Geometric Integration Theory
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Geometric Integration Theory

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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.
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Product details

  • Hardback | 340 pages
  • 155 x 235 x 22.86mm | 1,490g
  • Secaucus, United States
  • English
  • 2008 ed.
  • 2 Tables, black and white; 33 Illustrations, black and white; XVI, 340 p. 33 illus.
  • 0817646760
  • 9780817646769
  • 1,369,725

Back cover copy

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 student



Motivating 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.
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Table of contents

Basics.- Caratheodory's Construction and Lower-Dimensional Measures.- Invariant Measures and the Construction of Haar Measure..- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: The Area Formula, the Coarea Formula, and Poincare Inequalities..- The Calculus of Differential Forms and Stokes's Theorem.- to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.
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Review Text

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)
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Review quote

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 fur Mathematik, Vol. 162 (3), March, 2011)
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About Steven G. Krantz

Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri.
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