Primer on the Basics of Tensor Analysis and the Laplacian in Generalized Coordinates (original) (raw)
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A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. We show that once orthonormality is relaxed, a dual basis, together with the contravariant and covariant components, naturally emerges. Manipulating these components requires some skill that can be acquired more easily and quickly once a new notation is adopted. This notation distinguishes multi-component quantities in different coordinate systems by a differentiating sign on the index labelling the component rather than on the label of the quantity itself. This tiny stratagem, together with simple rules openly stated at the beginning of this guide, allows an almost automatic, easy-to-pursue procedure for what is otherwise a cumbersome algebra. By the end of the paper, the reader will be skillful enough to tackle many applications involving tensors of any rank in any coordinate system, without indexmanipulation obstacles standing in the way. V
Springer eBooks, 2019
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2006
The direct notation operates with scalars, vectors and tens ors as physical objects defined in the three dimensional space. A vector (first rank te sor)a is considered as a directed line segment rather than a triple of numbers (co ordinates). A second rank tensorA is any finite sum of ordered vector pairs A = a ⊗ b + . . . + c ⊗ d. The scalars, vectors and tensors are handled as invariant (inde pe nt from the choice of the coordinate system) objects. This is the reason for the us e of the direct notation in the modern literature of mechanics and rheology, e.g. [29 , 3 , 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of v ectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write
Textbook on Tensor Calculus and Differential Geometry
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The Metric Tensor 78 2.1.1 Fundamental Contravariant Tensor 83 2.1.2 Length of a Curve 92 2.2 Associated Tensors 94 2.2.1 Magnitude of a Vector 97 2.2.2 Angle Between Two Vectors 100 2.2.3 Orthogonality of Two Vectors 101 2.3 Some Loci 104 2.3.1 Coordinate Curve 105 2.3.2 Hypersurface 107 2.3.3 Orthogonal Ennuple 111 2.4 Affine Coordinates 113 2.5 Curvilinear Coordinates 114 2.5.1 Coordinate Surfaces 115 2.5.2 Coordinate Curves 116 2.5.3 Line Element 120 2.5.4 Length of a Vector 120 2.5.5 Angle between Two Vectors 122 2.5.6 Reciprocal Base System 124 2.5.7 Partial Derivative 129 2.6 Exercises 133 4. Riemannian Geometry .
Universitext, 2005
Part I Basic Tensor Algebra Tensor Spaces 3 X Contents 3.5 Einstein's contraction of the tensor product 3.6 Matrix representation of tensors 3.6.1 First-order tensors 3.6.2 Second-order tensors 3.7 Exercises 4 Change-of-basis in Tensor Spaces 65 4.1 Introduction 65 4.2 Change of basis in a third-order tensor product space ....... 65 4.3 Matrix representation of a change-of-basis in tensor spaces ... 4.4 General criteria for tensor character 4.5 Extension to homogeneous tensors 4.6 Matrix operation rules for tensor expressions 74 4.6.1 Second-order tensors (matrices) 4.6.2 Third-order tensors 77 4.6.3 Fourth-order tensors 78 4.7 Change-of-basis invariant tensors: Isotropic tensors 4.8 Main isotropic tensors 4.8.1 The null tensor 4.8.2 Zero-order tensor (scalar invariant) 4.8.3 Kronecker's delta 4.9 Exercises