Grassmann Algebra Volume 1: Foundations

Exploring extended vector algebra with Mathematica


Author: John Browne

Publisher: John M Browne

ISBN: 1479197637

Category: Mathematics

Page: 588

View: 5964

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.

Linear Algebra via Exterior Products


Author: Sergei Winitzki

Publisher: Sergei Winitzki

ISBN: 140929496X


Page: 285

View: 1403

This is a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the array-based formalism of vector and matrix calculations. This book makes extensive use of the exterior (anti-commutative, "wedge") product of vectors. The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. Without cumbersome matrix calculations, this text derives the standard properties of determinants, the Pythagorean formula for multidimensional volumes, the formulas of Jacobi and Liouville, the Cayley-Hamilton theorem, the Jordan canonical form, the properties of Pfaffians, as well as some generalizations of these results.

Exterior Analysis

Using Applications of Differential Forms


Author: Erdogan Suhubi

Publisher: Elsevier

ISBN: 0124159281

Category: Mathematics

Page: 779

View: 1899

Exterior analysis uses differential forms (a mathematical technique) to analyze curves, surfaces, and structures. Exterior Analysis is a first-of-its-kind resource that uses applications of differential forms, offering a mathematical approach to solve problems in defining a precise measurement to ensure structural integrity. The book provides methods to study different types of equations and offers detailed explanations of fundamental theories and techniques to obtain concrete solutions to determine symmetry. It is a useful tool for structural, mechanical and electrical engineers, as well as physicists and mathematicians. Provides a thorough explanation of how to apply differential equations to solve real-world engineering problems Helps researchers in mathematics, science, and engineering develop skills needed to implement mathematical techniques in their research Includes physical applications and methods used to solve practical problems to determine symmetry

New Foundations in Mathematics

The Geometric Concept of Number


Author: Garret Sobczyk

Publisher: Springer Science & Business Media

ISBN: 0817683852

Category: Mathematics

Page: 370

View: 4230

The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner. New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.

A New Branch of Mathematics

The "Ausdehnungslehre" of 1844 and Other Works


Author: Hermann Grassmann

Publisher: N.A


Category: Mathematics

Page: 555

View: 442

Advanced Linear Algebra


Author: Bruce Cooperstein

Publisher: CRC Press

ISBN: 1439829691

Category: Mathematics

Page: 364

View: 463

Advanced Linear Algebra focuses on vector spaces and the maps between them that preserve their structure (linear transformations). It starts with familiar concepts and then slowly builds to deeper results. Along with including many exercises and examples, each section reviews what students need to know before studying the material. The book first introduces vector spaces over fields as well as the fundamental concepts of linear combinations, span of vectors, linear independence, basis, and dimension. After covering linear transformations, it discusses the algebra of polynomials with coefficients in a field, concentrating on results that are consequences of the division algorithm. The author then develops the whole structure theory of a linear operator on a finite dimensional vector space from a collection of some simple results. He also explores the entire range of topics associated with inner product spaces, from the Gram–Schmidt process to the spectral theorems for normal and self-adjoint operators on an inner product space. The text goes on to rigorously describe the trace and determinant of linear operators and square matrices. The final two chapters focus on bilinear forms and tensor products and related material. Designed for advanced undergraduate and beginning graduate students, this textbook shows students the beauty of linear algebra. It also prepares them for further study in mathematics.



Author: Hermann Günther Grassmann

Publisher: American Mathematical Soc.

ISBN: 9780821890493

Category: Mathematics

Page: 411

View: 3793

The Ausdehnungslehre of 1862 is Grassmann's most mature presentation of his "extension theory". The work was unique in capturing the full sweep of his mathematical achievements. Compared with Grassmann's first book, Lineale Ausdehnungslehre, this book contains an enormous amount of new material, including a detailed development of the inner product and its relation to the concept of angle, the "theory of functions" from the point of view of extension theory, and Grassmann's contribution to the Pfaff problem. In many ways, this book is the version of Grassmann's system most accessible to contemporary readers. This translation is based on the material in Grassmann's "Gesammelte Werke", published by B. G. Teubner (Stuttgart and Leipzig, Germany). It includes nearly all the Editorial Notes from that edition, but the "improved" proofs are relocated, and Grassmann's original proofs are restored to their proper places. The original Editorial Notes are augmented by Supplementary Notes, elucidating Grassmann's achievement in modern terms. This is the third in an informal sequence of works to be included within the History of Mathematics series, co-published by the AMS and the London Mathematical Society. Volumes in this subset are classical mathematical works that served as cornerstones for modern mathematical thought.

Understanding Geometric Algebra

Hamilton, Grassmann, and Clifford for Computer Vision and Graphics


Author: Kenichi Kanatani

Publisher: CRC Press

ISBN: 1482259516

Category: Computers

Page: 208

View: 2771

Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton’s quaternion algebra, Grassmann’s outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres. With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today’s geometric algebra.

