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

Cover

Table of 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 Finite-Dimensional Real Vector Space; 3.1 Definitions.
  • 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.
  • 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.

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