Fundamentals of tensor calculus for engineers with a primer on smooth manifolds

This book presents the fundamentals of modern tensor calculus for students in engineering and applied physics, emphasizing those aspects that are crucial for applying tensor calculus safely in Euclidian space and for grasping the very essence of the smooth manifold concept. After introducing the sub...

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Bibliographic Details
Main Author Mühlich, Uwe (Author)
Format Electronic eBook
LanguageEnglish
Published Cham, Switzerland : Springer, 2017.
SeriesSolid mechanics and its applications ; v. 230.
Subjects
Calculus of tensors.
Manifolds (Mathematics)
elektronické knihy
electronic books
Online AccessPlný text

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MARC

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100 1 |a Mühlich, Uwe,  |e author. 
245 1 0 |a Fundamentals of tensor calculus for engineers with a primer on smooth manifolds /  |c Uwe Mühlich. 
264 1 |a Cham, Switzerland :  |b Springer,  |c 2017. 
300 |a 1 online resource 
336 |a text  |b txt  |2 rdacontent 
337 |a počítač  |b c  |2 rdamedia 
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490 1 |a Solid mechanics and its applications ;  |v volume 230 
505 0 |6 880-01  |a Preface; Acknowledgements; Contents; Selected Symbols; 1 Introduction; 1.1 Space, Geometry, and Linear Algebra; 1.2 Vectors as Geometrical Objects; 1.3 Differentiable Manifolds: First Contact; 1.4 Digression on Notation and Mappings; References; 2 Notes on Point Set Topology; 2.1 Preliminary Remarks and Basic Concepts; 2.2 Topology in Metric Spaces; 2.3 Topological Space: Definition and Basic Notions; 2.4 Connectedness, Compactness, and Separability; 2.5 Product Spaces and Product Topologies; 2.6 Further Reading; References; 3 The Finite-Dimensional Real Vector Space; 3.1 Definitions. 
505 8 |a 3.2 Linear Independence and Basis3.3 Some Common Examples for Vector Spaces; 3.4 Change of Basis; 3.5 Linear Mappings Between Vector Spaces; 3.6 Linear Forms and the Dual Vector Space; 3.7 The Inner Product, Norm, and Metric; 3.8 The Reciprocal Basis and Its Relations with the Dual Basis; References; 4 Tensor Algebra; 4.1 Tensors and Multi-linear Forms; 4.2 Dyadic Product and Tensor Product Spaces; 4.3 The Dual of a Linear Mapping; 4.4 Remarks on Notation and Inner Product Operations; 4.5 The Exterior Product and Alternating Multi-linear Forms; 4.6 Symmetric and Skew-Symmetric Tensors. 
505 8 |a 6.3 Gradient of a Scalar Field and Related Concepts in mathbbRN6.4 Differentiability in Euclidean Space Supposing Affine Relations; 6.5 Characteristic Features of Nonlinear Chart Relations; 6.6 Partial Derivatives as Vectors and Tangent Space at a Point; 6.7 Curvilinear Coordinates and Covariant Derivative; 6.8 Differential Forms in mathbbRN and Integration; 6.9 Exterior Derivative and Stokes' Theorem in Form Language; References; 7 A Primer on Smooth Manifolds; 7.1 Introduction; 7.2 Basic Concepts Regarding Analysis on Surfaces in mathbbR3; 7.3 Transition to Smooth Manifolds. 
505 8 |a 7.4 Tangent Bundle and Vector Fields7.5 Flow of Vector Fields and the Lie Derivative; 7.6 Outlook and Further Reading; References; Appendix Solutions for Selected Problems; Index. 
504 |a Includes bibliographical references and index. 
506 |a Plný text je dostupný pouze z IP adres počítačů Univerzity Tomáše Bati ve Zlíně nebo vzdáleným přístupem pro zaměstnance a studenty 
520 |a This book presents the fundamentals of modern tensor calculus for students in engineering and applied physics, emphasizing those aspects that are crucial for applying tensor calculus safely in Euclidian space and for grasping the very essence of the smooth manifold concept. After introducing the subject, it provides a brief exposition on point set topology to familiarize readers with the subject, especially with those topics required in later chapters. It then describes the finite dimensional real vector space and its dual, focusing on the usefulness of the latter for encoding duality concepts in physics. Moreover, it introduces tensors as objects that encode linear mappings and discusses affine and Euclidean spaces. Tensor analysis is explored first in Euclidean space, starting from a generalization of the concept of differentiability and proceeding towards concepts such as directional derivative, covariant derivative and integration based on differential forms. The final chapter addresses the role of smooth manifolds in modeling spaces other than Euclidean space, particularly the concepts of smooth atlas and tangent space, which are crucial to understanding the topic. Two of the most important concepts, namely the tangent bundle and the Lie derivative, are subsequently worked out. 
590 |a SpringerLink  |b Springer Complete eBooks 
650 0 |a Calculus of tensors. 
650 0 |a Manifolds (Mathematics) 
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655 9 |a electronic books  |2 eczenas 
776 0 8 |i Print version:  |a Mühlich, Uwe.  |t Fundamentals of tensor calculus for engineers with a primer on smooth manifolds.  |d Cham, Switzerland : Springer, 2017  |z 3319562630  |z 9783319562636  |w (OCoLC)973914631 
830 0 |a Solid mechanics and its applications ;  |v v. 230. 
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