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Tensors, Relativity, and Cosmology, Second Edition, combines relativity, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject that is followed by detailed mathematical derivations. The book includes a section on general relativity that gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes and Penrose processes), and considers the energy-momentum tensor for various solutions. In addition, a section on relativi

Calculus of tensors. --- Cosmology. --- General relativity (Physics). --- Relativity (Physics). --- Relativity (Physics) --- Astronomy --- Deism --- Metaphysics --- Gravitation --- Nonrelativistic quantum mechanics --- Space and time --- Absolute differential calculus --- Analysis, Tensor --- Calculus, Absolute differential --- Calculus, Tensor --- Tensor analysis --- Tensor calculus --- Geometry, Differential --- Geometry, Infinitesimal --- Vector analysis --- Spinor analysis

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Modern imaging techniques and computational simulations yield complex multi-valued data that require higher-order mathematical descriptors. This book addresses topics of importance when dealing with such data, including frameworks for image processing, visualization, and statistical analysis of higher-order descriptors. It also provides examples of the successful use of higher-order descriptors in specific applications and a glimpse of the next generation of diffusion MRI. To do so, it combines contributions on new developments, current challenges in this area, and state-of-the-art surveys.   Compared to the increasing importance of higher-order descriptors in a range of applications, tools for analysis and processing are still relatively hard to come by. Even though application areas such as medical imaging, fluid dynamics, and structural mechanics are very different in nature they face many shared challenges. This book provides an interdisciplinary perspective on this topic with contributions from key researchers in disciplines ranging from visualization and image processing to applications. It is based on the 5th Dagstuhl seminar on Visualization and Processing of Higher Order Descriptors for Multi-Valued Data.   This book will appeal to scientists who are working to develop new analysis methods in the areas of image processing and visualization, as well as those who work with applications that generate higher-order data or could benefit from higher-order models and are searching for novel analytical tools.

Mathematics. --- Visualization. --- Linear and Multilinear Algebras, Matrix Theory. --- Computational Science and Engineering. --- Image Processing and Computer Vision. --- Computer vision. --- Matrix theory. --- Computer science. --- Mathématiques --- Vision par ordinateur --- Informatique --- Visualisation --- Calculus of tensors -- Data processing. --- Information visualization. --- Engineering & Applied Sciences --- Computer Science --- Calculus of tensors --- Data processing. --- Data visualization --- Visualization of information --- Absolute differential calculus --- Analysis, Tensor --- Calculus, Absolute differential --- Calculus, Tensor --- Tensor analysis --- Tensor calculus --- Image processing. --- Algebra. --- Computer mathematics. --- Information science --- Visual analytics --- Geometry, Differential --- Geometry, Infinitesimal --- Vector analysis --- Spinor analysis --- Machine vision --- Vision, Computer --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Informatics --- Science --- Imagination --- Visual perception --- Imagery (Psychology) --- Optical data processing. --- Optical computing --- Visual data processing --- Bionics --- Electronic data processing --- Integrated optics --- Photonics --- Computers --- Computer mathematics --- Mathematics --- Mathematical analysis --- Math --- Optical equipment

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This book presents the science of tensors in a didactic way. The various types and ranks of tensors and the physical basis is presented. Cartesian Tensors are needed for the description of directional phenomena in many branches of physics and for the characterization the anisotropy of material properties. The first sections of the book provide an introduction to the vector and tensor algebra and analysis, with applications to physics,  at undergraduate level. Second rank tensors, in particular their symmetries, are discussed in detail. Differentiation and integration of fields, including generalizations of the Stokes law and the Gauss theorem, are treated. The physics relevant for the applications in mechanics, quantum mechanics, electrodynamics and hydrodynamics is presented. The second part of the book is devoted to  tensors of any rank, at graduate level.  Special topics are irreducible, i.e. symmetric traceless tensors, isotropic tensors, multipole potential tensors, spin tensors, integration and spin-trace formulas, coupling of irreducible tensors, rotation of tensors. Constitutive laws for optical, elastic and viscous properties of anisotropic media are dealt with.  The anisotropic media include crystals, liquid crystals and isotropic fluids, rendered anisotropic  by external orienting fields. The dynamics of tensors deals with phenomena of current research.  In the last section,  the 3D Maxwell equations are reformulated in their 4D version,  in accord with special relativity.

Physics. --- Mathematical Methods in Physics. --- Mathematical Applications in the Physical Sciences. --- Soft and Granular Matter, Complex Fluids and Microfluidics. --- Physical Chemistry. --- Appl.Mathematics/Computational Methods of Engineering. --- Chemistry, Physical organic. --- Mathematical physics. --- Engineering mathematics. --- Physique --- Physique mathématique --- Mathématiques de l'ingénieur --- Physics --- Physical Sciences & Mathematics --- Physics - General --- Physical chemistry. --- Amorphous substances. --- Complex fluids. --- Applied mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Complex liquids --- Fluids, Complex --- Amorphous substances --- Liquids --- Soft condensed matter --- Physical mathematics --- Chemistry, Theoretical --- Physical chemistry --- Theoretical chemistry --- Chemistry --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematics --- Mathematical and Computational Engineering. --- Chemistry, Physical organic --- Chemistry, Organic --- Chemistry, Physical and theoretical --- Calculus of tensors. --- Mathematical modelling.

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This is the fourth and revised edition of a well-received book that aims at bridging the gap between the engineering course of tensor algebra on the one side and the mathematical course of classical linear algebra on the other side. In accordance with the contemporary way of scientific publications, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this monograph addresses graduate students as well as scientists working in this field. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

Engineering. --- Continuum Mechanics and Mechanics of Materials. --- Differential Geometry. --- Linear and Multilinear Algebras, Matrix Theory. --- Mechanics. --- Matrix theory. --- Global differential geometry. --- Materials. --- Ingénierie --- Géométrie différentielle globale --- Mécanique --- Matériaux --- Chemical & Materials Engineering --- Engineering & Applied Sciences --- Materials Science --- Applied Mathematics --- Algebra. --- Differential geometry. --- Continuum mechanics. --- Mechanics of continua --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Differential geometry --- Mathematics --- Mathematical analysis --- Construction --- Industrial arts --- Technology --- Mechanics, Applied. --- Solid Mechanics. --- Classical Mechanics. --- Geometry, Differential --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Engineering mathematics. --- Tensor algebra. --- Calculus of tensors.