# Pure mathematics problem solving and relevance to theoretical physics

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• rhotonsix
In summary, when studying the theoretical aspects of physics, it is important to have a strong mathematical foundation. While working through books like Rudin's analysis can be helpful, it is not necessary to spend a lot of time solving difficult problems. A balance between pure and applied mathematics, such as in Hubbard's vector calculus/differential forms book, may be more beneficial in building a solid foundation for studying theoretical physics. Ultimately, the level of pure mathematics needed for this goal is still up for debate, but a good understanding of real analysis, topology, differential geometry, representation theory, and algebra can be helpful. It is also important to enjoy the process and not rush, as it is a significant investment of time.
rhotonsix
TL;DR Summary
How relevant is it to study and work through problems in a book like Rudin's analysis to build a mathematical foundation for studying theoretical physics
I am recreationally self studying physics and math with the ultimate goal of understanding theoretical/mathematical framework of classical mechanics, general relativity, and quantum mechanics (in addition to the actual physics). The question I have is how deep in the realm of pure math does one need for such a goal. For example I am working my way through baby Rudin, and reading and working out his proofs in the text is not an issue, but a number of the end chapter problems are extremely challenging. How useful is it if my goal is to study theoretical aspects of physics to spend a lot of time solving the problems in a book like Rudin’s? Would my time be better spent on texts like Hubbard’s vector calculus/differential forms book which strike a balance between the pure (proves all the major theorems of multivariable analysis in text rigorously) and applied (treats computation as well)? I am in no rush and have no problem spending the time to tackle the Rudin level problems but just unsure if it is counterproductive to my goal, which is to build a decent mathematical foundation for studying theoretical physics. Thanks in advance.

etotheipi
rhotonsix said:
Summary:: How relevant is it to study and work through problems in a book like Rudin's analysis to build a mathematical foundation for studying theoretical physics

I am recreationally self studying physics and math with the ultimate goal of understanding theoretical/mathematical framework of classical mechanics, general relativity, and quantum mechanics (in addition to the actual physics). The question I have is how deep in the realm of pure math does one need for such a goal. For example I am working my way through baby Rudin, and reading and working out his proofs in the text is not an issue, but a number of the end chapter problems are extremely challenging. How useful is it if my goal is to study theoretical aspects of physics to spend a lot of time solving the problems in a book like Rudin’s? Would my time be better spent on texts like Hubbard’s vector calculus/differential forms book which strike a balance between the pure (proves all the major theorems of multivariable analysis in text rigorously) and applied (treats computation as well)? I am in no rush and have no problem spending the time to tackle the Rudin level problems but just unsure if it is counterproductive to my goal, which is to build a decent mathematical foundation for studying theoretical physics. Thanks in advance.
In general, you are better off studying vectors, calculus, differential equations and using mathematics as a tool. It doesn't hurt to know rigorous mathematics, but it's a big investment in time.

How much physics do you already know?

Thanks for the reply. Makes sense. I took a couple of courses as an undergrad in mechanics and EM but its been 15 years and my career is completely unrelated so I am going back and reviewing - Taylor and Morin for mechanics, Griffith for EM. Again this is purely recreational so I am in no rush, just planning on taking my time building up my foundations on both the physics and math side of things and enjoy it as much as I can.

If you understand well the Mathematics used in Griffiths EM you will have a very solid foundation. That encompasses the trinity of subjects mentioned by @PeroK.

For me it is easier to get enthused by the physics before I get confused by the maths.

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For a general understanding of physics, Rudin probably isn't necessary. It certainly won't hurt you, but it won't really help you. But if you are specifically focusing on theoretical physics, I would think you would eventually want a strong understanding of real analysis, topology, differential geometry, representation theory, and algebra. Theoretical physicists basically end up with the equivalent understanding of perhaps at least a mathematics BS or even an MS. It' not uncommon these days for entering graduate students in theoretical physics to have had a math BS as well and to have taken some analysis courses.

I'd like to add something. I loved Griffiths' books, but I do not recommend them without a solid understanding of PDE. You should read Haberman's PDE book before attempting Griffiths in my opinion. Haberman really helps you understand the mysterious mathematical techniques presented in Griffiths with much more clarity. Griffiths kind of rushes through them.

Physics uses a lot of applied mathematics. So it is an open question on what the proper level of understanding of pure mathematics is. I think that most would agree that you do not need to go beyond an undergraduate level of understanding of pure math. Of course, you are doing this for fun, so follow your heart.

