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tensor analysis and differential geometry r.r. van hassel

Tensor Analysis and Differential Geometry by R.R. van Hassel

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Tensor Analysis and Differential Geometry written by R.R. van Hassel, Helmond This is an other great mathematics book cover the following topics.

Preface

Multilinear Algebra
Vector Spaces and Bases, Dual Space. The concept dual basis, Kronecker tensor, Linear Transformations. Index-gymnastics, Inner product, Reciproke basis, Special Bases and Transformation Groups, Tensors, General Definition, Continuation of the general considerations about rs-tensors. Contraction and ⊗., Tensors on Vector Spaces provided with an inner product, Mathematical interpretation of the "Engineering tensor concept", Symmetric and Antisymmetric Tensors, Vector Spaces with a oriented volume, The Hodge Transformation, Exercises, RRvH: Identification V and V∗, RRvH: Summary, The four faces of bilinear maps

Tensor Fields on Rn
Curvilinear Coordinates and Tangent Spaces, Definition of Tensor Fields on Rn, Alternative Definition, Examples of Tensor Fields, The Kronecker Tensor Field, Fundamental Tensor Fields, Volume Forms and Densities, Examples of Curvilinear Coordinates, Polar coordinates on R2, Cylindrical coordinates on R3, Spherical coordinates on R3, Differential Operations on Tensor Fields, The gradient, The Lie derivative, Christoffel symbols on Rn, The covariant derivative on Rn, The exterior derivative, Combinations of the exterior derivative and the Hodge transformation, Combinations of d and ∗ in R2, Combinations of d and ∗ in R3, Combinations of d and ∗ in Rn, The classical vector operations in R3, The gradient, The curl, The divergence, The Laplace operator, Exercises, RRvH: Overall example(s), Helicoidal coordinates

Differential Geometry
Differential geometry of curves in R3, Space curves, The Frenet formulas, Differential geometry of surfaces in R3, Surfaces, The first fundamental tensor field, The second fundamental tensor field, Curves at a surface, The covariant derivative at surfaces, Exercises 134RRvH: Christoffel symbols? 134Christoffel symbols

Manifolds
Differentiable Functions, Manifolds, Riemannian manifolds, Covariant derivatives, The curvature tensor

Appendices
The General Tensor Concept, The Stokes Equations in (Orthogonal) Curvilinear Coordinates, Introduction, The Stress Tensor and the Stokes equations in Cartesian Coordinates, The Stress Tensor and the Stokes equations in Arbitrary Coordinates, The Extended Divergence and Gradient in Orthogonal Curvilinear Coordinates, The Extended Gradient, The Extended Divergence, The theory of special relativity according Einstein and Minovski, Brief sketch about the general theory of special relativity, Lattices and Reciproke Bases. Piezoelectricity, Some tensors out of the continuum mechanics, Thermodynamics and Differential Forms

Index

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