Beginner friendly mathematical physics book

Hello,
Which is the best beginner friendly mathematical physics book that can help me understand undergraduate physics? I'm self teaching myself from the videos. Right now I've learnt upto higher school mathematics(trigonometry, calculus, vector algebra and matrices).

veil said:

Hello,
Which is the best beginner friendly mathematical physics book that can help me understand undergraduate physics? I'm self teaching myself from the videos. Right now I've learnt upto higher school mathematics(trigonometry, calculus, vector algebra and matrices).

Do you mean undergraduate physics for STEM students, or non-STEM students?

For STEM students, you will want to study calculus-based physics. When will you be self-studying Calc 1, 2, 3?

Apologies, I just noticed that you did say you have studied Calculus. Did that include 1-2-3? (derivatives, integrals, DiffEqs?)

Mary L Boas is the usual suggestion however @Orodruin has also published a book.

PhDeezNutz said:

@Orodruin has also published a book.

LOL, I have that textbook. It is *not* for beginners in undergrad.

berkeman said:

LOL, I have that textbook. It is *not* for beginners in undergrad.

Fair. I haven’t checked it out but I assumed it was.

In that case nearly everyone takes a math methods class where the book is

Either

Boas

Or

Kreyzig

veil said:

Hello,
Which is the best beginner friendly mathematical physics book that can help me understand undergraduate physics? I'm self teaching myself from the videos. Right now I've learnt upto higher school mathematics(trigonometry, calculus, vector algebra and matrices).

If you have good understanding of the mathematics (trig, calc, vectors, and matrices) then I would recommend going ahead with physics texts (start with classical mechanics). Usually undergrad physics texts introduce any additional needed mathematics as you go along.

Perhaps you can clarify what YOU mean by "mathematical physics." In my view, "mathematical" physics is physics that uses calculus, whereas "mathematical physics" is a much more advanced branch of study - where the basic physics is already well understood and more subtle concepts (symmetries, group theory, etc) are used to broaden and deepen understanding.

berkeman said:

Do you mean undergraduate physics for STEM students, or non-STEM students?

For STEM students, you will want to study calculus-based physics. When will you be self-studying Calc 1, 2, 3?

Apologies, I just noticed that you did say you have studied Calculus. Did that include 1-2-3? (derivatives, integrals, DiffEqs?)

Hi,

No I have not yet covered advanced calculus. I want to study for undergraduate STEM level physics.

gmax137 said:

If you have good understanding of the mathematics (trig, calc, vectors, and matrices) then I would recommend going ahead with physics texts (start with classical mechanics). Usually undergrad physics texts introduce any additional needed mathematics as you go along.

Perhaps you can clarify what YOU mean by "mathematical physics." In my view, "mathematical" physics is physics that uses calculus, whereas "mathematical physics" is a much more advanced branch of study - where the basic physics is already well understood and more subtle concepts (symmetries, group theory, etc) are used to broaden and deepen understanding.

Hi, I meant the mathematical methods to understand STEM undergraduate level physics.

Get a copy of “Div, Curl and All That.” After you master vector calculus, you are ready to start reading undergrad physics textbooks as gmax137 suggests. If you truly want a text covering “the mathematical methods to understand STEM undergraduate level physics,” then Boas, Riley or Arfken are, indeed, the books to get.

veil said:

Right now I've learnt upto higher school mathematics(trigonometry, calculus, vector algebra and matrices).

That should be enough for studying a standard introductory calculus-based general physics textbook, e.g. Halliday/Resnick/Walker. Or have you done that already, and are now planning to study intermediate level textbooks on individual subjects (classical mechanics, electromagnetism, thermodynamics, quantum mechanics, ...)?

jtbell said:

That should be enough for studying a standard introductory calculus-based general physics textbook, e.g. Halliday/Resnick/Walker. Or have you done that already, and are now planning to study intermediate level textbooks on individual subjects (classical mechanics, electromagnetism, thermodynamics, quantum mechanics, ...)?

I am already studying Halliday/Resnick/Walker. I want to simultaneously study necessary mathematics required for individual subjects (classical mechanics, thermodynamics etc). I want to study the advanced mathematics for next 5 months so that I can study individual subjects from next year onwards.

veil said:

No, I have not yet covered advanced calculus. I want to study for undergraduate STEM level physics.

It's still not clear to me what you've covered. You said you covered vector algebra and matrices. I covered those topics in high school physics and Algebra II. Those topics are quite a bit different than vector calculus and linear algebra (which is what most people here seem to be assuming you meant). Can you be more specific about what you mean by "calculus" and "advanced calculus"? Have you studied ordinary differential equations yet (beyond what's typically covered in calculus)?

If you haven't covered the equivalent of three semesters of calculus (essentially an entire calculus book), linear algebra, and ordinary differential equations, you want to get all that squared away before moving onto a book like Boas.

veil said:

I am already studying Halliday/Resnick/Walker.

