Looking for math books which "balance the need for rigor and formalism with an intuitive approach to the matter"

  • #1
Mr.Husky
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Hi everyone,
I recently started studying real analysis from baby Bruckner couple. It feels me like,
"I am running too fast to reach my destination but in the process of running, I decreased my oxygen level."

So, I stopped trying to complete uni coursework fast. But rather I started reading some good books like the following:-https://www.amazon.com/gp/product/3540709967/?tag=pfamazon01-20

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

https://www.amazon.com/dp/0526311916/?tag=pfamazon01-20
Now I am looking for math books which "balance the need for rigor and formalism with an intuitive approach to the matter". So if you know any books which explains things like this, then please recommend them. It may be on any field of Mathematics.

Thanks in advance!
 

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  • #2
Frabjous
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I am not sure if these are what you are looking for…
Naive Set Theory by Halmos
What is Mathematics by Courant
Geometry and Imagination by Hilbert
 
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  • #3
mpresic3
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I am familiar with the Whittaker text on rigid bodies. it is excellent. I would not call it a math textbook though. it is really physics, far removed from naive set theory or geometry. Whittaker is more applied math with special functions and complex variables, than really abstract math. Not familiar with Bruckner. Learned a bit about measure and integration from Bartle and Functional Analysis from Reid/Simon. Is it like any of those.
 
  • #4
Mr.Husky
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I am familiar with the Whittaker text on rigid bodies. it is excellent. I would not call it a math textbook though. it is really physics, far removed from naive set theory or geometry. Whittaker is more applied math with special functions and complex variables, than really abstract math. Not familiar with Bruckner. Learned a bit about measure and integration from Bartle and Functional Analysis from Reid/Simon. Is it like any of those.
@mpresic3, I gave the books which I am reading now. Both physics and Mathematics.
 
  • #6
Frabjous
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Your examples are all over the place. It also appears from this and other posts that you might be trying to learn too many things at the same time and not necessarily in the correct order.

You are studying a series of topics right now. What are the topics that you are going to replace them with? Which one’s do you need references to?

If answering these questions is difficult, the question you need to ask is
”I am trying to learn [x], I already know [y]; what is the list of subtopics that I need to learn and in what order should I address them?
 
  • #7
Mr.Husky
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Your examples are all over the place. It also appears from this and other posts that you might be trying to learn too many things at the same time and not necessarily in the correct order.

You are studying a series of topics right now. What are the topics that you are going to replace them with? Which one’s do you need references to?

If answering these questions is difficult, the question you need to ask is
”I am trying to learn [x], I already know [y]; what is the list of subtopics that I need to learn and in what order should I address them?
Hmm. Thanks caz. I already learned computational calculus. Now studying Spivak. I wanted to study non euclidean geometry. I found a gem by Marcel Berger on classical geometry which I am going through.

I tried Bruckner couple. But I was attracted by the writing style of Spivak. I thought it is worth reading. I am only studying four subjects now. I have some free time. And now in my country, October is the month to buy anything ( because of huge discounts ). I have money now. So ...
 
  • #8
Frabjous
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It may be on any field of Mathematics.
Specific requests are better.

I recently started studying real analysis from baby Bruckner couple. It feels me like,
"I am running too fast to reach my destination but in the process of running, I decreased my oxygen level."
Abbott Understanding Analysis

It may be on any field. Abstract algebra to probability
I haven’t done it, but it gets good reviews
https://www.probabilitycourse.com/

It’s old, so there are probably some things it doesn’t cover, but it is good
Birkhoff Survey of Modern Algebra
 
  • #9
Mr.Husky
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Specific requests are better
Please don't get me wrong caz. I need to learn about elliptic and hyper-elliptic functions. Do you know about them? If yes, what might be the prerequisites (complex analysis I suppose?) And some good references?

