Regarding Mathematics for Understanding Physics and Other Worries

[Matthewkind]

Hello. I'm an eighteen year old currently enrolled in a community college. I fell in love with physics when I was sixteen, so I haven't been on this path for very long. Before then, I was messing around with philosophy and fancied myself pretty smart. Oh, how I lament those childish prejudices, now. I didn't do very good in high school owing to circumstances regarding problems in my family and my overall lethargy. I have since then dedicated myself to studying physics and have studied a bit of algebra, geometry, trigonometry, and calculus. However, I wish to go back to the beginning and start fresh. I wish to study mathematics on my own while I am attending college. I have worked on a list of things to study in order to achieve my goals (ideally unifying the four fundamental forces, but let's be frank... that is a overly expectant thing). For those of you who have studied these materials, is this list logical? Please assist me in a list of things to study so that I know where to go.

---The List---
1. College algebra
2. Euclidean geometry
3. Trigonometry | Analytic geometry
4. Single-variable calculus
5. Vector calculus | Linear algebra
6. Multi-variable Calculus | Probability theory
7. Classical mechanics | ODEs | PDEs
8. Methods of Approximation | Set theory
9. Topology | Optics
10. Statistical mechanics | Thermodynamics
11. Electronics | Electromagnetism
12. Computational physics
13. Non-relativistic quantum mechanics
14. Quantum chemistry | Solid state physics
15. Nuclear physics | Plasma physics
16. Group theory | Lie group theory
17. Vector/Tensor analysis | Difference equations
18. Generating functions | Hilbert space
19. Introduction to the functional integral
20. Special relativity
21. Advanced quantum mechanics
22. Phenomenology | General relativity
23. Quantum field theory
24. String theory

Sorry about the inconvenience, but I would be eternally grateful if you could assist in this as I have resolutely determined this to be my life's works. I know I sound naive, but physics is a beautiful pursuit and I would like nothing more than to at least be of some value to the physics community. Of course, I do dream of making it big as I am youthful, but I am also somewhat pragmatic and realize that it would be overwhelmingly joyous to help my fellow physicist in the overall task of unifying gravity, electromagnetism, and the two nuclear forces.
Most of the list comes from that superstring theory site and this site I found a while back...

 

[micromass]

I'm no physicist by far, but your list has some gaps:

2. Euclidean geometry
3. Trigonometry | Analytic geometry

You should do Euclidean geometry together with Analytic geometry. In fact, you won't really need Euclidean geometry anyway. The only thing you need of (2) will be the things in preperation for analytic geometry. So you can scratch (2) of your list.

I recommend the two books of Serge Lang:
"Basic Mathematics" and "geometry" (although the first book contains everything you ever need). Read it and understand all the concepts very well.

Be sure to be familiar to vectors at this point (which at this stage is just a pointed arrow). Be able to work with them.

4. Single-variable calculus
5. Vector calculus | Linear algebra
6. Multi-variable Calculus | Probability theory

This is screwed up. Vector calculus should come after multi-variable calculus. Furthermore, it would be wise to learn the very basics of ODE's already. Also, you can already do some physics after your single variable calculus. This would be a wise thing to do as it allows you to make the calculus topics practical. So I suggest changing (4) to

4)Single-variable calculus
5) intro to ODE | intro to physics

As for books, you can find my recommendation for calculus books here: https://www.physicsforums.com/blog.php?b=3438 It is probably best to see single-variable calculus twice: once in a "dumbed down" context and once in a rigorous context. This is essentially how I did it. Why is this good?? Because studying the dumbed down version only will not give you enough tools. And studying the rigorous version immediately is very difficult.

There are many books for ODE. I really liked the Khan academy videos. Maybe studying Boyce-Diprima will be good here. You won't be able to fully grasp exact equations at this point, but don't worry about that. Studying ODE's right now is good because you need to be very familiar with them.

As for intro to physics: try to read the book by Halliday and Resnick. Or better yet: read the Feynan lectures (you will want to do this at some point), but you perhaps won't grasp everything in there.

After this, you're ready for multivariable calculus and linear algebra. Maybe study this concurrently:

Linear algebra | Multivariable calculus

There are many good linear algebra books out there. Here is a list of my favorites: https://www.physicsforums.com/blog.php?b=3206
Be sure to grasp the notions of "matrix", "linear transformation", "determinant" well before beginning multivariable. Things like the chain rule is much easier with knowledge of linear algebra.

After this, you can do "vector calculus" or "calculus on manifolds".

Furthermore, don't do probability theory until you really need it. This will be something for later.

