Which mathematical topics should I attempt to self-study over the Summer?

[ADCooper]

Just as a little background, by the time summer comes around, I will have taken mathematics classes up to basic differential equations, multivariable calculus, linear algebra, and some probability theory. In Physics, I will have taken up to classical mechanics and mathematical physics. I want to attempt to learn some topics to give myself an advantage so I won't just be lost in the fall when I have to overload with a bunch of upper level physics and math classes (including Advanced Calculus).

I am already planning to attempt to learn programming in C and also mathematical proofs in some way since I've never had to attempt them before (I have a few books I can use, just not sure if I want to just go for really basic math concept proofs or go straight for the intro to real analysis books).

My question is: are there any other topics I should look into over the summer? I'm a math and physics major, and I plan to eventually go to graduate school for physics. I was thinking maybe partial differential equations would be a good thing to add, but wasn't sure (if anyone could recommend a book for that, it'd be great).

 

[fourier jr]

even if you're planning to do physics in the end this will definitely help with the math courses you've got coming up, especially if you're not used to doing proofs:

 

[ADCooper]

Alrighty, I appreciate it! I'll definitely look into that one

 

[deluks917]

Arnold's ODE has a lot of beautiful physics in it. Maybe some complex analysis. A book I really enjoyed was Non-linear Oscilations, Dynamical systems and bifurcations of Vector Fields.

 

[ADCooper]

Arnold's ODE has a lot of beautiful physics in it. Maybe some complex analysis. A book I really enjoyed was Non-linear Oscilations, Dynamical systems and bifurcations of Vector Fields.

I'll definitely look into those. Any specific book for complex analysis? I'm assuming I should wait until I do a little bit of proving before I attempt complex analysis, or do you think its manageable without it?

Thanks a lot, I really appreciate recommendations

 

[fourier jr]

Alrighty, I appreciate it! I'll definitely look into that one

btw I don't know if you looked that one up but that book is ideal for self-study since most of the proofs are left undone, especially when you get farther in. In the early chapters he proves things a lot more thoroughly than the later ones, but as you work through it you can see a definite pattern.

re: complex analysis you probably have an actual course on that coming up, which probably comes after your first real analysis course.

 

[Oriako]

If you want to knock off two in one (and in fact learn something you may not get a chance to take a course in), I would highly recommend a book written by a professor at the University of Calgary, who I actually have the privilege of taking Algebra I and II with, called "An Introduction to Abstract Algebra". http://www.amazon.com/dp/0471331090/?tag=pfamazon01-20

It has a very good introductory section that introduces notation and any pre-requisite knowledge on Sets, Proof Techniques, Mappings, Relations, etc. before actually jumping into the content and it is highly recommended for self study. I'm sure you can find a cheaper version somewhere... that is an extremely overpriced copy listed on amazon.

You can learn proofs and also new content in an all in one package :P

 

[micromass]

even if you're planning to do physics in the end this will definitely help with the math courses you've got coming up, especially if you're not used to doing proofs:

Absolutely avoid the book by Suppes! Suppes does some things very unconventially. In fact, I have seen people who read Suppes and who were very confused later on. Suppes doesn't make any mistakes, but he gives definitions which are quite different from other math texts.

If you want a text on set theory then go for either "Naive set theory" by Halmos, or "introduction to set theory" by Hrbacek and Jech.

 

[Goldbeetle]

Read this: "A Book of Abstract Algebra: Second Edition" by
Charles Pinter

It is an introduction to abstract algebra and mathematical thinking. From a didactic viewpoint it is unbeatable!

Probability and statistics is also something very interesting, that you could learn.

 

[fourier jr]

Absolutely avoid the book by Suppes! Suppes does some things very unconventially. In fact, I have seen people who read Suppes and who were very confused later on. Suppes doesn't make any mistakes, but he gives definitions which are quite different from other math texts.

What exactly do you mean? That's the first bad thing I've read about that book.

 

[AuraCrystal]

You could use Jiri Lebl's book for real analysis (it's free! =D). (Bartle and Sherbert is also a good one.) As for basic proofs book, you could try Lay's Real Analysis w/ an introduction to proof but I don't know of any good basic proof books. Perhaps you could look over your high-school geometry book. Personally, I was fine going into Lebl/Bartle and Sherbert without reading over any proofs books. (Though we spent like 2 weeks covering logic.)

I also like Strauss PDE book. (From what I've read of it!) Also, get Schaum's Outline's PDE book. It's like $10 and it gives you lots of extra problems which is valuable--especially if you're using a book which is skimpy on problems like Strauss. And like others have said, complex analysis would be good too. You might also benefit from studying vector analysis , depending on how thorough your Multivariable Calc. class was on those topics.

