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Tensor Spaces and Exterior Algebra

Tensor Spaces and Exterior Algebra PDF Author: Takeo Yokonuma
Publisher: American Mathematical Soc.
ISBN: 9780821827963
Category : Mathematics
Languages : en
Pages : 148

Book Description
This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. to facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. in particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.

Tensor Spaces and Exterior Algebra

Tensor Spaces and Exterior Algebra PDF Author: Takeo Yokonuma
Publisher: American Mathematical Soc.
ISBN: 9780821827963
Category : Mathematics
Languages : en
Pages : 148

Book Description
This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. to facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. in particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.

Linear Algebra Via Exterior Products

Linear Algebra Via Exterior Products PDF Author: Sergei Winitzki
Publisher: Sergei Winitzki
ISBN: 140929496X
Category : Science
Languages : en
Pages : 286

Book Description
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.

Contact Geometry and Nonlinear Differential Equations

Contact Geometry and Nonlinear Differential Equations PDF Author: Alexei Kushner
Publisher: Cambridge University Press
ISBN: 0521824761
Category : Mathematics
Languages : en
Pages : 472

Book Description
Shows novel and modern ways of solving differential equations using methods from contact and symplectic geometry.

Geometric Multivector Analysis

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

Book Description
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.

Exterior Calculus: Theory and Cases

Exterior Calculus: Theory and Cases PDF Author: Carlos Polanco
Publisher: Bentham Science Publishers
ISBN: 9814998796
Category : Mathematics
Languages : en
Pages : 141

Book Description
Exterior calculus is a branch of mathematics which involves differential geometry. In Exterior calculus the concept of differentiations is generalized to antisymmetric exterior derivatives and the notions of ordinary integration to differentiable manifolds of arbitrary dimensions. It therefore generalizes the fundamental theorem of calculus to Stokes' theorem. This textbook covers the fundamental requirements of exterior calculus in curricula for college students in mathematics and engineering programs. Chapters start from Heaviside-Gibbs algebra, and progress to different concepts in Grassman algebra. The final section of the book covers applications of exterior calculus with solutions. Readers will find a concise and clear study of vector calculus and differential geometry, along with several examples and exercises. The solutions to the exercises are also included at the end of the book. This is an ideal book for students with a basic background in mathematics who wish to learn about exterior calculus as part of their college curriculum and equip themselves with the knowledge to apply relevant theoretical concepts in practical situations.

Differential Forms and Connections

Differential Forms and Connections PDF Author: R. W. R. Darling
Publisher: Cambridge University Press
ISBN: 9780521468008
Category : Mathematics
Languages : en
Pages : 288

Book Description
Introducing the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--this textbook covers both classical surface theory, the modern theory of connections, and curvature. With no knowledge of topology assumed, the only prerequisites are multivariate calculus and linear algebra.

Exterior Analysis

Exterior Analysis PDF Author: Erdogan Suhubi
Publisher: Elsevier
ISBN: 0124159281
Category : Technology & Engineering
Languages : en
Pages : 780

Book Description
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

Algebra I

Algebra I PDF Author: N. Bourbaki
Publisher: Springer Science & Business Media
ISBN: 9783540642435
Category : Mathematics
Languages : en
Pages : 750

Book Description
An exposition of the fundamentals of general, linear and multilinear algebra. The first chapter introduces the basic objects: groups, actions, rings, fields. The second chapter studies the properties of modules and linear maps, and the third investigatesalgebras, particularly tensor algebras.

Geometry from a Differentiable Viewpoint

Geometry from a Differentiable Viewpoint PDF Author: John McCleary
Publisher: Cambridge University Press
ISBN: 0521116074
Category : Mathematics
Languages : en
Pages : 375

Book Description
A thoroughly revised second edition of a textbook for a first course in differential/modern geometry that introduces methods within a historical context.

Mathematics For Physics: An Illustrated Handbook

Mathematics For Physics: An Illustrated Handbook PDF Author: Adam Marsh
Publisher: World Scientific
ISBN: 9813233931
Category : Science
Languages : en
Pages : 301

Book Description
This unique book complements traditional textbooks by providing a visual yet rigorous survey of the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The exposition is pedagogical but compact, and the emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints. Certain topics which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations, are avoided. The primary physical models targeted are general relativity, spinors, and gauge theories, with notable chapters on Riemannian geometry, Clifford algebras, and fiber bundles.