Maths and physics books for undergraduate level self-study

[no-cheating]

Hello

The short story is that I'd like to go through university level maths and physics because:

• I like maths and physics
  
• Deepening knowledge of maths would be useful in my programming career

Formal education

The longer story is that while being a kid, I loved mathematics. Took extra lessons, went on a competitions (with some awards), generally spent a lot of time studying maths in my free time. In secondary school I was also introduced to physics and I liked it even more than maths

Then in high school, because of some other troubles I had as a kid, I grew a dislike to science and stopped doing all that in my free time. I still got great grades in both subjects, but wasn't interested in them enough to work on anything more than what school demanded from me.

Still for my career I choose computer science, which was 30-40% about maths. The maths on my university was at high level, hard and very theoretical. I passed everything without any huge problems, but I didn't feel my level of understanding the material was thorough. After university I haven't been coming back to maths at all.

What I want now?

Now after some years I matured and am feeling that my kindness for mathematics and physics is growing again. I'd like to know more in both of those areas than what I know so far. So I'm decided to educate myself in both of those topics.

I enjoy self-studying, so I'm planning to just get some textbooks and practice on my own. I'm not afraid of putting a lot of time into it. AlsoI I like to not only be able to do things, but understand how they work in-detail, so I prefer more theoretical approaches.

My level

Iwent through university maths, but as I said I don't feel I understand it thoroughly, so I'd like to go through it again, this time in a very thorough way.

With physics I stopped at high-school level.

Question

What books would you recommend to go through for this kind of education?

 

[fresh_42]

You mention two goals here, which I think lead partly into two very different directions. Roughly speaking: physics is about analysis in a broad sense, and programming typically about discrete math with a more algebraic emphasis. You also mentioned that your programming work includes 30-40% math, which is quite a lot! This makes me wonder if you speak about numeric algorithms, which again is a third direction. If you meant databases and their design, we end up in a fourth direction.

I like to recommend OpenStax at this point, not because those books are of extraordinary quality, but as a cheap way to determine where you really stand at. Another good read are the links in here: https://www.physicsforums.com/threads/self-teaching-gcse-and-a-level-maths.933639/#post-5896947
It is just a collection of related links, not that all of them would actually apply to your situation. However, it's a start.

Physics starts normally with classical mechanics, electro- and thermodynamics, and there should be plenty of sources available on the internet. The books on OpenStax should be ok, if you left the subject while in high school. The question about mathematics is more complicated as described above, so it presumably requires some decision from your side which way to go: calculus and linear algebra for physics, discrete math for programming, numeric for algorithms, set theory and logic for databases. Mathematics which accompanies physics is probably the longest way to go.

 

[no-cheating]

You mention two goals here, which I think lead partly into two very different directions. Roughly speaking: physics is about analysis in a broad sense, and programming typically about discrete math with a more algebraic emphasis.

Then I'd say the primary reason is just learning for fun and for broadening my horizons, and that any relation to or help in my programming career is only a secondary reason.

You also mentioned that your programming work includes 30-40% math, which is quite a lot!

I wrote it in a very unclear way, but what I really meant was that in my university 30-40% of the work was on mathematics. Since leaving university I haven't used too much maths at work (though I've been using a lot of skills that I developed while studying maths).

 

[fresh_42]

In this case linear algebra could be a good starting point, simply because it is needed very much in physics, can be useful in programming, and definitely gives you a glimpse how algebra works. From there, the paths towards functional analysis (physics), group, ring and field theory (abstract algebra), or algorithmic considerations (programming) are all open and base on linear algebra. A book I like is https://www.amazon.com/dp/8184896336/?tag=pfamazon01-20 but you could as well serach for lecture notes on the internet. Many professors at universities nowadays upload their scripts, which are usually a lot thinner than an entire book.

If you just want to keep your brain busy, I'd suggest group theory or basic topology. They both can well be read on e.g. commuter trains.

 

[no-cheating]

Thanks. Then I'll start with linear algebra. I see people recommending the following books. Are there any notable differences between them?

