A Course in Mathematics for Students of Physics: Volume 2 by Paul Bamberg (PDF)

Description

This textbook, available in two volumes, has been developed from a course taught at Harvard over the last decade. The course covers principally the theory and physical applications of linear algebra and of the calculus of several variables, particularly the exterior calculus. The authors adopt the ‘spiral method’ of teaching, covering the same topic several times at increasing levels of sophistication and range of application. Thus the reader develops a deep, intuitive understanding of the subject as a whole, and an appreciation of the natural progression of ideas. Topics covered include many items previously dealt with at a much more advanced level, such as algebraic topology (introduced via the analysis of electrical networks), exterior calculus, Lie derivatives, and star operators (which are applied to Maxwell’s equations and optics). This then is a text which breaks new ground in presenting and applying sophisticated mathematics in an elementary setting. Any student, interpreted in the widest sense, with an interest in physics and mathematics, will gain from its study.

User’s Reviews

Editorial Reviews: Review “This textbook (Vol. 1 & 2) is a well-written and well illustrated introduction to the mathematics of modern physics. The combination of mathematics and physical applications provides an excellent learning environment for students of physics and for scientists wishing to upgrade their education. This textbook emphasizes the geometric visualization of mathematical equations, and tries to develop in the student an intuitive feel for the mathematics. Overall, this book is a welcomed introduction to the mathematics required in modern physics.” La Physique au Canada”The beauty of the book lies in the examples used to illustrate the mathematical concepts.” Roger Waxler, Siam Review Book Description A textbook covering the theory and physical applications of linear algebra and the calculus of several variables. From the Back Cover This textbook has been developed from a course taught at Harvard over the last decade. The course covers principally the theory and physical applications of linear algebra, and of the calculus of several variables, particularly the exterior calculus.The authors adopt the ‘spiral method’ of teaching covering the same topic several times at increasing levels of sophistication and range of application. Thus the student develops a deep intuitive understanding of the subject as a whole and an appreciation of the natural progression of ideas.This the second volume, opens with an introduction to algebraic topology, introduced by the analysis of electrical networks or mathematically speaking, the topology of one-dimensional complexes.Chapters 15-18 develop the exterior differential calculus as a continuous version of the discrete theory of complexes. Facts of the exterior calculus are presented: exterior algebra, k-forms, pullback, exterior derivative and Stokes’ theorem.Chapter 16 presents another physical theory, electrostatics. The authors argue that the dielectric properties of the vacuum determine Euclidean geometry in three-dimensional space. The basic facts of potential theory are presented.Chapters 17 and 18 continue and conclude the study of the exterior differential calculus, developing the notions of vector fields and flows, interior products and Lie derivatives, and applying them to magnetostatics. The star operator is discussed in a general context.Chapter 19 can be thought of as the culmination of the course. It applies the results of the preceding chapters to the study of Maxwell’s equations and the associated wave equations.The last two chapters covercomplex analysis and elementary asymptotics, and the book ends with a sophisticated treatment of thermodynamics. Read more

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

⭐I really wanted to like this book. The idea behind the book is great: use circuit and electromagnetic theory to introduce some advanced mathematical ideas, which is the reason I bought the book. I know circuit theory at the undergraduate level (material in Desoer and Kuh, for example) and have a PhD in math (functional analysis) and wanted to learn about the connection between circuit theory and algebraic topology. The number of errors in this book makes that effort a most frustrating experience. I have read and consulted hundreds of advanced math books and an equal number of journal papers (and have authored over 50 journal articles) and this is by far the worst proofread book or article I have encountered. The innumerable errors make one question whether the authors proofread at all. The presentation is confusing and befuddling at times but that is more subjective.”Mature” mathematicians who are willing to put in the time will learn some interesting connections and college math instructors will benefit from seeing this material. An undergraduate student will probably not find this book useful or learn very much from it by reading it independently. On the other hand, this may be a good book for a reading course or an undergraduate seminar for well prepared and motivated students under the guidance of a qualified instructor.Bottom line: Buy it if you find a cheap copy, you probably will learn something.

⭐The right item came in the described condition on time.

⭐excellent

⭐This is an incredible introduction to topology, diff geometry, and cohomology in physics. As far as i have found this is the only and most complete physics text on differential forms in algebraic topology, diff geometry, de rham and homological algebra, etc that holds you by the hand and details everything inbetween. TONS of examples and exercises!The main fault of this text is that there are SEVERAL errors (figures not matching up with the text, wrong symbols or expressions, typos, etc)But this is entirely minor as the devoted student that works through the text with pen and paper will be quick to recognize these errors and correct them.

⭐In short, I find this kind of a book very rare indeed.If you are doing physics and electronics at the undergraduate level, this book will open your eyes to a whole new unified approach to several on-the-surface different topics. I wonder why many course designer’s haven’t looked at this book and realized how accessible it has made some of the relatively modern concepts. The authors laudably attend on each concept with a passion to make the reader confident of grasping at least a few different ways of looking at it, keeping the core well in view all the time. It is also to their credit to have kept the beauty in the ideas intact with a good balance of abstraction and concrete instances.In particular, the authors treatment of exterior calculus is an eye opener if you are new to the topic. For a student only exposed to traditional methods, it is a revealer to see the laws of linear electrical circuits as well as Maxwell’s equations of electromagnetism being expressed in precisely the same language. It is totally to the credit of the authors to have presented the concepts in such a simple to understand progression. For example, they make you see clearly why you have understood the divergence theorem or Stokes theorem of conventional vector calculus if you’ve grasped the essence of the calculus of functions of a single variable. Its a fantastic voyage folks, and you’ve got some of the best guides methinks.All those who find physics and mathematics a drag at college should grab this book and be enlightened. I wish they fix some of the errors in the book in future editions, but the errors don’t at all hinder the learning.Ten thumbs up!

⭐Still its the best book that condensed a lot of modern mathematics needed for theoretical physics.Pedagogically, the book builds on the same topics to teach more complicated things as it advances.

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