You will also want to learn calculus of variations at some time.

7. Classical mechanics | ODEs | PDEs

This is good. But be sure to be already familiar with ODE's at this point. And you already need to have had an introduction to physics. Starting immediately with things like Landau and Lifschitz will kill you.

8. Methods of Approximation | Set theory
9. Topology | Optics

I will only comment on the mathematics from now on. It is ideal that you already saw some set theory before. Maybe before linear algebra even. Many physicists don't care about set theory, which is fine. But if you want to understand the mathematics behind things, then you better learn set theory soon.

You won't understand topology at this point. Topology generalizes things called "metric spaces". So you first need to study metric spaces. This is usually seen in a real analysis course. I don't know if you really want to study real analysis.
Alternatively, you could study a text that only deals with metric spaces.

If you want to understand Hilbert spaces, then you also need to know real analysis. So maybe it's best to bite the real analysis bullet anyway. I recommend you to learn about Hilbert spaces before Quantum theory. Quantum theory will make much more sense with it.

A good text is "introduction to functional analysis" by Kreyszig. It requires only a calculus background. But it doesn't deal with Lebesgue integration (if you ever want this). This might be a good book to read before you embark on topology.

Read some probability theory before you study statistical mechanics. If you want to do it the right way, then you will want to see measure theoretic probability.

I would also insert some differential geometry in there. Preferably before you see general relativity and after you did topology and group theory. Lie group theory should be done after differential geometry.

Finally, buy the book "road to reality" by Penrose. You won't understand this book at all. But it will provide you with a road map to modern physics. If you study the topics as they appear in the book separately, then you should be ok.

 

[lisab]

Gerard 't Hooft, who won a Nobel Prize in physics, compiled this list:

http://www.staff.science.uu.nl/~hooft101/theorist.html

It includes links as well as topics.

 

[micromass]

Gerard 't Hooft, who won a Nobel Prize in physics, compiled this list:

http://www.staff.science.uu.nl/~hooft101/theorist.html

It includes links as well as topics.

I was looking for that...

John Baez has a similar list, do you know where to find it??

 

[Matthewkind]

So I should start with the book by Serge Lang: "Basic Mathematics" and that will encompass algebra and trigonometry? After that, I should familiarize myself with vectors? I believe I understand the concept of vectors. They're quantities with both a magnitude and a direction, correct? Like velocity is a vector because you need to specify a direction. Furthermore, a change in velocity occurs when an object changes its direction. So I should familiarize myself with the mathematical operations concerning vectors, then?

Next I'd be learning basic conceptual single variable calculus. Then I should come back and study the rigorous single variable calculus, correct? Afterwards, I should study some ordinary differential equations? Okay, so just correct me at any point if I am making a mistake.

At this point I should read that book by Halliday and Resnick regarding an introduction to physics. After that, I will study the Feynman lectures. Then I should study linear algebra and multi-variable calculus. Then calculus on manifolds? Then calculus of variations?

After these things, should I jump into classical mechanics and partial differential equations? Will I be ready for Landau and Lifschitz then?

As regards set theory, I've looked EVERYWHERE for a [free] comprehensible text. All I get is extremely technical PDFs that make me a bit dizzy.

At this point I would be on...real analysis which should also encompass metric spaces, correct?

Could you perhaps assist me in rearranging this list so that I have something to guide me?

 

[micromass]

So I should start with the book by Serge Lang: "Basic Mathematics" and that will encompass algebra and trigonometry? After that, I should familiarize myself with vectors? I believe I understand the concept of vectors. They're quantities with both a magnitude and a direction, correct? Like velocity is a vector because you need to specify a direction. Furthermore, a change in velocity occurs when an object changes its direction. So I should familiarize myself with the mathematical operations concerning vectors, then?

Next I'd be learning basic conceptual single variable calculus. Then I should come back and study the rigorous single variable calculus, correct? Afterwards, I should study some ordinary differential equations? Okay, so just correct me at any point if I am making a mistake.

At this point I should read that book by Halliday and Resnick regarding an introduction to physics. After that, I will study the Feynman lectures. Then I should study linear algebra and multi-variable calculus. Then calculus on manifolds? Then calculus of variations?

After these things, should I jump into classical mechanics and partial differential equations? Will I be ready for Landau and Lifschitz then?

As regards set theory, I've looked EVERYWHERE for a [free] comprehensible text. All I get is extremely technical PDFs that make me a bit dizzy.

At this point I would be on...real analysis which should also encompass metric spaces, correct?

