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Rating: 
Summary: The authoritative reference in the field.
Review: First published in 1969 and initially intended as a reference for mature mathematicians and as a textbook for able students, this remarkable book constitutes the ultimate treatise on the subject still nowadays.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).
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