Which is less mathematical than Haliday/Resnick/Krane.

It's not clear to me why you want to go so much faster in the math than in the physics.

Vanadium 50 said:

It's not clear to me why you want to go so much faster in the math than in the physics.

I have the same question.

This approach (learning "all" the math first) is not typical. When I was in school, the physics and math classes went hand-in-hand, each reinforcing the other. The math classes (taught by math professors) were more rigorous and "proofy" while the physics professors re-taught the math from a more operational perspective. The proofs showed the math to be "true" while the physics showed it to be "useful."

In the US, the usual pre-requisite math courses for the core intermediate-level physics courses (above Halliday/Resnick/Walker) are single- and multi-variable calculus, ordinary differential equations, and linear algebra.

However, there are probably variations in different countries (and even among different universities in the US), so you should go by whatever prerequisites are specified for the actual courses that you are going to take, according to the university's web site or course catalog.

"Mathematical physics," or else "methods of theoretical physics," is a standard term for upper-class undergrad or first-year postgrad mathematical methods required for all theoretical physics studies. I know it's a highfaluting term, but usually misleading when referring to lower study levels.

So, you won't find one single-volume book on that topic. You better look for individual texts, in particular on calculus and linear algebra at college level. You mentioned Halliday & Resnick. A good mathematics companion might be Thomas Calculus, especially the edition with analytic geometry and early introduced transcendental functions.

vela said:

It's still not clear to me what you've covered. You said you covered vector algebra and matrices. I covered those topics in high school physics and Algebra II. Those topics are quite a bit different than vector calculus and linear algebra (which is what most people here seem to be assuming you meant). Can you be more specific about what you mean by "calculus" and "advanced calculus"? Have you studied ordinary differential equations yet (beyond what's typically covered in calculus)?

If you haven't covered the equivalent of three semesters of calculus (essentially an entire calculus book), linear algebra, and ordinary differential equations, you want to get all that squared away before moving onto a book like Boas.

I covered the topics given in this link.
https://ncert.nic.in/pdf/syllabus/desm_s_Mathematics.pdf

gmax137 said:

I have the same question.

This approach (learning "all" the math first) is not typical. When I was in school, the physics and math classes went hand-in-hand, each reinforcing the other. The math classes (taught by math professors) were more rigorous and "proofy" while the physics professors re-taught the math from a more operational perspective. The proofs showed the math to be "true" while the physics showed it to be "useful."

Vanadium 50 said:

Which is less mathematical than Haliday/Resnick/Krane.

It's not clear to me why you want to go so much faster in the math than in the physics.

If this is not the right approach I am willing to change that. How should I study advanced concepts(classical mechanics, quantum mechanics, electromagnetism etc) and the necessary math required for this?

A good starting pointg would be the syllabus and degree plans from a local college.

veil said:

Hello,
Which is the best beginner friendly mathematical physics book that can help me understand undergraduate physics? I'm self teaching myself from the videos. Right now I've learnt upto higher school mathematics(trigonometry, calculus, vector algebra and matrices).

If you are teaching yourself, I highly recommend the Open University course books for the course “The Physical World S207”. These are books which you can genuinely learn from and include all the answers to the exercises. They start at a very basic level but do build up to a good level by the end and will cover all basic material you would find in year one of an undergraduate course. The main thing is that these books are designed to be books that you can learn from independently.

I can understand why you want to learn math first. I think math is the main obstacle most student face when learning physics.

In any case, Linear algebra, (vector) calculus, and ordinary differential equations are enough for introductory and intermediate classical mechanics, electromagnetism, and thermodynamics (there are some partial diff equations here and there, but most likely you will not deal with them in their full glory).

You will need to understand complex variables for quantum mechanics ( I like complex variables for scientists and engineers by Paliouras and Meadows), but there is a long way before QM, so no need to hurry.

resurgance2001 said:

If you are teaching yourself, I highly recommend the Open University course books for the course “The Physical World S207”. These are books which you can genuinely learn from and include all the answers to the exercises. They start at a very basic level but do build up to a good level by the end and will cover all basic material you would find in year one of an undergraduate course. The main thing is that these books are designed to be books that you can learn from independently.

Thanks for this information. Just for a confirmation these are the books you have mentioned?
https://www.goodreads.com/series/144730-s207-the-physical-world

gmax137 said:

I have the same question.

This approach (learning "all" the math first) is not typical. When I was in school, the physics and math classes went hand-in-hand, each reinforcing the other. The math classes (taught by math professors) were more rigorous and "proofy" while the physics professors re-taught the math from a more operational perspective. The proofs showed the math to be "true" while the physics showed it to be "useful."

This is the big problem of physics didactics at the university level. In the first semesters you haven't yet learnt the math needed in the physics lectures. So you need to learn the math on a level that you can use it in the physics courses, while the math lectures have to make it solid with all the proofs etc.