The following is the paragraph from chapter 0 of "Theory of top by Arnold sommerfeld and Felix Klein".
the complete analytic treatment of the top with a moving support point leads to hyperelliptic functions, while the general motion of the top with a fixed support point is represented by elliptic
functions
This is the reason I need to. Other than that, I wish I can get a taste of number theory.
 
  • #10
PeroK
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Please don't get me wrong caz. I need to learn about elliptic and hyper-elliptic functions. Do you know about them? If yes, what might be the prerequisites (complex analysis I suppose?) And some good references?

The following is the paragraph from chapter 0 of "Theory of top by Arnold sommerfeld and Felix Klein".

This is the reason I need to. Other than that, I wish I can get a taste of number theory.
What you need is access to a university library. Then you can read ten mathematics books a day, if you want!
 
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  • #11
Mr.Husky
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What you need is access to a university library. Then you can read ten mathematics books a day, if you want!
Great suggestion. But it is not available for me. However I have access to a engineering college library (through my brother) which will be less helpful for me.
 
  • #12
Mr.Husky
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By the way, @PeroK , can you answer my questions in post #9?
 
  • #16
Mr.Husky
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And, you're planning to read graduate texts in mathematics?
Sorry @PeroK , I don't know that elliptic functions belong to graduate level math. I thought these will be present in a complex analysis book. So I am wrong. But Is there any way that I can study about them?
 
  • #17
PeroK
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Sorry @PeroK , I don't know that elliptic functions belong to graduate level math. I thought these will be present in a complex analysis book. So I am wrong. But Is there any way that I can study about them?
Everything is online these days.

https://en.wikipedia.org/wiki/Elliptic_function

https://en.wikipedia.org/wiki/Hyperelliptic_curve

The analysis book (by Lieb and Loss) you have been reading is explicity from a graduate studies in mathematics series.

Differential geometry, for example, is an advanced topic because it has several prerequisites in terms of undergraduate material. And the same for complex analysis.

The gist of this thread, as far as I can tell, is that you backed off studying undergraduate mathematics and are now looking for books at all levels.
 
  • #18
Mr.Husky
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Everything is online these days.

https://en.wikipedia.org/wiki/Elliptic_function

https://en.wikipedia.org/wiki/Hyperelliptic_curve

The analysis book (by Lieb and Loss) you have been reading is explicity from a graduate studies in mathematics series.

Differential geometry, for example, is an advanced topic because it has several prerequisites in terms of undergraduate material. And the same for complex analysis.

The gist of this thread, as far as I can tell, is that you backed off studying undergraduate mathematics and are now looking for books at all levels.
@PeroK , As I already said in post #5, I planned to go for lieb and loss after Spivak. Isn't it enough? I really don't want to go through the undergrad or grad curriculum as I mentioned in my post #1. I am now interested in learning dynamics of top. But the chapter 0 of the book I choosen mentioned those functions.

I am learning math whenever I need it. Now, complex analysis. However, I am not skipping undergraduate mathematics.
 
  • #19
PeroK
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@PeroK , As I already said in post #5, I planned to go for lieb and loss after Spivak. Isn't it enough? I really don't want to go through the undergrad or grad curriculum as I mentioned in my post #1. I am now interested in learning dynamics of top. But the chapter 0 of the book I choosen mentioned those functions.

I am learning math whenever I need it. Now, complex analysis. However, I am not skipping undergraduate mathematics.
It's not possible to learn mathematics at the speed you want to. And I'm not sure that Sommerfeld and Klein book is introductory physics. It's difficult to know what to say. It all seems a bit mad to me!
 
  • #20
Mr.Husky
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It's not possible to learn mathematics at the speed you want to. And I'm not sure that Sommerfeld and Klein book is introductory physics. It's difficult to know what to say. It all seems a bit mad to me!
Well, for sure the books by sommerfeld and Klein are not introductory physics. They are research monographs explaining theory of spinning top accurately with a balance on theory and applications. What I need to know is analytical mechanics on the physics side. But math is the problem.

