Help for Student Seeking Math Books

[hitchhikerr]

Hello,

As a background I am a community college student about to transfer to a university trying to better myself and catch up with the competition. I have taken multivariable calc, a basic linear algebra class, and an ordinary differential equations class, and would like to further myself at my own pace for mainly personal interest in mathematics, but also to prepare myself and turn over a new leaf in my approach to academics.

My main interest in posting this question was to find some books with which i could personally advance myself at my own pace on the subjects proceeding the ones i have already covered. Thanks to a great cc professor i had in an honors class, I've had rough introductions to tensors, topology and the dual space (and by rough introductions i mean the absolute basics). Tensors are a particular interest of mine as i am possibly looking to go into physics or engineering degree.

Any advice would be much appreciated!
Also excuse any grammar or syntax issues, as I've always hated english class haha.

 

[dydxforsn]

Well, you are well on pace if you're in between basics and your upper level undergraduate classes. I mean, it's hard to recommend books with such a vague pretense, but I would simply recommend buying the standard undergraduate texts in the various subjects and just start reading! Try to find out what your school is using, but there are some very standard texts in the various subjects taught to undergraduates (Thermal Physics by Kittel for Thermal/Statistical Mechanics, Classical Dynamics of Particles and Systems by Thorton, Introduction to Electrodynamics by Griffiths, etc.)

I would for sure recommend The Feynman Lectures on Physics for you (they're at about your level and they're very satisfying takes on the most of the material you will experience in a physics undergraduate setting.)But on a personal level I would say that you should look into a few specific topics that I found to be among the hardest in my own experience at the undergraduate level. The three things are as follows (all of which I could have just learned when I was at the point you are and gotten some hardship out of the way):

1.) Tensors - You seem to already have a very good start here, seeing as how you are already specifically interested in these complicated mathematical objects. I personally found these very hard to deal with, and I would say that a good book that you would much enjoy on this subject would be A Student's Guide to Vectors and Tensors by Fleisch.

2.) Fourier Series and Fourier Transforms - You usually run into your first formal treatments of these objects in either an advanced Cal II class (you sometimes see the Fourier Series taught immediately after Taylor series in the sequences and series section of your calculus book), or your first Partial Differential Equations class (seeings as how all of your solutions are infinite series representations of some function solution.) Fourier Series is just a way to expend period functions in terms of an infinite series of sines and cosines in much the same way you found that you could expand functions in terms of power series. This topic is going to show up in ALL of your undergraduate junior and senior level courses from mechanics to optics to quantum mechanics. So get a head start on it :) As far as books, I first learned about this topic in Applied Partial Differential Equations by Haberman. You really need to be comfortable with this "frequency representation" of functions. There's also a Student's Guide to Fourier Transforms, but having not read it before I'm not sure how good it is. My gut feeling is that it's decent.

3.) The Calculus of Variations - First shows up (unless you're sufficiently advanced in mathematics, which it sounds like you may have already come across it) in mechanics as a new way of doing mechanics without Newton's laws (but it's completely equivalent). Best way I can explain this "Calculus of Variations" is it's basically like the simple idea in calculus of minimizing and maximizing functions, only you're doing it to "functionals". So basically you are minimizing and maximizing the shapes of functions themselves instead of just finding the minimum or maximum value of some set function like you do in calculus. I really don't know of any good texts in this other than what you will find in your junior level mechanics textbook. I know Feynman talks about it in the first volume of his lecture series.

Also, I can't stress how important the Calculus of Variations is.. Don't ignore this topic, it increasingly becomes ever present in everything you do in physics pretty much from that first time you probably see it in mechanics till you die..It's really hard to recommend non textbooks (I'm assuming you don't want to spend and arm and a leg) to physics students like ourselves, because other books have just such an aptitude to be terrible and uninformative :/ Wish I could have recommended more books for you. I do think you should just find out what textbooks your school is using and just read them on your own. It's really nice if you've read your book before you've even gone in the classroom, this helps immensely.

 

[hitchhikerr]

Thank you so much for replying! I must admit that although I have come across functionals it was only as an introduction to the purely mathematical concept of the dual space being the set of all linear functionals etc so it seems as though the calculus of variations seems as something i should definitely delve into. Also that vectors and tensors book seems to be exactly the kind of text i was looking for (from what appears on amazon that is haha). However with the applied pde book, would it be considered over my head without a background in analysis or is it more of a rigorous continuation into topics not covered by an ODE class?

 

[dydxforsn]

Heh, the Student's Guide to Vectors and Tensors book was actually pretty good, I was recommended to read it by one of my professors whom had come across it (I was like the tensors riddler in class), it's actually pretty legit. It also has a great treatment of contravarient and covarient vectors which you will also run into with formal treatments of special relativity just to name one case. Though honestly the first half of the book is just stuff you've seen in vector calculus (multi variable calculus, calculus 3, etc. whatever you want to call the typical course.)

But to answer your question, it's a really good introduction to PDE (the Haberman book that is). It's not excessively bogged down in analysis topics, it was my PDE book when I was a junior and it was prereq for our senior level electromagnetism course, but it should probably have been a prereq for most undergraduate courses. Why? Not really because you need to be very good at PDE, rather it is a first solid treatment where you run into Fourier series and the Fourier transform.
The book really only requires that you have taken a standard Ordinary Differential Equations course, because the main technique for solving PDEs is just to decompose them into say two to three ordinary differential equations. I would recommend that you are at the level now where it is appropriate, you will get out of it what the author is intending.

 

[hitchhikerr]

Awesome, thanks again for the concise and knowledgeable answers!