Could you perhaps assist me in rearranging this list so that I have something to guide me?

Note that I am a mathematician, so my remarks should be seen in that light. If you're going to be an experimental physicist then learning things like real analysis are definitely overkill. You have to judge for yourself how much mathematics is enough for you.
However, if you want to understand the math behind physics, then you need to study the math.

Here is my proposed list for you:

1) Basic mathematics (algebra, trig, geometry). Lang is a good outline. You might need to check other sources if you find it too dense.
2) Single variable calculus (nonrigorous): derivatives, integrals, sequences. | Algebra-based physics (if you're thirsty for physics)
3) "How to prove it" by Velleman (covers logic and set theory)
4) Single variable calculus (rigorous) | ODE's | intro to physics (this can be done simultaneously)
5) Linear algebra | Multi-variable calculus (this can be done simultaneously)
6) Vector calculus , calculus on manifolds, advanced calculus (whatever it's called) | ODE's and PDE's

Try to read the Feynman lectures concurrently with other texts. First you want to see it explained in the other text, and then check what Feynman says about it.

From physics point-of-view, you're likely ready for Classical Mechanics now (try the book by Taylor). I'll comment on the math side of things now:

7) Functional analysis from the point-of-view of Kreyszig (which covers metric spaces and normed spaces) | Non-measure probability theory | Abstract algebra (for group theory)
8) Topology | Calculus of variations (make sure you now some physics before starting this)
9) Real Analysis (including measures, so don't pick Rudin as book, I recommend Aliprantis and Burkinshaw) | Differential Geometry
10) Probability theory | Complex analysis

 

[Matthewkind]

I actually planned on being a theoretician. My main concern is either being of some use in unifying the four fundamental forces or else I want to unify the fundamental forces myself. I know this sounds arrogant, but please understand that I am merely setting a nearly impossible goal for myself so that I am always studying and always striving.

 

[Matthewkind]

Also, if I go to a physics university, will their courses be enough to understand a topic? I want to be a particle physicist, I suppose and study string theory. That's ideal since I believe the theory has promise. The idea of limiting distance-scales as a means by which to unify the two pillars of modern physics conceptually is pretty genius. So I want to learn more about string theory and maybe come up with my own advances in physics. :3
Thank you very much for helping me. Please forgive me for the inconveniences.

 

[micromass]

I actually planned on being a theoretician. My main concern is either being of some use in unifying the four fundamental forces or else I want to unify the fundamental forces myself. I know this sounds arrogant, but please understand that I am merely setting a nearly impossible goal for myself so that I am always studying and always striving.

This doesn't sound arrogant at all, but you need to be prepared for hard times and failures! You will see very soon that physics and math can be very difficult. You need to put in hard work.

Also: you want to be a theoretician. Good. But you absolutely need knowledge on experiments. You need to have done some experiments yourself. Only studying theory is not good enough. So you need to do some labs yourself. Preferably in a class room setting.

Also, there are some other skills which you didn't mention. For example, programming. Knowing how to program can be very important. So you probably want to learn some programming language at some point.

Knowing LaTeX is also very important (and very easy to learn).

 

[micromass]

Also, if I go to a physics university, will their courses be enough to understand a topic?.

Well, what does it mean to "understand" something. I'm a PhD student now, but I can still go back to calculus and linear algebra texts and see things that I only understand now. In my opinion, you only understand a topic once you did research in it. Only following a class is not sufficient.

However, a class is a start. And you need to know a lot. So don't start dwindling on a topic. Read the book until you have a sufficient knowledge of the topic (that is: until you can do the problems in the book or until you did an exam about it) and then move on to another topic. Understanding will come later.

 

[jtbell]

Go to the Web sites of some of the universities that you might apply to after your community college studies. Look at the course requirements for a physics major. The pre-requisite courses will give you an idea of what math you need to take, and what sequence you'll taking the courses in. You'll often find suggested course schedules that show the sequence.

Starting from where you are right now, focus first on getting ready for calculus if necessary. Then focus on calculus, differential equations and linear algebra. Those are the foundation for the core intermediate to upper-level physics courses that an undergraduate physics major takes (after the introductory sequence): classical mechanics, electricity and magnetism, thermodynamics / statistical mechanics, quantum mechanics. It will take you a few years to get past that level, so don't over-think right now about what you'll do at that point.