Geometric Modeling and Mesh Generation from Scanned Images


Author: Yongjie Jessica Zhang

Publisher: CRC Press

ISBN: 1482227770

Category: Computers

Page: 354

View: 1759

Cutting-Edge Techniques to Better Analyze and Predict Complex Physical Phenomena Geometric Modeling and Mesh Generation from Scanned Images shows how to integrate image processing, geometric modeling, and mesh generation with the finite element method (FEM) to solve problems in computational biology, medicine, materials science, and engineering. Based on the author’s recent research and course at Carnegie Mellon University, the text explains the fundamentals of medical imaging, image processing, computational geometry, mesh generation, visualization, and finite element analysis. It also explores novel and advanced applications in computational biology, medicine, materials science, and other engineering areas. One of the first to cover this emerging interdisciplinary field, the book addresses biomedical/material imaging, image processing, geometric modeling and visualization, FEM, and biomedical and engineering applications. It introduces image-mesh-simulation pipelines, reviews numerical methods used in various modules of the pipelines, and discusses several scanning techniques, including ones to probe polycrystalline materials. The book next presents the fundamentals of geometric modeling and computer graphics, geometric objects and transformations, and curves and surfaces as well as two isocontouring methods: marching cubes and dual contouring. It then describes various triangular/tetrahedral and quadrilateral/hexahedral mesh generation techniques. The book also discusses volumetric T-spline modeling for isogeometric analysis (IGA) and introduces some new developments of FEM in recent years with applications.

Plato's Ghost

The Modernist Transformation of Mathematics


Author: Jeremy Gray

Publisher: Princeton University Press

ISBN: 9781400829040

Category: Mathematics

Page: 528

View: 3477

Plato's Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions. Plato's Ghost evokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Richard Dedekind, Henri Lebesgue, Henri Poincaré, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naïve set theory and the revived axiomatic method--debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism. Plato's Ghost is essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics.

Group Theory

Birdtracks, Lie's, and Exceptional Groups


Author: Predrag Cvitanović

Publisher: Princeton University Press

ISBN: 9781400837670

Category: Mathematics

Page: 288

View: 5079

If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional. The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, "birdtracks" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as "negative dimensional" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

Geometric Calculus

According to the Ausdehnungslehre of H. Grassmann


Author: Giuseppe Peano

Publisher: Springer Science & Business Media

ISBN: 1461221323

Category: Mathematics

Page: 150

View: 3065

Calcolo Geometrico, G. Peano's first publication in mathematical logic, is a model of expository writing, with a significant impact on 20th century mathematics. Kannenberg's lucid and crisp translation, Geometric Calculus, will appeal to historians of mathematics, researchers, graduate students, and general readers interested in the foundations of mathematics and the development of a formal logical language. The book has never been reprinted in its entirety, and only two chapters have ever been translated into English. Readers of this valuable translation will gain insight into the work of a distinguished mathematician and founder of mathematical logic.

Relativity and Scientific Computing

Computer Algebra, Numerics, Visualization


Author: Friedrich W. Hehl,Roland A. Puntigam,Hanns Ruder

Publisher: Springer Science & Business Media

ISBN: 3642957323

Category: Science

Page: 389

View: 5950

For this set of lectures we assumed that the reader has a reasonable back ground in physics and some knowledge of general relativity, the modern theory of gravity in macrophysics, and cosmology. Computer methods are present ed by leading experts in the three main domains: in numerics, in computer algebra, and in visualization. The idea was that each of these subdisciplines is introduced by an extended set of main lectures and that each is conceived as being of comparable 'importance. Therefpre we believe that the book represents a good introduction into scientific I computing for any student who wants to specialize in relativity, gravitation, and/or astrophysics. We took great care to select lecturers who teach in a comprehensible way and who are, at the same time, at the research front of their respective field. In numerics we had the privilege of having a lecturer from the National Center for Supercomputing Applications (NCSA, Champaign, IL, USA) and some from other leading institutions of the world; visualization was taught by a visualization expert from Boeing; and in com puter algebra we took recourse to practitioners of different computer algebra systems as applied to classical general relativity up to quantum gravity and differential geometry.

Problems and Solutions in Introductory Mechanics


Author: David J. Morin

Publisher: Createspace Independent Publishing Platform

ISBN: 9781482086928

Category: Science

Page: 352

View: 6204

This problem book is ideal for high-school and college students in search of practice problems with detailed solutions. All of the standard introductory topics in mechanics are covered: kinematics, Newton's laws, energy, momentum, angular momentum, oscillations, gravity, and fictitious forces. The introduction to each chapter provides an overview of the relevant concepts. Students can then warm up with a series of multiple-choice questions before diving into the free-response problems which constitute the bulk of the book. The first few problems in each chapter are derivations of key results/theorems that are useful when solving other problems. While the book is calculus-based, it can also easily be used in algebra-based courses. The problems that require calculus (only a sixth of the total number) are listed in an appendix, allowing students to steer clear of those if they wish. Additional details: (1) Features 150 multiple-choice questions and nearly 250 free-response problems, all with detailed solutions. (2) Includes 350 figures to help students visualize important concepts. (3) Builds on solutions by frequently including extensions/variations and additional remarks. (4) Begins with a chapter devoted to problem-solving strategies in physics. (5) A valuable supplement to the assigned textbook in any introductory mechanics course.

