Explorations in Mathematical Physics: The Concepts Behind an Elegant Language 2006th Edition by Don Koks (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Explorations in Mathematical Physics contains an eclectic mix of topics related to mathematical physics. This is not a textbook and more an overview of a collection of topics with often unique insights into key areas in less formal language. The book is very different in approach to most mathematical physics books in that it is not a textbook but more like a book of wisdom on how to think about many topics, which are not necessarily related. The book covers linear algebra, differential geometry, some probability theory as well as some Fourier analysis. The main physical theories underlying the mathematical content are quantum mechanics and relativity, both special and later general.As mentioned this is not really a textbook as much as a sophisticated treatment of several topics in mathematical physics catered for the interested reader. I have read the book on and off for years but finally went through it recently. It has no exercises so it is not good for testing what one has really learnt. As such when I first was using the book it was with less familiarity with formal methods in relativity. Trying to learn tensor formalism or geometric algebra from this book is a challenge. However if one already has a sense of these topics from studying spinors as a precursor to geometric algebra and quaternions or one is familiar with differential geometry then this book really is excellent. The author’s covering of the Rindler wedge is better than I’ve read elsewhere. Its coverage of basis changes in linear algebra are highly illuminating. It coverage of Fourier transforms are novel. The author does a great job of providing intuition to the math behind modern physical theories such that one comes away with a better picture of how to think about these geometric theories. The coverage of tensors and differential geometry is excellent and in particular the Riemann Christoffel Tensor and parallel transport are more clear from this book than a formal textbook.Overall I have a high regard for the author’s ability to communicate effectively in this book. The issues with the book are that I think one needs already history with the subjects to get the most out of this as the learning math and physics from this book is challenging in the absence of exercises. Obviously just learning math/physics from a textbook is rather dry but other mathematical physics have a range of essential topics they try to cover, usually erring on the side of being like math textbooks, with exercises to solidify understanding. This book tries to give those who have studied physics a new resource to think about mathematical techniques and their relevance in physics but with the physics first and without any exercises. So I think this can be read on many levels and that the most fruitful use is when one already has familiarity with the physical theories and with some mathematical background and one gleans the author’s own take on these subjects, which is highly illuminating.

⭐I own about a dozen textbooks on different subjects in mathematics covering topics like Fourier analysis, mathematical statistics and mathematical physics. I also have textbooks on quantum mechanics and solid state physics that use a lot of that math. This all came about through graduate school studies and my current job. Having studied much of this material in school and on the job, I thought I understood the basics of it all. Yet when I started reading this book I discovered some eye-opening insights and had several aha-of-course! moments that kept me reading on.As Dr. Koks explains, the book is more a narrative than a textbook. Thus, there are no problem sets, no useless end-of-chapter summaries, no extraneous color-coded exercise boxes. Instead, Dr. Koks gets straight to the point of explaining the whys of several mathematical phenomena in conversational, common-sense prose. For example, Dr. Koks discusses the familiar exponential function and its relation to radioactive decay by drawing a parallel to a group of persons each flipping a coin; this analogy intuitively shows, in a manner equations can’t, why radioactive decay follows exponential behavior. Another example I found useful was his point about the relationship between the reciprocal lattice and a vector cobasis. I’ve worked with reciprocal lattices but never realized that point before. And his explanation of how a probability density evolves from a histogram is superb, invoking nothing more than first-semester calculus. Yet I’ve never seen a comparable explanation in any of the statistical books I’ve read.The above examples are from the first few chapers of the book, which I’ve read. The latter chapters cover aspects of tensors, special and general relativity, field theory, and cosmology. Having embarked on self-study of these subjects out of academic curiosity, I know I’ll be turning to this book for clarification on a lot of these subjects. But even though I’ve noted that the book is not a textbook per se, the number of equations is still on par with any mathematics textbook, and I wouldn’t recommend this book for the mathematically faint of heart. Yet the equations aren’t there to intimidate but instead are used to make the author’s points. With a bit of effort they can be readily understood.If you’re studying mathematical physics (say, using Hassani), quantum mechanics, special or general relativity, or just love a clear exposition on the subtleties of many of the mathematical concepts underlying physics, you will want this book for its lucid discussions that are usually missing from mainstream textbooks. The book is accessible to advanced undergraduates, and as a working professional with an engineering Ph.D. I glean all sorts of insights from it. I highly recommend _Explorations in Mathematical Physics_.

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