Tensor Structures And Applications
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Tensor Structures and Applications
"Tensor Structures and Applications" "Tensor Structures and Applications" offers a comprehensive and rigorous exploration of tensors, guiding the reader from foundational mathematics to advanced, real-world applications. The book opens with a thorough treatment of the mathematical underpinnings, framing tensors as a natural extension of vectors and matrices through multilinear algebra, formal classification by order and type, and detailed conventions for representation and notation. Further foundational chapters bridge these ideas into the realm of differential geometry, introducing the calculus of tensor fields on manifolds, metric tensors, and their indispensable role in Riemannian geometry and general relativity. Building on these theoretical structures, the text presents an in-depth treatment of tensor decompositions, factorization strategies, and the algorithms that power high-dimensional data analysis. Special attention is given to the unique computational challenges posed by tensor methods, covering efficient data storage, parallel and distributed computing, automatic differentiation, and best practices in leading software libraries. These chapters serve both as a roadmap and a toolbox for researchers and practitioners working with complex, high-order data in numerical and machine learning contexts. The latter sections of the book survey an impressive array of contemporary applications, from signal processing and computer vision—where tensors enable sophisticated tasks such as multi-modal filtering and high-dimensional image analysis—to scientific computing, continuum mechanics, quantum information, and engineering simulations. "Tensor Structures and Applications" concludes by charting emerging research directions, including geometric deep learning, topological data analysis, and multiway network modeling, providing readers with both a solid foundation and a forward-looking perspective in this rapidly evolving field.
Tensor Analysis With Applications In Mechanics
The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions.A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems.This book is a clear, concise, and self-contained treatment of tensors, tensor fields, and their applications. The book contains practically all the material on tensors needed for applications. It shows how this material is applied in mechanics, covering the foundations of the linear theories of elasticity and elastic shells.The main results are all presented in the first four chapters. The remainder of the book shows how one can apply these results to differential geometry and the study of various types of objects in continuum mechanics such as elastic bodies, plates, and shells. Each chapter of this new edition is supplied with exercises and problems — most with solutions, hints, or answers to help the reader progress. An extended appendix serves as a handbook-style summary of all important formulas contained in the book.
Tensor Algebra and Tensor Analysis for Engineers
Author: Mikhail Itskov
language: en
Publisher: Springer Science & Business Media
Release Date: 2007-05-04
There is a large gap between engineering courses in tensor algebra on one hand, and the treatment of linear transformations within classical linear algebra on the other. This book addresses primarily engineering students with some initial knowledge of matrix algebra. Thereby, mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises provided in the book are accompanied by solutions enabling autonomous study. The last chapters deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and might therefore be of high interest for PhD-students and scientists working in this area.