Geometric Algebra for Physicists


Author: Chris Doran,Anthony Lasenby

Publisher: Cambridge University Press

ISBN: 1139643142

Category: Science

Page: 578

View: 6242

Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.

Seven Elements That Have Changed The World

Iron, Carbon, Gold, Silver, Uranium, Titanium, Silicon


Author: John Browne

Publisher: Hachette UK

ISBN: 0297868063

Category: Political Science

Page: 288

View: 9985

'Fascinating and enjoyable ... enthused with insight' - Brian Cox Uranium, carbon, iron, titanium, gold, silver and silicon - former BP CEO John Browne explains how seven elements are shaping the 21st century, for good and for bad. Humans have put the Earth's resources to extraordinary use, but not always for the benefit of humankind. SEVEN ELEMENTS vividly describes how iron, carbon, gold, silver, uranium, titanium and silicon have shaped the world around us - for good and for bad. This book takes you on an adventure of human passion, ingenuity and discovery, but it is a journey that is far from over: we continue to find surprising new uses for each of these seven key elements. Discover how titanium pervades modern consumer society, how natural gas is transforming the global energy sector and how an innovative new form of carbon could be starting a technological revolution. SEVEN ELEMENTS is a unique mix of science, history and politics, interwoven with the author's extensive personal and professional experience.

The Glass Closet

Why Coming Out Is Good Business


Author: John Browne

Publisher: Harper Collins

ISBN: 0062316982

Category: Biography & Autobiography

Page: 240

View: 5578

Part memoir and part social criticism, The Glass Closet addresses the issue of homophobia that still pervades corporations around the world and underscores the immense challenges faced by LGBT employees. In The Glass Closet, Lord John Browne, former CEO of BP, seeks to unsettle business leaders by exposing the culture of homophobia that remains rampant in corporations around the world, and which prevents employees from showing their authentic selves. Drawing on his own experiences, and those of prominent members of the LGBT community around the world, as well as insights from well-known business leaders and celebrities, Lord Browne illustrates why, despite the risks involved, self-disclosure is best for employees—and for the businesses that support them. Above all, The Glass Closet offers inspiration and support for those who too often worry that coming out will hinder their chances of professional success.

Geometric Algebra for Computer Science

An Object-Oriented Approach to Geometry


Author: Leo Dorst,Daniel Fontijne,Stephen Mann

Publisher: Elsevier

ISBN: 0080553109

Category: Computers

Page: 664

View: 4713

Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down. * The first book on Geometric Algebra for programmers in computer graphics and entertainment computing * Written by leaders in the field providing essential information on this new technique for 3D graphics * This full colour book includes a website with GAViewer, a program to experiment with GA

Practical Graph Analytics with Apache Giraph


Author: Roman Shaposhnik,Claudio Martella,Dionysios Logothetis

Publisher: Apress

ISBN: 1484212517

Category: Computers

Page: 315

View: 6204

Practical Graph Analytics with Apache Giraph helps you build data mining and machine learning applications using the Apache Foundation’s Giraph framework for graph processing. This is the same framework as used by Facebook, Google, and other social media analytics operations to derive business value from vast amounts of interconnected data points. Graphs arise in a wealth of data scenarios and describe the connections that are naturally formed in both digital and real worlds. Examples of such connections abound in online social networks such as Facebook and Twitter, among users who rate movies from services like Netflix and Amazon Prime, and are useful even in the context of biological networks for scientific research. Whether in the context of business or science, viewing data as connected adds value by increasing the amount of information available to be drawn from that data and put to use in generating new revenue or scientific opportunities. Apache Giraph offers a simple yet flexible programming model targeted to graph algorithms and designed to scale easily to accommodate massive amounts of data. Originally developed at Yahoo!, Giraph is now a top top-level project at the Apache Foundation, and it enlists contributors from companies such as Facebook, LinkedIn, and Twitter. Practical Graph Analytics with Apache Giraph brings the power of Apache Giraph to you, showing how to harness the power of graph processing for your own data by building sophisticated graph analytics applications using the very same framework that is relied upon by some of the largest players in the industry today.

Beyond Business

An Inspirational Memoir From a Visionary Leader


Author: John Browne

Publisher: Weidenfeld & Nicolson

ISBN: 0297859188

Category: Biography & Autobiography

Page: 288

View: 7350

An inspirational memoir from a remarkable leader. Once a lacklustre organisation, BP became one of the world's biggest, most successful and most admired companies in the new millennium. John Browne, the company's CEO for 12 years, invented the oil 'supermajor' and led the way on issues such as climate change, human rights and transparency. In BEYOND BUSINESS, Browne brings to life what he learned about leadership in a tough industry. His story encompasses the insights gained as he transformed a national company, challenged an entire industry and prompted political and business leaders to change. He takes us across the world on adventures that include going toe-to-toe with both tyrants and elected leaders, and involve engineering feats which in many ways rival those of going to the moon. And he shares his views on the true purpose of business and the leadership needed to tackle the grand challenges of our era. It is also a story of failure and human frailty, as Browne reveals how his private and public lives collided at frightening speed in full view of the world, prompting his abrupt resignation as CEO of BP.