By the way, Landau and Lifschitz is graduate-school level. Don't even think about them until you get through the core undergraduate courses. Typical textbooks at that level are:

Mechanics - Marion or Symon
E&M - Griffiths or Purcell
Thermo - Schroeder or Zemansky
QM - Griffiths or Sakurai

(check out the "Science Books" forum here for discussion about these and other textbooks)

Most physics majors in the USA probably get their introduction to QM in a second-year "Introduction to Modern Physics" type course which also covers relativity and some atomic and nuclear physics. After that, they take a full-blown QM course.

If you look at university web sites, you'll often find pages for these courses, put up by the professors, with lists of reading assignments, homework problems, and/or lecture notes.

 

[Matthewkind]

So what programming languages to I need to know? While compiling a list of what to learn, I have all of the branches of physics, and mostly all of the necessary mathematics, but I don't have a clue what sort of programming language I would need...

 

[Matthewkind]

1. Basic mathematics [algebra, analytic geometry, trigonometry]
2. Non-rigorous single-variable calculus | algebra-based physics
3. "How to Prove it" by Velleman [covering logic and set theory]
4. Rigorous single-variable calculus | Ordinary differential equations | Introduction to physics [calculus-based]
5. Linear algebra | Multi-variable calculus
6. Vector calculus | Calculus on manifolds | Ordinary differential equations [advanced] | Partial differential equations
7. Classical mechanics
8. Optics | LaTeX [programming]
9. Functional analysis from the point-of-view of Kreyszig [metric spaces and normed spaces]
10. Non-measure probability theory | Abstract algebra
11. Statistical mechanics | Thermodynamics
12. Electronics | Electromagnetism
13. Non-relativistic quantum mechanics
14. Quantum chemistry | Solid-state physics
15. Nuclear physics | Plasma physics
16. Topology | Calculus of variations
17. Real analysis | Differential geometry
18. Probability theory | Complex analysis
19. Special relativity
20. Advanced quantum mechanics
21. Phenomenology
22. General relativity
23. Quantum field theory
24. Superstring theory

Something like this, correct? ^

 

[root.kit]

So what programming languages to I need to know? While compiling a list of what to learn, I have all of the branches of physics, and mostly all of the necessary mathematics, but I don't have a clue what sort of programming language I would need...

FORTRAN is generally regarded as being the 'scientific' programming language. However; probably not for you yet as you're not a professional.

I'd advice you to start with Python as it's probably the easiest programming language to get started with and it's actually quite useful.

C, however, is the programming language when it comes to really understand computing and programming at its core; I would almost dare to state that you don't really, really understand programming before you know C. In and out.

Then, it'd be wise to get hold of an object-oriented language, as they are also widely used today. C++, although a completely new paradigm, is linked to your (to come) knowledge of C, so it should be a relatively simple step.

Also, knowing LaTeX (although not a programming language) and how to use Mathematica is obviously strongly adviced.

 

[lisab]

I was looking for that...

John Baez has a similar list, do you know where to find it??

Is this it?

http://math.ucr.edu/home/baez/books.html

 

[micromass]

Is this it?

http://math.ucr.edu/home/baez/books.html

Yes, thank you!

To the OP: any programming language is good. Just learn the skills of programming and the way they think. I'm fond of Lisp, so I recommend that. But python is also good for starters.

 

[Matthewkind]

All right, so...this may be an unnecessary fear, but... I don't have to learn ALL of those that are listed, do I...? I'm not really that good with computers, you see... I can only program DM which is... well, a pathetic game-making programming language and I'm not even very good at that. So then, what am I to do? I find myself in the position of Buridan's *** if I only need to learn one...

 

[root.kit]

Go with Python.

 

[Matthewkind]

Yikes... I'm already having trouble in college algebra! I'm really going to have to step-up my game. For some reason I keep getting the wrong answers on the 'Equations Quadratic in Form' section. I think it might have something to do with how when you raise something to the second power, that something can be both positive or negative... Maybe that's it. Bluh. It's the smallest things I keep overlooking! I must seem pretty silly. XD

 

[Matthewkind]

Don't get me wrong, I understand the exponent rules. It's just that I keep rushing through problems, I guess. I need to slow down... ;_;

 

[micromass]

Don't worry, it's normal. It is crucial to make a LOT of exercises to get used to it.

In my experience, you really need to drill college algebra. Make many exercises of the same type. It's very boring, but it's necessary.

 

[Mépris]

From what I see, throughout an undergraduate physics program, as a student gets more acquainted with advanced mathematics, the physics courses often have to be taken again (classical mechanics for e.g) but this time, they are more mathematically sophisticated.

Would it be a good idea to learn the mathematics required first and only then start with physics? (while of course, staying in touch with the maths, so as not to forget it) If so/not, why?