Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds
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 p...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Cham
Springer Nature
2017
Springer Springer International Publishing AG Springer International Publishing |
| Edition | 1 |
| Series | Solid Mechanics and Its Applications |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9783319562643 3319562649 9783319562636 3319562630 |
| ISSN | 0925-0042 2214-7764 |
| DOI | 10.1007/978-3-319-56264-3 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Mühlich, Uwe |
| Author_xml | – sequence: 1 fullname: Muhlich, Uwe |
| BackLink | https://cir.nii.ac.jp/crid/1130000793944951936$$DView record in CiNii |
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| Copyright | Springer International Publishing AG 2017 |
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| DOI | 10.1007/978-3-319-56264-3 |
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| Notes | Includes bibliographical references and index |
| OCLC | 983204384 |
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| PageCount | 134 |
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| PublicationSeriesTitle | Solid Mechanics and Its Applications |
| PublicationSeriesTitleAlternate | Solid Mechanics, Applicat. |
| PublicationYear | 2017 |
| Publisher | Springer Nature Springer Springer International Publishing AG Springer International Publishing |
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| RelatedPersons | Gladwell, Graham M. L. Klarbring, Anders Barber, J. R. |
| RelatedPersons_xml | – sequence: 1 givenname: Graham M. L. surname: Gladwell fullname: Gladwell, Graham M. L. organization: University of Waterloo, Waterloo, Canada – sequence: 2 givenname: J. R. surname: Barber fullname: Barber, J. R. organization: University of Michigan, Ann Arbor, USA – sequence: 3 givenname: Anders surname: Klarbring fullname: Klarbring, Anders organization: Mechanical Engg, A Building, Linköping University, Linköping, Sweden |
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| Snippet | Presents the fundamentals of modern tensor calculus for students in engineering and applied physics, emphasizing those aspects that are crucial for applying... |
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| SubjectTerms | Calculus of tensors Classical and Continuum Physics Classical mechanics Solid mechanics Engineering Manifolds (Mathematics) Mathematical Applications in the Physical Sciences Mathematical Methods in Physics Mathematical physics Solid Mechanics |
| TableOfContents | 6.3 Gradient of a Scalar Field and Related Concepts in mathbbRN -- 6.4 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 Fields -- 7.5 Flow of Vector Fields and the Lie Derivative -- 7.6 Outlook and Further Reading -- References -- Appendix Solutions for Selected Problems -- Index Intro -- 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 -- 3.2 Linear Independence and Basis -- 3.3 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 -- 4.7 Generalized Kronecker Symbol -- 4.8 The Spaces Λk mathcalV and Λk mathcalV* -- 4.9 Properties of the Exterior Product and the Star-Operator -- 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 |
| Title | Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds |
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