Geometric Multivector Analysis

From Grassmann to Dirac

Andreas Rosén

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English

Springer Nature Switzerland AG
20 November 2019

Algebra; Calculus & mathematical analysis; Differential calculus & equations; Integral calculus & equations; Differential & Riemannian geometry

Series: Birkhauser Advanced Texts / Basler Lehrbucher

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.

By: Andreas Rosén
Imprint: Springer Nature Switzerland AG
Country of Publication: Switzerland
Edition: 1st ed. 2019
Dimensions: Height: 235mm, Width: 155mm,
Weight: 881g
ISBN: 9783030314101
ISBN 10: 3030314103
Series: Birkhauser Advanced Texts / Basler Lehrbucher
Pages: 465
Publication Date: 20 November 2019
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Prelude: Linear algebra.- Exterior algebra.- Clifford algebra.- Mappings of inner product spaces.- Spinors in inner product spaces.- Interlude: Analysis.- Exterior calculus.- Hodge decompositions.- Hypercomplex analysis.- Dirac equations.- Multivector calculus on manifolds.- Two index theorems.

Andreas Rosén is a Professor at the Chalmers University of Technology and the University of Gothenburg, Sweden. His research mostly concerns Partial Differential Equations, and uses techniques from harmonic analysis and operator theory.

Reviews for Geometric Multivector Analysis: From Grassmann to Dirac

The book is carefully prepared and well presented, and I recommend the book ... for students who have just mastered vector calculus and Maxwellian electromagnetism. (Hirokazu Nishimura, zbMATH 1433.58001, 2020)