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Linear Algebra" by Kenneth M. Hoffman
• "Linear Algebra" by Serge Lang
• "Linear Algebra" by Werner H. Greub (the one you recommended too)

And what about calculus? Should I wait with working on that until I'm done with linear algebra? From what I know that's used in physics quite extensively. I worked throughout single and multi variable calculus in university, but that was always the topic that I didn't feel I fully grasp.

And what will be the point I could actually start doing some physics? What are the mathematical prerequisites that I should finish before doing that? And what are good books for self-studying physics? Some years ago I've already planned to do that and after some research I've bought all the volumes of 4th edition of "Physics" by Resnick/Halliday/Krane. Still have it, though they are left in Poland - currently I'm living in Mexico.

 

[fresh_42]

I don't know the other books, only that Serge Lang belonged to a group of mathematicians (N. Bourbaki) whose goal it was to build up mathematical knowledge very systematically and by the use of form (-ulas) in comparison to wordy explanations. I like this approach very much, but it's not everybody's taste.

From what I learned here is, that people consider Spivak as the standard book on calculus. I learned it in my own language and the English book I have is a bit too sophisticated and abstract to start with.

Physics and mathematics often go hand in hand, so I see no problem to read both, especially as we nowadays can quickly look up things which at least answer the "what is it" question, so one can study physics and learn the mathematical part accordingly. I usually found what I wanted to know on Wikipedia, nLab, or in lecture notes found via Google without much effort.

What's more difficult to manage in my opinion is, that physics and mathematics use a different language. Both is mathematics, but it's a kind of different dialect, and this can cause confusion. E.g. I'm no big fan of coordinates and indices and physics is full of it. That's no surprise, since physics is measurement and to measure something you need scales and orientation. Mathematics is more structural. On the other hand, you've found a great place here to come and ask if you have trouble to understand something.

 

[no-cheating]

Thanks for the explanations.

I think I'll start with linear algebra alone and only after completing some book/course I'll start other things. What do you think could be the next steps after linear algebra?

 

[fresh_42]

If you want to understand physics, you need calculus and linear algebra to start. What's needed for mechanics isn't too complicated, so it could go hand in hand. But the basic of calculus are necessary, and in all variants: $\mathbb{R}\; , \;\mathbb{R}^n\; , \;\mathbb{C}\,.$

 

[no-cheating]

Do I also need both multi variable calculus for physics early, or just single variable for calculus is enough to start with? Spivak covers only single variable, right?

 

[fresh_42]

Do I also need both multi variable calculus for physics early,..

Not sure, but gradients come in pretty soon. But as I said, you can look it up easily.

... or just single variable for calculus is enough to start with? Spivak covers only single variable, right?

I don't know, I only mentioned it as others here on PF recommend it frequently. I can't imagine that it's only single variable calculus, maybe it has more than one volume. If you do a forum search (Science and Math Textbooks) on "Spivak", you will find dozens of posts.

 

[no-cheating]

Alright. That's enought to get me started. I'll browse the forums along to your recommendations. Thanks for all the help. I appreciate it very much.

 

[no-cheating]

I did some research, including reading this forum, and decided to start with studying linear algebra and calculus (single-variable). I have few questions though:

1. Is it better to start with linear algebra and then do whole calculus course or start with single-variable calclus, then linear algebra, then multi-variable calculus?
2. What are the best books for linear algebra if I want to learn both the theory (for further study of maths) and applications (that might come in handy for computer science)? The books I see people recommending the most are the following, but I find it hard to grasp how much they are oriented towards theory and applications, nor how much they are suited to my level (I've already completed basic university maths, but without deep understanding of the subjects).
   • "Linear Algebra" by by Stephen H. Friedberg, Lawrence E. Spence,, Arnold J. Insel
   • "Linear Algebra" by Kenneth Hoffman
   • "Linear Algebra Done Right" by Sheldon Axler
     
   • "Linear Algebra" by Georgi E. Shilov
3. I see 2 most recommended books for calculus are Spivak's and Apostol's. I've read Spivak is more conversational and entertaining, which I think is the attitude I'd prefer. But then Spivak doesn't cover multi-variable calculus and Apostol course is said to cover some linear algebra, which I also plan to study.
   • Do you think Apostol would be a better choice in this situation?
     