As for physics, Griffiths E/M book is a good intermediate book and I like his Quantum book as well. When you say you've had classical mechanics, do you mean at the intermediate or introductory level? (The classic book for the intermediate level is Thornton & Marion but personally I prefer Taylor.) And if you're interested, Hartle is a good book on General Relativity. (You need SR and Lagrangian dynamics though.)

 

[ADCooper]

Thanks to everyone for the recommendations! Tons of stuff to look into now. One more question! Anyone have any books specifically good for improving with geometry? I mean, I have a lot of the basics down, but I feel like its probably my weakest subject math-wise.
(Preferably not Euclid's Elements)

As for physics, Griffiths E/M book is a good intermediate book and I like his Quantum book as well. When you say you've had classical mechanics, do you mean at the intermediate or introductory level? (The classic book for the intermediate level is Thornton & Marion but personally I prefer Taylor.) And if you're interested, Hartle is a good book on General Relativity. (You need SR and Lagrangian dynamics though.)

Thanks for covering basically everything! For classical mechanics, I think just the basics. I've had physics I, and I'm taking Classical Mechanics next semester (which is why I'm aiming this self study for the fall), but we'll probably (judging by syllabus I've seen) be using:

Mechanics, Third Edition (L.D. Landau and E.M. Lifgarbagez)
Classical Mechanics, Third Edition (Goldestein & Poole)

I'll also be taking a class in analytical mechanics the semester after that, so is that what you consider intermediate mechanics?

 

[Lavabug]

For analytical/classical mechanics, Landau/Goldstein are the standards in my course as well but they're terrible texts to learn from. Use Morin's "Intro to Classical Mechanics" to for elementary Lagrangian mechanics and rigid body motion, and Calkin's "Lagrangian & Hamiltonian Mechanics" for anything theory-related. They have way more solved examples and far clearer explanations than Goldstein or Landau's texts.

Out of all these the only ones I would keep would be Calkin due to the brevity and tons of theory/proofy problems which is very unusual (though it doesn't cover small oscillations or rigid body motion). Morin's is a nice reference book but its overkill for any single course.

 

[Deveno]

my recommendation is a bit off the beaten path:

https://www.amazon.com/dp/0486679667/?tag=pfamazon01-20

it's well-written, and desgined for the undergraduate student who hasn't taken advanced analysis or abstract algebra yet. it's mostly concerned with low-dimensional examples, the kind you can draw a picture of (and the pictures are funny!).

a good intutive grasp of topology is good for anyone learning higher-level math and physics, and this introduction is gentle. it's something you can self-study, the exposition is remarkably clear. it has a proof of the Jordan Curve Theorem (a difficult proof of this occupied a good portion of my complex analysis course), and the sections on Vector Fields and qualitative analysis of differential equations alone are worth the price of the book.

this is not one of the "benchmark" texts in a given field, it's NOT encyclopedic, and doesn't go into a lot of depth. what it DOES do, is prepare you to be unafraid of a lot of difficult subjects, by giving you a taste of what can be done. do not under-estimate the power of the Euclidean plane, it has much to teach (and if you really "get" how to think through 2 dimensions, more than 2 is just "the same, but more of them").

and it's fun to read. i wish more math books were written with the subtle humor you can find throughout this text. how many textbooks can you name that have an index entry for "man in the moon"?

 

[Nano-Passion]

my recommendation is a bit off the beaten path:

https://www.amazon.com/dp/0486679667/?tag=pfamazon01-20

it's well-written, and desgined for the undergraduate student who hasn't taken advanced analysis or abstract algebra yet. it's mostly concerned with low-dimensional examples, the kind you can draw a picture of (and the pictures are funny!).

a good intutive grasp of topology is good for anyone learning higher-level math and physics, and this introduction is gentle. it's something you can self-study, the exposition is remarkably clear. it has a proof of the Jordan Curve Theorem (a difficult proof of this occupied a good portion of my complex analysis course), and the sections on Vector Fields and qualitative analysis of differential equations alone are worth the price of the book.

this is not one of the "benchmark" texts in a given field, it's NOT encyclopedic, and doesn't go into a lot of depth. what it DOES do, is prepare you to be unafraid of a lot of difficult subjects, by giving you a taste of what can be done. do not under-estimate the power of the Euclidean plane, it has much to teach (and if you really "get" how to think through 2 dimensions, more than 2 is just "the same, but more of them").

and it's fun to read. i wish more math books were written with the subtle humor you can find throughout this text. how many textbooks can you name that have an index entry for "man in the moon"?

Thanks for sharing this.

I haven't read this book but from what you have written and the reviews I have read it seems to be a great introductory book! Thanks for sharing! I'll take a look at it after DFQ ( I sincerely hope I remember hehe).

 

[AuraCrystal]

ADCooper: What books are you using for analytical mechanics? And those books are like...graduate level. (They're the two classic ones.)