Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds 1st Edition by Uwe Mühlich ISBN 3319562649 9783319562643

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Product details:

ISBN 10:  3319562649
ISBN 13: 978-3319562643
Author:      Uwe Mühlich

Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds 1st Table of contents:

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

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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

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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

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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 ℳ and Λk ℳ*
4.9 Properties of the Exterior Product and the Star-Operator
4.10 Relation with Classical Linear Algebra
References

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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

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6. Tensor Analysis in Euclidean Space

6.1 Differentiability in ℝ and Related Concepts Briefly Revised
6.2 Generalization of the Concept of Differentiability
6.3 Gradient of a Scalar Field and Related Concepts in ℝ^N
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 ℝ^N and Integration
6.9 Exterior Derivative and Stokes’ Theorem in Form Language
References

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7. A Primer on Smooth Manifolds

7.1 Introduction
7.2 Basic Concepts Regarding Analysis on Surfaces in ℝ³
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

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