ENGLISH

Geometric Multivector Analysis

By Andreas Rosén

Book information

Publisher
Birkhäuser
Year
2020
ISBN
978-3-030-31410-1
Language
english
Format
PDF
Filesize
8 MB (8465170 bytes)
Series
Birkhäuser Advanced Texts Basler Lehrbücher
Edition
1
Pages
471\471
Time added
2019-11-10 15:02:31
geometric analysis

Description

This book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions. The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics. The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis. Front Matter ....Pages i-xiii Prelude: Linear Algebra (Andreas Rosén)....Pages 1-22 Exterior Algebra (Andreas Rosén)....Pages 23-71 Clifford Algebra (Andreas Rosén)....Pages 73-103 Rotations and Moobius Maps (Andreas Rosén)....Pages 105-151 Spinors in Inner Product Spaces (Andreas Rosén)....Pages 153-184 Interlude: Analysis (Andreas Rosén)....Pages 185-207 Multivector Calculus (Andreas Rosén)....Pages 209-254 Hypercomplex Analysis (Andreas Rosén)....Pages 255-284 Dirac Wave Equations (Andreas Rosén)....Pages 285-341 Hodge Decompositions (Andreas Rosén)....Pages 343-382 Multivector and Spinor Bundles (Andreas Rosén)....Pages 383-422 Local Index Theorems (Andreas Rosén)....Pages 423-449 Back Matter ....Pages 451-465

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