Geometric multivector analysis : from Grassmann to Dirac

• 作者： Rosen, Andreas, author.
• 其他作者：
  • SpringerLink (Online service)
• 其他題名：
  • Birkhauser advanced texts basler lehrbucher.
• 出版： Cham : Springer International Publishing :Imprint: Birkhauser
• 叢書名： Birkhauser advanced texts basler lehrbucher,
• 主題： Algebras, Linear. , Linear and Multilinear Algebras, Matrix Theory. , Global Analysis and Analysis on Manifolds. , Partial Differential Equations. , Integral Equations. , Differential Geometry.
• ISBN： 9783030314118 (electronic bk.) 、 9783030314101 (paper)
• FIND@SFXID: CGU
• 資料類型: 電子書
• 內容註: 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.
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• 摘要註: 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 Mobius 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.
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