I committed to understand rigid body dynamics ( theory of spinning top, stability of bicycle ) because I want to know how research is been done before existence of quantum mechanics. And also I can learn huge loads of mathematics by the way.

Do you think, Is it good to go on with sommerfeld and Klein? I think nobody is working on this classical stuff now. Should I change my plans to some modern stuff?
 
  • #21
PeroK
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Well, for sure the books by sommerfeld and Klein are not introductory physics. They are research monographs explaining theory of spinning top accurately with a balance on theory and applications. What I need to know is analytical mechanics on the physics side. But math is the problem.

I committed to understand rigid body dynamics ( theory of spinning top, stability of bicycle ) because I want to know how research is been done before existence of quantum mechanics. And also I can learn huge loads of mathematics by the way.

Do you think, Is it good to go on with sommerfeld and Klein? I think nobody is working on this classical stuff now. Should I change my plans to some modern stuff?
I don't see how you can be studying this at high school - even if you were a complete genius. It doesn't seem realistic to me. I can't really say more than that or what to recommend - except to take a more conventional approach.
 
  • #22
Frabjous
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felt under prepared for Griffiths em.
There is a lot about functions here: https://dlmf.nist.gov/

After reading the preface (https://www.springer.com/cda/content/document/cda_downloaddocument/9780817647209-p1.pdf?SGWID=0-0-45-721413-p173798518) I think that you should pursue something other than Klein and Sommerfeld. You should be aiming to build a solid foundation. To my mind, you should be aiming toward intermediate mechanics, e&m and quantum mechanics with the prerequisite math (vector calc, diffeq, pde’s, linear algebra, complex analysis).
 
  • #23
mpresic3
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Great suggestion. But it is not available for me. However I have access to a engineering college library (through my brother) which will be less helpful for me.
I was ready to suggest you need a tutor, a well qualified HS teacher, someone with more scientific or work experience to learn from and advise you. It seems a older brother engineer would do nicely. Does he have any suggestions?
You would go broke buying all the books you are hoping to read. It is good you have access to a tehcnical library. An engineering library contains most of the same books as a university library, and most universities contain engineering libraries. If you can find the books, do not worry they did not come from a particular library.


I committed to understand rigid body dynamics ( theory of spinning top, stability of bicycle ) because I want to know how research is been done before existence of quantum mechanics. And also I can learn huge loads of mathematics by the way.

Do you think, Is it good to go on with sommerfeld and Klein? I think nobody is working on this classical stuff now. Should I change my plans to some modern stuff?
This is good. There was a lot of good physics that was done before a century ago and this is not studied enough today. I found many of the older books have good nuggets that are today forgotten. In addition, many schools want to bring students to the fronteir of their subjects, which tend to be modern too fast. If you enjoy the classical stuff, there is no rush to get to the newest physics. Too my mind, TV shows seem to treat quantum computing, loop quantum gravity, and superstrings, like it's the only physics today worth studying. This is not the case.
Also, you can get a lot of interesting physics outside of physics books. Much of what I learned on the subject of rigid body dynamics and orbital trajectories comes from aerospace engineering, electrical engineering and celestial mechanics, and space dynamic textbooks. These textbooks may go lighter on the formal proofs in mathematics and be more intuitive, and may be what you are looking for. Something like Space Dynamics by Thomson, Introduction to Astrodynamics, by Bate, Mueller, White, and a few others.
However, I think you should maintain your interest, and keep an open mind regarding the early physics, and relax. You have to continue to do well in your current school coursework too. You will most likely not be looking for a technical job for a few years, anyway, and you have plenty of time. See if you can get someone experienced and older to help you, as they no doubt found and may be continuing to find concepts difficult that you will encounter.
 
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  • #24
Mr.Husky
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I don't see how you can be studying this at high school - even if you were a complete genius. It doesn't seem realistic to me. I can't really say more than that or what to recommend - except to take a more conventional approach.
I am not studying some advanced graduate level topics now @PeroK . I don't know why you thought it is impossible for a high school.