There are some good books trying to close that gap. My favorite is unfortunately available only in German (Grossmann, Mathematischer Einführungskurs für die Physik). A good English one is

C. W. Wong, Introduction to mathematical physics, Oxford University Press (2013)

The only major sin he commits is to introduce special relativity in the outdated ##\mathrm{i} c t## convention :-((.

vanhees71 said:

This is the big problem of physics didactics at the university level. In the first semesters you haven't yet learnt the math needed in the physics lectures. S

For a while, my thinking was that this could be finessed by starting with geometric optics, which is usually quite light om calculus. That would give the students the benefit of having had a week or so of calculus instruction before launching into mechanics. Never got the department to agree, but at a colloquium dinner I discovered that another department got the same idea and tried it.

The math faculty hated it.
The students hated it.
The dean hated it.
The benefits never materialized in physics,.

For good or ill, math departments expect the physics departments to be the primary teachers of calculus. Then they come in and do it again.

Of course, finally at university you aim at teaching physics in a somewhat more logical manner than in high-school. That's why you start with the most simple thing first, i.e., Newtonian point-particle mechanics (the vanilla Newtonian approach in Experimental Physics 1 and analytical mechanics in Theoretical Physics 2; Theoretical Physics 1 covers "mathematical methods", indeed providing the "emergency kit" of calculus for the students to survive till the math lectures cover the material in a more rigorous way).

That's why you don't start with geometrical optics at the university. In German high schools it's indeed what's done for decades now. Already we started (and I got my "Abitur" already in 1990) with geometrical optics into physics. I definitely didn't hate it, and a lot of experiments have been done to demonstrate how lenses, prisms, etc. work.

Why everybody hated geometrical optics is not so clear to me. In my experience, what everybody really hates is thermodynamics, but in my case this was cured when it was presented as statistical physics rather than a sequence of Legendre transformations between thermodynamical potentials ;-)).

vanhees71 said:

(the vanilla Newtonian approach in Experimental Physics 1 and analytical mechanics in Theoretical Physics 2; Theoretical Physics 1 covers "mathematical methods").

This sounds really weird to me.

Why? It works well for decades!

andresB said:

You will need to understand complex variables for quantum mechanics ( I like complex variables for scientists and engineers by Paliouras and Meadows), but there is a long way before QM, so no need to hurry.

I don't remember any "complex variables" (which means complex analysis) in undergraduate QM. Complex numbers, yes, but I don't remember anything that was unfamiliar from college trig and calculus.

Anyway, I agree with the recommendations of "Div, Grad, Curl and All That" and (some older, cheaper edition of) Boas.

EDIT: OK, I see that contour integrals are used in 1D potential scattering. You can learn that in an afternoon from the chapter in Boas.

One possibly relevant text is "Basic Training in Mathematics: A Fitness Program for Science Students" by Shankar. Use Amazon's Look Inside feature to view at the table of contents, and to read Shankar's rational for creating the book. As new texts go, the paperback version is relatively inexpensive. Shankar also has written a quantum mechanics text that is very well known.

From
https://news.yale.edu/2019/07/16/ramamurti-shankar-named-gibbs-professor-physics

"Shankar designed and taught, for nearly a decade, Physics 301a, a one-semester course aimed at first-years and sophomores, to bridge the gap between their mathematical knowledge and the expectations of their teachers. The book he wrote based on his lectures, “Basic Training in Mathematics,” now serves as the text for the course at Yale and elsewhere. His Physics 200 and 201 lectures, available online as part of Open Yale Courses and other platforms like iTunes and YouTube, have been viewed over 20 million times."

vanhees71 said:

Why? It works well for decades!

I'm sure it works well, given the success of the German system. It's just that I don't understand the distribution of topics you are presenting.

The history is that in Frankfurt Walter Greiner started with theoretical physics in the 1st semester, which was a novum in these days. That's why there had to be "Mathematical Methods" in the 1st-semester theory lecture. This is something you have to offer anyway, because the math lectures, which IMHO should be taken by the physicists together with the mathematicians, cannot deliver the necessary mathematical tools as quickly as needed in the theoretical-physics lectures.

Math must be taught in a rigorous manner with theorems and strict proofs. "Mathematical Methods" just provides plausibility arguments for the calculational machinery needed for theoretical physics. Thus the lecture mainly concentrates on vector algebra and vector calculus (assuming that the students are familiar with differentiation and integration for functions with one real variable, which is not always fulfilled either nowadays though, because the German high-school teaching is in a monotonic decline particularly in math) and on the calculational side of the subject, i.e., usually no rigorous proofs for theorems (e.g., Stokes's and Gauss's integral theorems) are given but plausibility arguments.

Usually that's done with examples from physics, i.e., classical mechanics and some electrostatics. This also has the advantage that early on you learn to think in the typical physicists' way in terms of mathematics as a language to formulate and work with the physical laws.