- Taschenbuch: 700 Seiten
- Verlag: Springer; Auflage: Reprint of the 1st ed. Berlin, Heidelberg, New York 1969 (5. Januar 1996)
- Sprache: Englisch
- ISBN-10: 3540606564
- ISBN-13: 978-3540606567
- Größe und/oder Gewicht: 15,5 x 4 x 23,5 cm
- Durchschnittliche Kundenbewertung: 1 Kundenrezension
- Amazon Bestseller-Rang: Nr. 218.606 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
- Komplettes Inhaltsverzeichnis ansehen
Geometric Measure Theory (Classics in Mathematics) (Englisch) Taschenbuch – 5. Januar 1996
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From the reviews: "...Federer's timely and beautiful book indeed fills the need for a comprehensive treatise on geometric measure theory, and his detailed exposition leads from the foundations of the theory to the most recent discoveries...The author writes with a distinctive style which is both natural and powerfully economical in treating a complicated subject. This book is a major treatise in mathematics and is essential in the working library of the modern analyst."Bulletin of the London Mathematical Society
Über den Autor und weitere Mitwirkende
Biography of Herbert Federer
Herbert Federer was born on July 23, 1920, in Vienna. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley. Affiliated to Brown University, Providence since 1945, he is now Professor Emeritus there.
The major part of Professor Federer's scientific effort has been directed to the development of the subject of Geometric Measure Theory, with its roots and applications in classical geometry and analysis, yet in the functorial spirit of modern topology and algebra. His work includes more than thirty research papers published between 1943 and 1986, as well as this book.
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Then another couple of years later, I tried again. Since I had already done some postdoctoral research and a bit of teaching, I thought I would now have a chance. But I got through only about the first 2 pages before realising I didn't have a clue. And on a 4th attempt, I made it to the phrase "jointly characterized" on page 9 (i.e. the second page) before realising I didn't have a clue.
It was only a couple of years ago, after writing a 1500 page book (now 2015 pages) on the foundations of differential geometry that I finally managed to get the gist of the first 8 pages. I still don't really get anything beyond that. But in 1985, I remember being told in Canberra that Leon had led a working seminar for graduate students, to work through Federer's GMT, where Leon explained multiple chapters to the satisfaction of the students. Naturally I was annoyed at missing this golden opportunity. From that day until the present, I still have only the dimmest clue what the symbols in Federer's book mean.
Nevertheless, in my own efforts to present a coherent picture of just the alternating algebra part of the first chapter of the Federer book, I came to the conclusion that Federer was doing it the "wrong" way. He gives two ways of defining general tensors. One uses characterisation with a strong category theory flavour. The other definition uses a quotient space of equivalence classes of characteristic functions with finite support on linear space products. I now consider these to be the worst style of definitions possible for tensors. The rest of the book is much the same. Whenever it is possible to give a plain definition, Federer gives a convoluted incomprehensible definition.
Be that as it may, Federer's GMT is still the Mount Everest of geometric measure theory, where many climbers fail to reach the summit, and many fail to return to base camp. If you want to make someone give up hope of a future in mathematics, give them a copy of this book. Every year or so, I still leaf through a few pages of this book with regret. All I needed was another 30 IQ points, and I could have had a chance. I could have been a contender!
It is written in an "economical" style, which means that you may well spend several hours in reading one single page (and there are 654 of them!). As a matter of fact, the author himself states in the preface that just chapter 2 is enough for a one-year graduate course.
The contents are: 1 Grassmann Algebra. 2 General Measure Theory: Measures and measurable sets; Borel and Suslin sets; measurable functions; Lebesgue integration; linear functionals; product measures; invariant measures; covering theorems; derivates; Caratheodory's construction. 3 Rectifiability: Differentials and tangents; area and coarea of Lipschitzian maps; structure theory. 4 Homological Integration Theory: Differential forms and currents; deformations and compactness; slicing; homology groups; normal currents of dimension n in R^n. 5 Applicatios to the Calculus of Variations: Integrands and minimizing currents; regularity of solutions of certain differential equations; excess and smoothness; further results on area-minimizing currents.
Each chapter could have been published as a separate monograph for they are +100 pages long!
To read this book you must have a solid background in analysis, topology, differential geometry, and algebra, plus having mastered some introductory text on the subject, like Morgan's. Eventhough it is hard, the effort is worth it because it shows how to relate some concepts of analysis by means of algebraic or topological techniques.
Includes extensive references though it lacks some motivation and explanations.
Please take a look at the rest of my reviews (just click on my name above).