Grassmann's Space Analysis

Grassmann's Space Analysis PDF Author: Edward Wyllys Hyde
Publisher:
ISBN:
Category : Ausdehnungslehre
Languages : en
Pages : 94

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Geometric Multivector Analysis

Geometric Multivector Analysis PDF Author: Andreas Rosén
Publisher: Springer Nature
ISBN: 3030314111
Category : Mathematics
Languages : en
Pages : 471

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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.

A History of Vector Analysis

A History of Vector Analysis PDF Author: Michael J. Crowe
Publisher: Courier Corporation
ISBN: 0486679101
Category : Mathematics
Languages : en
Pages : 306

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Prize-winning study traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers to the final acceptance around 1910 of the modern system of vector analysis.

Grassmann Algebra Volume 1: Foundations

Grassmann Algebra Volume 1: Foundations PDF Author: John Browne
Publisher: John M Browne
ISBN: 1479197637
Category : Mathematics
Languages : en
Pages : 589

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Grassmann Algebra Volume 1: Foundations Exploring extended vector algebra with Mathematica Grassmann algebra extends vector algebra by introducing the exterior product to algebraicize the notion of linear dependence. With it, vectors may be extended to higher-grade entities: bivectors, trivectors, … multivectors. The extensive exterior product also has a regressive dual: the regressive product. The pair behaves a little like the Boolean duals of union and intersection. By interpreting one of the elements of the vector space as an origin point, points can be defined, and the exterior product can extend points into higher-grade located entities from which lines, planes and multiplanes can be defined. Theorems of Projective Geometry are simply formulae involving these entities and the dual products. By introducing the (orthogonal) complement operation, the scalar product of vectors may be extended to the interior product of multivectors, which in this more general case may no longer result in a scalar. The notion of the magnitude of vectors is extended to the magnitude of multivectors: for example, the magnitude of the exterior product of two vectors (a bivector) is the area of the parallelogram formed by them. To develop these foundational concepts, we need only consider entities which are the sums of elements of the same grade. This is the focus of this volume. But the entities of Grassmann algebra need not be of the same grade, and the possible product types need not be constricted to just the exterior, regressive and interior products. For example quaternion algebra is simply the Grassmann algebra of scalars and bivectors under a new product operation. Clifford, geometric and higher order hypercomplex algebras, for example the octonions, may be defined similarly. If to these we introduce Clifford's invention of a scalar which squares to zero, we can define entities (for example dual quaternions) with which we can perform elaborate transformations. Exploration of these entities, operations and algebras will be the focus of the volume to follow this. There is something fascinating about the beauty with which the mathematical structures that Hermann Grassmann discovered describe the physical world, and something also fascinating about how these beautiful structures have been largely lost to the mainstreams of mathematics and science. He wrote his seminal Ausdehnungslehre (Die Ausdehnungslehre. Vollständig und in strenger Form) in 1862. But it was not until the latter part of his life that he received any significant recognition for it, most notably by Gibbs and Clifford. In recent times David Hestenes' Geometric Algebra must be given the credit for much of the emerging awareness of Grass­mann's innovation. In the hope that the book be accessible to scientists and engineers, students and professionals alike, the text attempts to avoid any terminology which does not make an essential contribution to an understanding of the basic concepts. Some familiarity with basic linear algebra may however be useful. The book is written using Mathematica, a powerful system for doing mathematics on a computer. This enables the theory to be cross-checked with computational explorations. However, a knowledge of Mathematica is not essential for an appreciation of Grassmann's beautiful ideas.

An Elementary Exposition of Grassmann's Ausdehnungslehre

An Elementary Exposition of Grassmann's Ausdehnungslehre PDF Author: Joseph Victor Collins
Publisher:
ISBN:
Category : Ausdehnungslehre
Languages : en
Pages : 60

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Elements of Mechanism

Elements of Mechanism PDF Author: Peter Schwamb
Publisher:
ISBN:
Category :
Languages : en
Pages : 318

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The Filtration of Public Water-supplies

The Filtration of Public Water-supplies PDF Author: Allen Hazen
Publisher:
ISBN:
Category : Filters and filtration
Languages : en
Pages : 462

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Comprising the leading principles and practices connected with the propagation, culture, and improvement of fruits, nuts, ornamental trees, shrubs, and plants in the United States and Canada

Comprising the leading principles and practices connected with the propagation, culture, and improvement of fruits, nuts, ornamental trees, shrubs, and plants in the United States and Canada PDF Author: Joseph L. Budd
Publisher:
ISBN:
Category : Fruit-culture
Languages : en
Pages : 546

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Elementary Mechanics

Elementary Mechanics PDF Author: Francis M. Hartmann
Publisher:
ISBN:
Category : Mechanics
Languages : en
Pages : 171

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An Elementary Treatise on Phototopographic Methods and Instruments

An Elementary Treatise on Phototopographic Methods and Instruments PDF Author: John Adolphus Flemer
Publisher:
ISBN:
Category : Photogrammetry
Languages : en
Pages : 620

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A Text-Book on Roofs and Bridges

A Text-Book on Roofs and Bridges PDF Author: Mansfield Merriman, Henry S. Jacoby
Publisher:
ISBN:
Category :
Languages : en
Pages : 290

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