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Author: Peeter Joot
Publisher: Createspace Independent Publishing Platform
ISBN: 9781987598971
Category :
Languages : en
Pages : 282
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Book Description
This book introduces the fundamentals of geometric algebra and calculus, and applies those tools to the study of electromagnetism. Geometric algebra provides a structure that can represent oriented point, line, plane, and volume segments. Vectors, which can be thought of as a representation of oriented line segments, are generalized to multivectors. A full fledged, but non-commutative (i.e. order matters) mul- tiplication operation will be defined for products of vectors. Namely, the square of a vector is the square of its length. This simple rule, along with a requirement that we can sum vectors and their products, essentially defines geometric algebra. Such sums of scalars, vectors and vector products are called multivectors. The reader will see that familiar concepts such as the dot and cross product are related to a more general vector product, and that algebraic structures such as complex numbers can be represented as multivectors. We will be able to utilize generalized complex exponentials to do rotations in arbitrarily oriented planes in space, and will find that simple geometric algebra representations of many geometric transformations are possible. Generalizations of the divergence and Stokes' theorems are required once we choose to work with multivector functions. There is an unfortunate learning curve required to express this gen- eralization, but once overcome, we will be left with a single powerful multivector integration theorem that has no analogue in conventional vector calculus. This fundamental theorem of geo- metric calculus incorporates Green's (area) theorem, the divergence theorem, Stokes' theorems, and complex residue calculus. Multivector calculus also provides the opportunity to define a few unique and powerful Green's functions that almost trivialize solutions of Maxwell's equations. Instead of working separately with electric and magnetic fields, we will work with a hybrid multivector field that includes both electric and magnetic field contributions, and with a mul- tivector current that includes both charge and current densities. The natural representation of Maxwell's equations is a single multivector equation that is easier to solve and manipulate then the conventional mess of divergence and curl equations are familiar to the reader. This book is aimed at graduate or advanced undergraduates in electrical engineering or physics. While all the fundamental results of electromagnetism are derived from Maxwell's equations, there will be no attempt to motivate Maxwell's equations themselves, so existing familiarity with the subject is desirable.
Author: Peeter Joot
Publisher: Createspace Independent Publishing Platform
ISBN: 9781987598971
Category :
Languages : en
Pages : 282
Get Book Here
Book Description
This book introduces the fundamentals of geometric algebra and calculus, and applies those tools to the study of electromagnetism. Geometric algebra provides a structure that can represent oriented point, line, plane, and volume segments. Vectors, which can be thought of as a representation of oriented line segments, are generalized to multivectors. A full fledged, but non-commutative (i.e. order matters) mul- tiplication operation will be defined for products of vectors. Namely, the square of a vector is the square of its length. This simple rule, along with a requirement that we can sum vectors and their products, essentially defines geometric algebra. Such sums of scalars, vectors and vector products are called multivectors. The reader will see that familiar concepts such as the dot and cross product are related to a more general vector product, and that algebraic structures such as complex numbers can be represented as multivectors. We will be able to utilize generalized complex exponentials to do rotations in arbitrarily oriented planes in space, and will find that simple geometric algebra representations of many geometric transformations are possible. Generalizations of the divergence and Stokes' theorems are required once we choose to work with multivector functions. There is an unfortunate learning curve required to express this gen- eralization, but once overcome, we will be left with a single powerful multivector integration theorem that has no analogue in conventional vector calculus. This fundamental theorem of geo- metric calculus incorporates Green's (area) theorem, the divergence theorem, Stokes' theorems, and complex residue calculus. Multivector calculus also provides the opportunity to define a few unique and powerful Green's functions that almost trivialize solutions of Maxwell's equations. Instead of working separately with electric and magnetic fields, we will work with a hybrid multivector field that includes both electric and magnetic field contributions, and with a mul- tivector current that includes both charge and current densities. The natural representation of Maxwell's equations is a single multivector equation that is easier to solve and manipulate then the conventional mess of divergence and curl equations are familiar to the reader. This book is aimed at graduate or advanced undergraduates in electrical engineering or physics. While all the fundamental results of electromagnetism are derived from Maxwell's equations, there will be no attempt to motivate Maxwell's equations themselves, so existing familiarity with the subject is desirable.
