Want to improve math skills, need some suggestions

[Polymath89]

Hey guys this is my first post, glad to have found this awesome forum.

As the title suggests I want to improve my math skills and I'd highly appreciate if some of you could recommend me some books and in general tell me which areas I should focus on.
I feel that my current math skills are rather superficial( I'm an economics student...^^), although I've covered calculus and linear algebra in my studies, I feel that I don't have a deep understanding of those areas at all, basically what we did was just take derivatives, do integrals or form matrix inverses, but never really solve problems.
I'd like to improve my math skills for several reasons, first of all I find math really interesting and important, I highly envy people who have really mastered math and can apply it to a lot of areas in life. Second, I'm interested in physics and want to study it in my spare time. I'm really interested in astronomy and would like to be able to read rather advanced material later on. I know that this will take me a lot of years to get there, since I can't spend as much time on learning physics as I'd like to, but hopefully I will one day get there. And last, but probably the most important reason, is that I will need really good quantitative skills for the master's degree that I plan to take. I have a little more than a year to get my math skills to a respectable level and I'm willing to invest 1-1.5 hrs per day.
So how do I best approach this? I think my current math level is comparable to that of a good math student coming out of high school. I figured that I'd start with a good calculus book that you can use as a self-study book and a calculus based physics textbook. Do any of you have some book recommendations? And in which order should I tackle linear algebra, differential equations and stochastic calculus( contrary to my math skills, I feel that I have a solid foundation in probability theory and statistics)?


Best regards

 

[micromass]

Welcome to the forums Polymath89!

I'm glad you want to learn math on a deeper level. Let's see if I can recommend some books for that.

First of all, modern math is being done with proofs, so the first thing you will need to do is get a good grasp on proofs. A very popular book is

https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

In this book, you will learn logic, set theory and the basic structure of proofs. This is the language in which mathematics is written. An alternative book, which is entirely free and at least as good as Velleman, is

http://www.people.vcu.edu/~rhammack/BookOfProof/

After this, you're ready for calculus. The best calculus text out there is without a doubt

https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20

You absolutely must read this. However, the book is pretty hard and the exercises are quite difficult. So perhaps you might like an easier calculus book first, like

https://www.amazon.com/dp/0387962018/?tag=pfamazon01-20

As for linear algebra, I like these two excellent books:

https://www.amazon.com/dp/3540780602/?tag=pfamazon01-20

https://www.amazon.com/dp/1441930817/?tag=pfamazon01-20

As you can see, the second book is a follow-up on the first book, which is significantly harder.

As for physics, I think the usual suspects are good enough. This are books like

https://www.amazon.com/dp/0471401943/?tag=pfamazon01-20

The order in which to read these books doesn't matter. If you want to do physics, then you can start immediately with Halliday & Resnick. If you want to do math, then read the proof book first. Afterwards, you can go to calculus and linear algebra (doesn't really matter which one you do first).

 

[Obis]

I'd suggest this book for algebra :

http://books.google.lt/books/about/Algebra_and_Geometry.html?id=tVTd78vYzPkC&redir_esc=y

I believe this will not only teach you algebra very well, but will also help seeing mathematics as a whole.

 

[Polymath89]

Thanks a lot you two for your replies. Micromass, I really appreciate your effort. I think I will start with the proof book first and then start with calculus and the physics book.

I have one more question though. Although I want to improve my math skills, I'd like to do this from a more applied math standpoint. How important is it, for example for a physicist, to have a very broad understanding of pure, rigorous mathematics( e.g number theory)?

 

[nonequilibrium]

Well that dichotomy doesn't quite capture everything. On one hand you have applied/applicable/practical math, like calculus, statistics, fluid dynamics, probability theory which are of course imporant for such fields as physics, and on the other extreme side of the spectrum you have things like mathematical logic and number theory. These latter things are absolutely irrelevant to a physicist (to be clear "logic" is more than just the concept of proving, the mathematical branch of "logic" is really quite abstract and advanced). But the thing is, for physics you don't have everything you need with just the former side of the spectrum. You also need stuff from the middle, from such fields as analysis, algebra, topology, geometry. For example quantum mechanics works with infinite dimensial vector spaces with certain properties that make up a Hilbert space, which is a large piece of (functional) analysis, and that's just the basics, if you want to get into advanced theoretical QM you'll need more advanced notions of functional analysis (regarding properties of so-called Banach spaces of operators on the aforementioned Hilbert spaces). For general relativity you need a lot of differential geometry, which is advanced geometry. For QM and pretty much any of the more recent fields like particle physics you also need (abstract) algebra: fundamental theories about nature got really, really abstract since the 20th century. Notions of topology are dispersed all throughout.

