Explorations in Mathematical Physics: The Concepts Behind an Elegant Language

by Don Koks

Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature? This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You will see how the accelerated frames of special relativity tell us about gravity. On the journey, you will discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology. The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods. Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

Genres: Physics, Mathematics, Science

554 pages, Hardcover

First published September 15, 2006

Reviews

[Jamie · Rating 5 out of 5]

It’s has ceased to be surprising to me how close and how far off modern education delivers. Inner and outer products, geometric algebra, tensors, topology, all these must wait because presumably these topics are too advanced. It’s a disservice to youthful minds feeding them pop science void of substantive value. A much better approach once rudimentary mathematical mechanics are covered, or along side, is jumping directly to ‘advanced’ applications, shoring up intervening knowledge and intuition along the way. That’s exactly where this book shines.

[Keith · Rating 4 out of 5]

I wish this excellent survey of mathematica physics had been available when I was an undergrad.

[Jagatheesan Jag · Rating 5 out of 5]

TL;DR: It’s an excellent book, and I enjoyed every page.

Exploration in Mathematical Physics by Don Koks is like a survey of mathematics topics used in physics. It’s not just mathematics. The author related the topic of mathematics that is being explained to its physics application.

The book is dense, with a lot of topics covered. The book goes through linear algebra, probabilities, differential geometry, and calculus of variations, among others on the mathematics side, with an exploration of Newtonian mechanics, electromagnetics, field theories, special/general relativity, and so on.

There are no exercises in the book. There is some sprinkling of “XYZ left to the reader.” However, I found that the equations usually skip intermediate steps, and the author goes straight to the main result. Taking this as an advantage, I studied the book by working out the missing steps. That helped me a lot to understand what was going on and correct my misunderstandings.

Since I’m a self-learner and not a physics or math graduate, it took me nearly two months to earnestly study the book with the occasional aid of AI tools like ChatGPT, which is surprisingly helpful in some cases. At times when I couldn’t derive the steps, I took screenshots of the pages and asked it to derive the steps. Sometimes, it gives the correct working. It usually hallucinates but points to the right keywords to investigate further. That being said, having some calculus and linear algebra background before studying this book would certainly help.

In my previous studies, I merely applied equations to solve problems. Studying this book, I appreciated seeing how elegant mathematics gives rise to these equations. It felt like a serendipitous encounter to see how the language of mathematics explains so many physical observations beautifully and predicts how Nature works, which is later confirmed by experiments.

To sum up, the book was a thoroughly enjoyable study.

[Parviz Abilly · Rating 5 out of 5]

Excellent

[Minger]

have to crack the book, not the usual math-free coffee table science for relative laypeople.