Author: David Hestenes
Publisher: Springer Science & Business Media
ISBN: 9789027725615
Category : Mathematics
Languages : en
Pages : 340
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Book Description
Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.
Author: John W. Arthur
Publisher: John Wiley & Sons
ISBN: 0470941634
Category : Science
Languages : en
Pages : 320
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Book Description
This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison. Professors can request a solutions manual by email:
[email protected]
Author: Eduardo Bayro-Corrochano
Publisher: Springer Nature
ISBN: 3030349780
Category : Mathematics
Languages : en
Pages : 609
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Book Description
This book presents a unified mathematical treatment of diverse problems in the general domain of robotics and associated fields using Clifford or geometric alge- bra. By addressing a wide spectrum of problems in a common language, it offers both fresh insights and new solutions that are useful to scientists and engineers working in areas related with robotics. It introduces non-specialists to Clifford and geometric algebra, and provides ex- amples to help readers learn how to compute using geometric entities and geomet- ric formulations. It also includes an in-depth study of applications of Lie group theory, Lie algebra, spinors and versors and the algebra of incidence using the universal geometric algebra generated by reciprocal null cones. Featuring a detailed study of kinematics, differential kinematics and dynamics using geometric algebra, the book also develops Euler Lagrange and Hamiltoni- ans equations for dynamics using conformal geometric algebra, and the recursive Newton-Euler using screw theory in the motor algebra framework. Further, it comprehensively explores robot modeling and nonlinear controllers, and discusses several applications in computer vision, graphics, neurocomputing, quantum com- puting, robotics and control engineering using the geometric algebra framework. The book also includes over 200 exercises and tips for the development of future computer software packages for extensive calculations in geometric algebra, and a entire section focusing on how to write the subroutines in C++, Matlab and Maple to carry out efficient geometric computations in the geometric algebra framework. Lastly, it shows how program code can be optimized for real-time computations. An essential resource for applied physicists, computer scientists, AI researchers, roboticists and mechanical and electrical engineers, the book clarifies and demon- strates the importance of geometric computing for building autonomous systems to advance cognitive systems research.
Author: Leo Dorst
Publisher: Elsevier
ISBN: 0080553109
Category : Juvenile Nonfiction
Languages : en
Pages : 664
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Book Description
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
Author: Eduardo Bayro-Corrochano
Publisher: Springer Science & Business Media
ISBN: 1849961085
Category : Computers
Languages : en
Pages : 527
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Book Description
This useful text offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. Its accessible style is enhanced by examples, figures and experimental analysis.
Author: Miroslav Josipović
Publisher: Springer Nature
ISBN: 3030017567
Category : Mathematics
Languages : en
Pages : 241
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Book Description
This book enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. High school students have the potential to explore it, and undergraduate students can master it. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.
Author: Eduardo Bayro Corrochano
Publisher: Springer Science & Business Media
ISBN: 1461201594
Category : Mathematics
Languages : en
Pages : 607
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Book Description
The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math ematics and physics. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from 1966 to 1967. He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education. Geometric algebra provides a rich, general mathematical framework for the develop ment of multilinear algebra, projective and affine geometry, calculus on a manifold, the representation of Lie groups and Lie algebras, the use of the horosphere and many other areas. This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers.
Author: Eduardo Bayro-Corrochano
Publisher: Springer
ISBN: 3319748300
Category : Technology & Engineering
Languages : en
Pages : 753
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Book Description
The goal of the Volume I Geometric Algebra for Computer Vision, Graphics and Neural Computing is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry. By treating a wide spectrum of problems in a common language, this Volume I offers both new insights and new solutions that should be useful to scientists, and engineers working in different areas related with the development and building of intelligent machines. Each chapter is written in accessible terms accompanied by numerous examples, figures and a complementary appendix on Clifford algebras, all to clarify the theory and the crucial aspects of the application of geometric algebra to problems in graphics engineering, image processing, pattern recognition, computer vision, machine learning, neural computing and cognitive systems.
Author: Larry C. Grove
Publisher: John Wiley & Sons
ISBN: 1118030931
Category : Mathematics
Languages : en
Pages : 228
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Book Description
An authoritative, full-year course on both group theory and ordinary character theory--essential tools for mathematics and the physical sciences One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups. Challenging yet accessible, Groups and Characters features: * An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups * A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics * Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index * Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers.