The flip side of the question is how much physics do you want to learn? The level of mathematical sophistication of a senior undergraduate understanding is less than what is needed for a graduate level understanding.

The problem is that physics uses a lot of different types of math. If you try to learn it all beforehand, you’ll never get to the physics. My advice is decide on the physics topic that you want to learn and then figure out what math you need to learn for it. Repeat.

BTW, there are books that try to split the difference . Off the top of my head, Flanders on Differential Forms, Sommerfeld on Partial Differential Equations, Burke on Applied Differential Geometry.

Last edited:
mpresic3 and PeroK

Sometimes older math books are more applicable to physics. For example Modern Analysis by Whittaker and Watson.

mpresic3
caz said:
Sometimes older math books are more applicable to physics. For example Modern Analysis by Whittaker and Watson.

Loomis and Sternberg's calculus book is a good example of an old school rigorous math book with a lot of physics insights as well.

Given your mathematical bent you might enjoy “Invitation to Classical Analysis” by Duren.

From the intro:

This is a book for undergraduates. To be more precise, it is designed for students who have learned the basic principles of analysis, as taught to undergraduates in advanced calculus courses, and are prepared to explore substantial topics in classical analysis. And there is much to explore: Fourier series, orthogonal polynomials, Stirling’s formula, the gamma function, Bernoulli numbers, elliptic integrals, Bessel functions, Tauberian theorems, etc. Yet the modern undergraduate curriculum typically does not encompass such topics, except perhaps by way of physical applications. In effect the student struggles to master abstract concepts and general theorems of analysis, then is left wondering what to do with them.

It was not always so. Around 1950 the typical advanced calculus course in American colleges contained a selection of concrete topics such as those just mentioned. However, the development could not be entirely rigorous because the underlying theory of calculus had been deferred to graduate courses. To remedy this unsatisfactory state of affairs, the theory of calculus was moved to the undergraduate level. Textbooks by Walter Rudin and Creighton Buck helped transform advanced calculus to a theoretical study of basic principles. Certainly much was gained in the process, but also much was lost. Various concrete topics, natural sequels to the abstract theory, were crowded out of the curriculum.

The purpose of this book is to recover the lost topics and introduce others, making them accessible at the undergraduate level by building on the theoretical foundation provided in modern advanced calculus courses.

Thank you all for the replies and book recommendations. I do have Loomis/Sternberg and have enjoyed the chapters I have read in that text - I will definitely read further and check out the other references you all suggested.

## 1. What is pure mathematics and how is it relevant to theoretical physics?

Pure mathematics is a branch of mathematics that deals with abstract concepts and theories, rather than real-world applications. It is closely related to theoretical physics as it provides the mathematical framework and tools for understanding and formulating theories in physics. Many mathematical concepts, such as calculus, differential equations, and group theory, are essential in theoretical physics and help physicists make predictions and solve complex problems.

## 2. How does solving pure mathematics problems help in understanding theoretical physics?

Solving pure mathematics problems requires a high level of abstract thinking and problem-solving skills. These skills are also crucial in theoretical physics, where physicists often have to deal with complex and abstract concepts. By practicing pure mathematics, physicists can develop their analytical and critical thinking abilities, which are essential for understanding and formulating theories in theoretical physics.

## 3. Can pure mathematics be used to solve real-world problems in theoretical physics?

Yes, pure mathematics can be used to solve real-world problems in theoretical physics. Many mathematical concepts, such as differential equations and group theory, have been successfully applied to model and explain physical phenomena. In fact, some of the most groundbreaking discoveries in physics, such as the theory of relativity and quantum mechanics, were made possible by using pure mathematics.

## 4. Are there any specific areas of pure mathematics that are particularly relevant to theoretical physics?

Yes, there are several areas of pure mathematics that are particularly relevant to theoretical physics. These include calculus, differential equations, linear algebra, group theory, and topology. These mathematical concepts are used to describe and analyze various physical phenomena, from the motion of objects to the behavior of subatomic particles.

## 5. How can studying pure mathematics benefit someone interested in pursuing a career in theoretical physics?

Studying pure mathematics can benefit someone interested in pursuing a career in theoretical physics in several ways. Firstly, it provides a strong foundation in abstract thinking and problem-solving skills, which are essential in theoretical physics. Additionally, understanding pure mathematics allows physicists to better understand and apply mathematical concepts in their research. Finally, studying pure mathematics can also open up opportunities for interdisciplinary research and collaborations with mathematicians and other scientists.

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