Mathematics Applied to Physics by E. Roubine (PDF)

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⭐I present brief survey of the material sandwiched between these covers: three chapters entail analysis: complex variables, theory of distributions, exterior differential forms (fifty pages each). Part of the text is devoted to differential and integral equations: ordinary, partial and Integral (50, 80 and 30 pages, respectively). Chapters 7, 8 and 9 are written in French, thus the text (although not the math) remains partly inaccessible to me (that is, pages 348 to 463, Numerical Approximations and Optimization, including Calculus of Variations). Concluding chapters present probability (40 pages) and quantum mechanics (40 pages). The Preface: “…for students who already have good grounding, for professors to brush-up or add to their knowledge, to modernize or assist them in their research…”I survey the highlights:(1) Deschamps, authors the chapter of Differential Forms. This should be required reading for all undergraduate students. Concluding on these words: “There is no need to memorize the many special formulas of vector analysis, no need to restrict oneself to three- dimensional spaces. Although Exterior Algebra is part of tensor algebra, it is,in fact, simpler and seems to apply directly to more situations in the physical world.” (This claim he substantiates throughout). (2) De Jager, authors the Chapter on Distributions. Much here of value. Start at an elementary vantage point, proceed to an advanced level, then, culminating in examples which includes solution to Klein-Gordon equation.(3) Fritz John presents partial and integral equations. Here made most comprehensible, especially in light of physical examples. These examples include: two dimensional steady fluid flow, waves on tractionless surface of elastic material, forced periodic oscillations of pendulum, gravity waves in channel of finite depth. A mix of theory and applications.(4) A brief section (in the chapter on probability) discusses diffusion and Brownian motion, advanced yet lucid.An interesting section of this chapter (ten pages) is devoted to a discussion of information theory.(5) Conclude with Yamanouchi’s quantum mechanics. The focus: Lie groups and their representation. This is essay is required reading ! Thus, here presented, a panoramic view of (advanced) mathematics applied to physics.Highly recommended for collateral reading and enrichment.

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