Customer Reviews

Differential Analysis: Differentiation, Differential Equations and Differential Inequalities

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This book deals with some foundational properties of real differential calculus (one or several variables). It is not comprehensive but it is a model to follow and admire (since T. M. Flett died well before his work was finished and edited by J. M. Pym). It reminds me of the more elaborate Burckel's treatise on complex analysis An Introduction to Classical Complex Analysis: Volume 1 (Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften / Mathematische Reihe) (v. 1). Its main purpose is not empty erudition, but rather a quest for unveiling what the main concepts and theorems mean. This work digs out the truth from the methods that inspired some of the creators of differential calculus: Lagrange, Cauchy, Peano, Lipschitz, Weierstrass, Dini, Arzela, Fréchet, Picard, Perron, Osgood... Technical discussions of old proofs or attempts of proofs, are amazing and brings mathematics back to life. Unfortunately, the book is too short and somewhat irregular, since, it covers (in five chapters) only functions of one real variable, (real) ordinary differential equations (ODE), and Fréchet, Gâteux and Hadamard basic differential calculus on a topological vector spaces. I wasn't even aware of the existence of Hadamard differential, and my knowledge of Gâteaux differential is quite shallow, so I cannot express but a naive appreciation of both. But, Fréchet calculus on Banach spaces, as well as ODE and one variable calculus is so universally widespread now, that even I know something about that, and I have really enjoyed the first three chapters. Great mathematics, deep insight, a book that opens our mind and improve our previous background on the subject. I'm not a pro, but I'm sure learning (and teaching) mathematical analysis could be much easier if teachers had a serious historical background and a more critical view about the meaning and the technicalities of most basic results. One example? Read Flett's proof of Rolle's theorem, where he only uses the completeness of real numbers, following an idea of Ampère (yes, that Ampère you know) and then linking it to a theorem of Paul Levy.