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Advancing Mathematical Knowledge 
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#1
Feb1713, 08:58 AM

P: 27

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 semistudied 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. 


#2
Feb1713, 09:04 AM

Mentor
P: 18,011

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. 


#3
Feb1713, 09:15 AM

P: 27

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 


#4
Feb1713, 09:44 AM

Mentor
P: 18,011

Advancing Mathematical 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 


#5
Feb1713, 09:52 AM

P: 27

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.



#7
Feb1713, 10:06 AM

P: 27

and Floyd..



#8
Feb1713, 10:08 AM

P: 690

jason 


#9
Feb1713, 10:20 AM

C. Spirit
Sci Advisor
Thanks
P: 5,397




#10
Feb1713, 10:24 AM

P: 27

'Semistudied' 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=81 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=11 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. 


#11
Feb1713, 11:11 AM

P: 20

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. 


#12
Feb1713, 11:33 AM

C. Spirit
Sci Advisor
Thanks
P: 5,397

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.



#13
Feb1713, 01:03 PM

P: 690

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 


#14
Feb1713, 03:45 PM

P: 27

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. 


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