My school already started teaching differentiation. I just learned how to compute integrals, limits and derivatives and started calculus by Spivak. I am now starting 3rd unit on series. Is it undoable for you?

My school teach calculus based physics. I kept kleppner and kolenkow as reference. Surprisingly I completed whole book except last 4 chapters on special relativity because I have other better resources for that. Is it undoable for you?

Many of my friends started Euclid elements(Dover book). I already proved many theorems in that during grade 10 (age 14). Rather I found Marcel Berger. It is not written theorem-proof style. Just flow paragraphs with plenty of pictures and puzzles. So it is a fun read.

From what I heard is there are books on complex analysis which can be read after Spivak. Looking to read Greenleaf's complex Analysis. I still cannot figure out what you thought about me as a high school student. I am not genius.


After reading the preface (https://www.springer.com/cda/content/document/cda_downloaddocument/9780817647209-p1.pdf?SGWID=0-0-45-721413-p173798518) I think that you should pursue something other than Klein and Sommerfeld. You should be aiming to build a solid foundation. To my mind, you should be aiming toward intermediate mechanics, e&m and quantum mechanics with the prerequisite math (vector calc, diffeq, pde’s, linear algebra, complex analysis).
@caz , I don't want to go through undergraduate studies. Can you please expand on why I should pursue something other than Klein and sommerfeld (just for knowledge). Do you think is there will be any benefit/good/ useful to learn qm early. My brother have atomic physics max born.

Thanks everyone for your replies!
 
  • #25
Mr.Husky
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I was ready to suggest you need a tutor, a well qualified HS teacher, someone with more scientific or work experience to learn from and advise you. It seems a older brother engineer would do nicely. Does he have any suggestions?
You would go broke buying all the books you are hoping to read. It is good you have access to a tehcnical library. An engineering library contains most of the same books as a university library, and most universities contain engineering libraries. If you can find the books, do not worry they did not come from a particular library.



This is good. There was a lot of good physics that was done before a century ago and this is not studied enough today. I found many of the older books have good nuggets that are today forgotten. In addition, many schools want to bring students to the fronteir of their subjects, which tend to be modern too fast. If you enjoy the classical stuff, there is no rush to get to the newest physics. Too my mind, TV shows seem to treat quantum computing, loop quantum gravity, and superstrings, like it's the only physics today worth studying. This is not the case.
Also, you can get a lot of interesting physics outside of physics books. Much of what I learned on the subject of rigid body dynamics and orbital trajectories comes from aerospace engineering, electrical engineering and celestial mechanics, and space dynamic textbooks. These textbooks may go lighter on the formal proofs in mathematics and be more intuitive, and may be what you are looking for. Something like Space Dynamics by Thomson, Introduction to Astrodynamics, by Bate, Mueller, White, and a few others.
However, I think you should maintain your interest, and keep an open mind regarding the early physics, and relax. You have to continue to do well in your current school coursework too. You will most likely not be looking for a technical job for a few years, anyway, and you have plenty of time. See if you can get someone experienced and older to help you, as they no doubt found and may be continuing to find concepts difficult that you will encounter.
Thanks for your suggestions @mpresic3 !
 
  • #26
PeroK
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My school already started teaching differentiation. I just learned how to compute integrals, limits and derivatives and started calculus by Spivak. I am now starting 3rd unit on series. Is it undoable for you?

My school teach calculus based physics. I kept kleppner and kolenkow as reference. Surprisingly I completed whole book except last 4 chapters on special relativity because I have other better resources for that. Is it undoable for you?
This makes no sense. K & K uses vector calculus and other advanced mathematics, yet you have just started calculus. How are you making sense of what you are studying? You may be ploughing through advanced material without even a grasp of the basics.

The others on this thread may be doing you no favours by recommending a host of advanced topics. I'm sceptical that your programme of study is in your own best interests.

