Geometric Integration Theory (Cornerstones) 2008th Edition by Steven G. Krantz (PDF)

Description

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. Motivating key ideas with examples and figures, this book is a comprehensive introduction ideal for both self-study and for use in the classroom. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

User’s Reviews

Editorial Reviews: Review From the reviews:”This is a graduate textbook with the main purpose of introducing geometric measure theory through the notion of currents. … One of the most important features of this text is that it is self-contained … . The book also contains an Appendix … as well as extended list of references, making it a good text for a graduate course, as well as for an independent or self study.” (Mihaela Poplicher, The Mathematical Association of America, March, 2009)”The book under review succeeds in giving a complete and readable introduction to geometric measure theory. It can be used by students willing to learn this beautiful theory or by teachers as a basis for a one- or two-semester course.” (Andreas Bernig, Mathematical Reviews, Issue 2009 m)“The authors present main fields of applications, namely the isoperimetric problem and the regularity of minimal currents. The exposition is detailed and very well organized and therefore the book should be quite accessible for graduate students.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 162 (3), March, 2011) From the Back Cover This textbook introduces geometric measure theory through the notion of currents. Currents―continuous linear functionals on spaces of differential forms―are a natural language in which to formulate various types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis.Key features of Geometric Integration Theory:* Includes topics on the deformation theorem, the area and coarea formulas, the compactness theorem, the slicing theorem and applications to minimal surfaces* Applies techniques to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics* Provides considerable background material for the studentMotivating key ideas with examples and figures, Geometric Integration Theory is a comprehensive introduction ideal for use in the classroom and for self-study. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for graduate students and researchers.

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

⭐In my opinion, this is the best introduction to geometric measure theory for the beginner, and should be more well-known than it is. Morgan’s book is decent and better-known, but somewhat sketchy. Federer’s classic text is extremely difficult to read.No prerequisites are assumed except for some basic analysis and perhaps measure theory. Everything is proven from the ground up. The book is very readable.To someone beginning to learn geometric measure theory, I would recommend they obtain this book as well as Simon’s “Lectures on Geometric Measure Theory’. Simon’s book is very hard to obtain, and is written by a typewriter, but is surprisingly readable.

⭐Many people talk about the Classical book by Federer, but for someone who wants to learn the Geometric Measure Theory Federer’s book is hard to read. Krantz makes things easier to understand. His style is unique and only a few writers have it. This is one of the books that you can read without having one extra amount of papers to complement the reading. More people should know about it.

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