That being said you probably shouldn't spend your time reading books about analysis, algebra, geometry, topology. Well, you should read a basic book about analysis and algebra, as advised above by micro, but you don't need to go very far: usually physics authors don't assume you know the advanced math already and usually give a shorter (and consequently also less rigourous) introduction and summary of the necessary math, but I wanted to make you aware of it before you start, so that you understand that only the truly purest fields of math (like logic and number theory) can be ignored for physics purposes (and even they have some applications, but you won't bump into those unless you specifically go search for them).

Have fun!

 

[micromass]

Thanks a lot you two for your replies. Micromass, I really appreciate your effort. I think I will start with the proof book first and then start with calculus and the physics book.

I have one more question though. Although I want to improve my math skills, I'd like to do this from a more applied math standpoint. How important is it, for example for a physicist, to have a very broad understanding of pure, rigorous mathematics( e.g number theory)?

To be clear. I recommended those books from the point of view that you would like a deeper understanding of math. In that case, the books are very good.
If you would just like to know physics, then books like the proof book or Spivak are unnecesary. They won't help you understand physics more and you could do without them.

 

[nonequilibrium]

Presumably the OP wants both: math for itself and its importance in physics, and he would like these two to go hand in hand. More specifically, I'd imagine he'd like to learn rigourously about for example analysis and algebra: the rigour itself might not be "necessary" for understanding physics (as many physicists don't, more specifically experimentalists), but it gives a better grasp, whereas for example rigour in number theory is pretty useless.

 

[Polymath89]

Yes, I want pure math to a certain extent. I've had problems with the notation in pure math books up to this point, because I'm not really used to the extremely compact argumentation so to say. Also my proof skills are basically non-existent, which always bothered me a little, so I will definitely read Velleman's book.
Furthermore as mr. vodka said, I think it's imortant to build a really solid foundation in topics such as calculus, that are required for more advanced stuff and I think that the pure math component is part of a solid foundation.
Thanks a lot guys, really appreciate your help.

 

[Nano-Passion]

For me to help you out, I need you to answer the following questions.

Are you interested in math for the sake of math, or are you seeking to study math to learn and understand physics?

What area of physics tends would you like to work your way up to?

 

[Polymath89]

For me to help you out, I need you to answer the following questions.

Are you interested in math for the sake of math, or are you seeking to study math to learn and understand physics?

What area of physics tends would you like to work your way up to?

Well I'm more interested in learning math than physics. I'd like to learn physics because I find it very interesting and because, based on what I've heard, it's basically the science that applies math to the real world in a way that no other science does. As I said, although I find pure math interesting and want to improve in that area, overall I'd like to have my focus on applied math.

As for what I want to do in physics: I honestly don't really know that much about all the branches in physics yet. I'm interested in cosmology and would also like to be able to dig deeper into special and general relativity and quantum mechanics. Honestly I have no idea whether I'm even going to be able to acquire the skills necessary to understand those areas considering the amount of time that I can only invest, but at least I'd like to try and enjoy the ride no matter where it takes me.

 

[Nano-Passion]

Well I'm more interested in learning math than physics. I'd like to learn physics because I find it very interesting and because, based on what I've heard, it's basically the science that applies math to the real world in a way that no other science does. As I said, although I find pure math interesting and want to improve in that area, overall I'd like to have my focus on applied math.

As for what I want to do in physics: I honestly don't really know that much about all the branches in physics yet. I'm interested in cosmology and would also like to be able to dig deeper into special and general relativity and quantum mechanics. Honestly I have no idea whether I'm even going to be able to acquire the skills necessary to understand those areas considering the amount of time that I can only invest, but at least I'd like to try and enjoy the ride no matter where it takes me.

Your jumping a bit back and forth, so it seems that your almost interested equally in both. Alright well the suggestions that Micromass gave should do quite good.

There is an alternate route however, and it really just depends on how much physics interests you. You can pick up an advanced book in physics and then learn the prerequisite material on a as-needed basis -- this give a lot of motivation behind the math.

Whenever you see a notation or something you don't understand-- just post it up on the forum asking for what math you need to study.

You can try the book by Griffith https://www.amazon.com/dp/013805326X/?tag=pfamazon01-20 and then work your way up to QM.

Or if you want, you can pick up an intro to general relativity book and start from there, the math for general relativity is pretty neat and I think you might be really interested in this route. Just look at this page to see what makes you feel more comfortable to start with. Some need more math, some need less. https://www.amazon.com/dp/013805326X/?tag=pfamazon01-20