Advancing Mathematical Knowledge

Dazed&Confused

Repeatedly in the past few months any attempt at learning more advanced physics has ended in hitting a brick wall: my maths knowledge is really not up to par. To put this into perspective, I am an A Level student but feel as though the current syllabus for the sciences is really not enough. I crave more knowledge. So far I have semi-studied books like ''University Physics'' by Young and Freedman but anything more in depth and maths becomes a real issue. It is as if the book becomes foreign to me.

To remedy this, I have studied ''Engineering Mathematics'' and attempted ''Mathematics for Engineers and Scientists''. While these books are good for practice and looking at the applied side to it, they are not rigorous studies and because of this I just end up in frustration having not understood the topic at its fundamental level.

So the reason I come here is to ask for any strategy for studying maths predominantly from textbooks for its own sake and to further my understanding of physics. Any suggestions would be very much appreciated. Thank you in advance.

Also I would like to note that obtaining the textbooks is not a problem.

micromass

It seems like the "math methods" books aren't working out for you. If you want to understand mathematics at a fundamental level, then you have no other choice than to study from mathematics books. I have to say that if you want to apply the mathematics, then methods books are more than enough, math books will be overkill. But if you want to grasp the mathematical concepts and if you're ok with books not mentioning applications very much, then you should get a math book.

What topics do you want to get a better grasp of? I'm sure I (or others) can recommend some nice books.

Dazed&Confused

I have ''Calculus'' by Spivak for example and that seems like a great book for it as all the theorems are given proper proofs etc so something of that sort.
The topics themselves that I particularly would like to understand ( at least right now ) are statistics and statistical mechanics, numerical analysis, vectors, tensors, but really any book suggestion that actually teaches would be good in itself. I guess a better way to narrow it down is any topic which is used more in scientific fields than others i.e has more application. Maybe also a book which doesn't require too much knowledge from other fields but I know that can be difficult to find as maths topics are interdependent.

Well thanks again

micromass

I see your problem now. Things like tensors are fundamental in physics, but physics texts usually don't treat them very rigorously. This is reasonably because a rigorous treatment of tensors is nontrivial. It requires quite some background knowledge.

I would suggest that you learn some linear algebra. You are probably already comfortable with matrices, their operations, eigenvalues, diagonalization, etc. A good book to read would be "Linear Algebra" by Serge Lang. It even contains some things on tensor products (which are related to but not the same as tensors in physics).

As for statistics, I recommend the probability theory textbooks of Feller. They are very good. They don't really cover statistics though, but to understand statistics, you need to know probability theory first.

As for numerical analysis, I have no idea

Dazed&Confused

Thanks.. I will check those out. And yeah that is the issue I've been having so I guess for now I will postpone studying physics and focus more on studying the maths.

micromass

Be sure to keep listening to good music like Zeppelin too

Dazed&Confused

and Floyd..

jasonRF

Can you give us examples of physics books for which the math was a significant barrier? Also, what does "semi-studied" mean? Did you solve a bunch of problems? If not, then regardless of the math you might not be prepared to tackle the next level of physics.

I have heard this is a wonderful book. Has studying Spivak helped you with the math you are finding in the physics books you are trying to study?

Would you mind giving us a link to those books? I'm not sure I know what books you are referring to...


You should just realize that these two goals will lead you down different, but overlapping, paths. Linear algebra is important for both. Real analysis will likely not help for understanding the next level of physics you are trying to learn but is crucial for math. In any case, I think your interest in math will serve you well - keep it up!

jason

WannabeNewton

This. I cannot tell you how many times I've been confused out of my naive little mind because of the way a physics text explains a mathematical concept or phrases a mathematical statement and had to go to micromass to have all the confusion untangled (just yesterday this happened again lol).

Dazed&Confused

'Semi-studied' means I did some of the problems but not all. I have not studied Spivak though.

The maths books I was referring to are:

http://www.amazon.co.uk/Engineering-...1117631&sr=8-1

although this is still quite basic level I think and only really just gives a lot of practice

http://www.amazon.co.uk/Mathematical...1117683&sr=1-1

and sorry I got the title of this wrong.

As to giving examples I would say one was ''General Relativity'' by Hobson. The other physics book had a chapter on special relativity but really that was just arithmetic. I guess the problem was my limited understanding of vector calculus. I know the second book I gave links to has a chapter on that topic but I have not reached that far into it and in any case it is still more of an applied book than anything which was my original issue.

I guess the reason I want to have a fuller understanding of the maths is that first in itself is very interesting and useful but also then I can immediately learn to apply it when studying physics. Learning the maths solely to apply it in a subject will obviously see some understanding lacking... well at least that is what I think anyway.

Frimus

.
As for tensors in Hobson I suggest
A Student's Guide to Vectors and Tensors by Daniel Fleisch
It enters the "field" of tensors very nicely and the learning curve is not so steep.

WannabeNewton

That book is a perfect example of the kind of books the OP DOESN'T want i.e. informal, hand wavy definitions of things and overly, overly computational exercises. When it comes to a subject as important and delicate as tensors, it is probably best to acquire the necessary pre reqs and learn it from a proper math book.

jasonRF

I just looked at the General Relativity book. It is written for advanced undergrads and graduate students. It assumes that you know the material in something like the second math reference you listed. A quick google search shows that schools that use this (eg u. iowa, john's Hopkins) indeed use it for advanced undergrad courses (ones taken by students who know EM at level of Griffiths and mechanics at level of Taylor) or for true graduate courses and all that entails. You certainly must know vector calculus and be comfortable using it to solve problems, and you should probably know special relativity at least at the level of basic books (French "special relativity", or Resnick and Halliday's "relativity and early quantum theory" or perhaps some better book). Knowing relativity at the level of Griffiths EM book would likely be good - most students learning from Hobson probably already have done that.

Your background of simple intro physics and not even any vector calculus are clearly inadequate, whether or not you know everything in Spivak's "Calculus." I am not saying you shouldn't pursue pure math as well, but I do think that skipping the standard intermediate level physics isn't the best path for most of us. Indeed, I recently picked up a grad mechanics book (I only learned upper division undergrad mechanics) and found that I need to refresh my memory on the undergrad version before I can really learn the graduate. If I only knew intro physics level mechanics it would be waaay out of the question!

best of luck,

jason

Dazed&Confused

Thank's for your advice I will also check those books out. Tbh I didn't attempt to study the relativity book as I knew it was way out of my league. It was just an example I could think of where the mathematics involved was one potential barrier for further study. Spivak's ''Calculus'' was also just an example of a textbook style I found helpful.
I will definitely look at the books mentioned here and thank you again for you advice.