Keep up with Arabic. My goal is to do a chapter of Al-Kitaab every 1-1.5 weeks, so I’ll (a) Remeber what’s going on and (b) Be more prepared for the coming semester. I feel like everyone in there spends absurd amounts of time on the course, so any boost I can get will be worthwhile.

Not-So-Essential, but I’d Still Like to Do:

1. Get a head start on the courses I’m taking next semester. (a) Work through either Dummitt & Foote or Lang (perhaps both) to relearn some of the main concepts of Algebra. (b) Try to work my way a little bit through Stein’s Complex Analysis, to at least get a feel for some of the main ideas before I encounter them in class. (c) Start learning about Differential Geometry via do Carmo and the other books I purchased. I feel like if I can even get a feel for these subjects, the coming semester will be a lot more doable. I just need to focus and help myself learn.

2. Work on Problem Solving: I want to work through The Art and Craft of Problem Solving and Putnam and Beyond. I’ve never been that great of a contest-type problem solver, and while this is obviously not the most important kind of math, I still feel like there are some non-trivial methods and outlooks to be gained from practicing this kind of math. I want to get better at it. Also, I’d like to read through Polya’s How to Solve It.

3. Reread Petersburg

If I focus, I know I can do all of this. It’s just a matter of staying on my grind. Towards the end of this semester, I feel like I finally learned how to work. I hope that I can carry this newfound ability through the summer and into next year when it really counts.

This Summer

I’m using this summer to do math and Arabic. I think I’ll use this  blog to set goals and then try to meet them.

Also, I hope Rohan will add  me as a  friend or whatever you do on here.