   • If I study both volumes of Apostol, would I cover the whole linear algebra, or would I still need to fill in my linear algebra studies from another textbook?
   • If I choose Spivak what books should I read after it to cover multi-variable calculus (and linear algebra)?

 

[vela]

Is it better to start with linear algebra and then do whole calculus course or start with single-variable calclus, then linear algebra, then multi-variable calculus?

If you're looking into learning math to help learn intro physics, I'd say start with single-variable calculus followed by multi-variable calculus. You can learn linear algebra at the same time.

If you're turned off by the OpenStax books, as most of my students are, there are other free physics textbooks online. If you don't mind spending a little money, you can get an older but relatively recent edition of a standard textbook, like Young and Freedman or Knight.

 

[archaic]

I know you are asking for book recommendation but I think you would also like to know this.
There are free MIT lectures about single and multivariable calculus both accompanied with lecture notes, as well as physics and linear algebra lectures.

Spoiler

Tool to look for freely available MIT lectures.

Problems and their solutions (if available) are usually at the very bottom of each session (in the course's main page), do not miss them.
Single variable calculus given by Prof. David Jerison :
The youtube playlist.
Lecture notes.
https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-1-introduction-to-derivatives/

Multivariable calculus given by Prof. David Jerison :
The youtube playlist.
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/
Other lecture notes.
Exams with solutions.

Highlights of calculus by Prof. Gilbert Strang + his textbook :
The youtube playlist.
The course's main page.
The textbook.

Linear algebra by Prof. Gilbert Strang :
The youtube playlist.
Lecture notes.

Physics by Prof. Walter Lewin :
8.01x - MIT Physics I: Classical Mechanics.
8.01x - MIT Help Sessions.

8.02x - MIT Physics II: Electricity and Magnetism.
8.02x - MIT Help Sessions.

8.03 - MIT Physics III: Vibrations and Waves.
8.03 - MIT Help Sessions by Professor Wit Busza.

 

[MidgetDwarf]

I did some research, including reading this forum, and decided to start with studying linear algebra and calculus (single-variable). I have few questions though:

1. Is it better to start with linear algebra and then do whole calculus course or start with single-variable calclus, then linear algebra, then multi-variable calculus?
2. What are the best books for linear algebra if I want to learn both the theory (for further study of maths) and applications (that might come in handy for computer science)? The books I see people recommending the most are the following, but I find it hard to grasp how much they are oriented towards theory and applications, nor how much they are suited to my level (I've already completed basic university maths, but without deep understanding of the subjects).
   • "Linear Algebra" by by Stephen H. Friedberg, Lawrence E. Spence,, Arnold J. Insel
   • "Linear Algebra" by Kenneth Hoffman
   • "Linear Algebra Done Right" by Sheldon Axler
   • "Linear Algebra" by Georgi E. Shilov
3. I see 2 most recommended books for calculus are Spivak's and Apostol's. I've read Spivak is more conversational and entertaining, which I think is the attitude I'd prefer. But then Spivak doesn't cover multi-variable calculus and Apostol course is said to cover some linear algebra, which I also plan to study.
   • Do you think Apostol would be a better choice in this situation?
   • If I study both volumes of Apostol, would I cover the whole linear algebra, or would I still need to fill in my linear algebra studies from another textbook?
   • If I choose Spivak what books should I read after it to cover multi-variable calculus (and linear algebra)?

Almost all of those books require how to read and write proofs. I think they may be to hard for if you the mathematics courses you taken did not require you to do proofs., ie number theory, analysis, etc... From the algebra books on the list. I enjoyed Axler the most and found it the most readable. However, other people I know liked Friedberg, Intel, Spence better.

Shilov is more difficult. If i recall correctly, it starts out with a permutation in regards to determinants?

Apostol is more difficult than the other books on the list...

 

[MidgetDwarf]

I would also download the pdf of Hammock:Book of Proofs. It is free to do or you can buy a copy cheap. Read this and what ever other book in conjunction. Particularly the stuff logic, proof methods, sets, relations, functions, and cardinality.

 

[TurboDiesel]

Any passers by of a similar intent might find this useful.

Maybe not on its own, but check out:
Bamberg and Sternberg: "A Course in Mathematics for Students of Physics," Volumes I & II