Physical description 
1 online resource. 
Series 
Solid mechanics and its applications ; volume 230 

Solid mechanics and its applications ; v. 230.


Springer Engineering eBooks 2017 English+International

Contents 
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 FiniteDimensional Real Vector Space; 3.1 Definitions. 

4.7 Generalized Kronecker Symbol4.8 The Spaces Λk mathcalV and Λk mathcalV*; 4.9 Properties of the Exterior Product and the StarOperator; 4.10 Relation with Classical Linear Algebra; References; 5 Affine Space and Euclidean Space; 5.1 Definitions and Basic Notions; 5.2 Alternative Definition of an Affine Space by Hybrid Addition; 5.3 Affine Mappings, Coordinate Charts and Topological Aspects; References; 6 Tensor Analysis in Euclidean Space; 6.1 Differentiability in mathbbR and Related Concepts Briefly Revised; 6.2 Generalization of the Concept of Differentiability. 

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 Multilinear 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 Multilinear Forms; 4.6 Symmetric and SkewSymmetric Tensors. 

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. 

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. 
Bibliography 
Includes bibliographical references and index. 
Summary 
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. 
Other author 
SpringerLink issuing body.

Subject 
Calculus of tensors.


Manifolds (Mathematics)


Electronic books. 
ISBN 
9783319562643 (electronic bk.) 

3319562649 (electronic bk.) 

9783319562636 (print) 

3319562630 
Standard Number 
10.1007/9783319562643 