You haven't posted much homework or specific questions on here. I found only this:

https://www.physicsforums.com/threa...on-proportional-to-the-removed-force.1006520/

Which only makes me more sceptical. You're under no obligation to listen to me, but you have my opinion for what it's worth.
 
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  • #27
Mr.Husky
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This makes no sense. K & K uses vector calculus and other advanced mathematics, yet you have just started calculus. How are you making sense of what you are studying? You may be ploughing through advanced material without even a grasp of the basics.

The others on this thread may be doing you no favours by recommending a host of advanced topics. I'm sceptical that your programme of study is in your own best interests.

You haven't posted much homework or specific questions on here. I found only this:

https://www.physicsforums.com/threa...on-proportional-to-the-removed-force.1006520/

Which only makes me more sceptical. You're under no obligation to listen to me, but you have my opinion for what it's worth.
Sorry @PeroK , If my words seem to be arguing with you. K&K does vector analysis. But it explains vector operators well in chapter 5(5.12 I think). They use it in some examples. Somehow I managed with YouTube. Bit I didn't learned theory. I will learn it during when I will take a course on it.

Yes. The topics I choose to study are my own personal interests. And yesterday I asked about complex analysis book. Today I said I am looking for Greenleaf"s book. I got my own answer.

Thanks a lot @PeroK . Yes I did not post any questions here. The question I posted is from revision package from my school. After knowing that I can't actually solve them, I rereading K&K for a revision.

Hmm. This tread is showing me that even my existing plans are making me to hold my oxygen(fun). Now, I don't know what to do.

@PeroK , can you please answer my question in post #24.
I don't want to go through undergraduate studies. Can you please expand on why I should pursue something other than Klein and sommerfeld (just for knowledge). Do you think is there will be any benefit/good/ useful to learn qm early. My brother have atomic physics max born.
 
  • #28
PeroK
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@PeroK , can you please answer my question in post #24.
I think you need advice from someone who knows what level you have reached. And knows your work.
 
  • #29
Mr.Husky
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I think you need advice from someone who knows what level you have reached. And knows your work.
Which means there is nobody to help me.
 
  • #30
Mr.Husky
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Can anyone answer my question? I have knowledge roughly of a freshman.
I don't want to go through undergraduate studies. Can you please expand on why I should pursue something other than Klein and sommerfeld (just for knowledge). Do you think is there will be any benefit/good/ useful to learn qm early. My brother have atomic physics max born.
 
  • #31
Frabjous
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Can anyone answer my question? I have knowledge roughly of a freshman.
How far have you read into Klein/Sommerfeld?
 
  • #32
Mr.Husky
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How far have you read into Klein/Sommerfeld?
Nothing. Stopped after hearing elliptic functions.
 
  • #33
Frabjous
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Nothing. Stopped after hearing elliptic functions.
Exactly. You stopped at Chapter 0. You are not ready for it.

You have a freshman’s knowledge, but do not want to pursue intermediate classical, qm or em, although you might look at Susskind’s theoetical minimum books and lectures.
For classical, I would suggest looking at thermodynamics, waves, optics, fluid dynamics or relativity. If you are set on rigid bodies, pick up an engineering book on the subject.
For quantum, I would suggest starting with a “modern physics” book.
There is always Feynman.
Theoretical Concepts in Physics by Longair.
Dimensional Analysis by Bridgman
Gravity by Schutz
Nonlinear dynamics and chaos by Strogatz
Physics of the Earth by Stacey
An astronomy book
An Introduction to Error Analysis by Taylor
 
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  • #34
mathwonk
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You show extremely good taste in books. However the ones you have listed already would require years of study for most people. So instead of suggesting more, I recommend that you actually dive into some of the ones you already have, e.g. Spivak, and Hilbert-Cohn Vossen. You will be well repaid for the time spent reading them in depth and working as many exercises as possible.
 
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  • #35
Mr.Husky
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Thanks @caz and @mathwonk for your suggestions. I will keep them in mind. Thanks everyone here for helping me.
 
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