Length of 17 20 78 119 Levi Civita 9 1

Length of 17 20 78 119 levi civita 9 1

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Length of vector, 11, 16, 17, 20, 22–25, 57, 64, 78, 119 Levi-Civita, 9, 10 identity, 10, 107 symbol, 10 tensor, 58, 70, 96 Linear algebra, 1, 89, 117, 118 combination, 50 operation, 158 Lower index, 54, 61–63, 73, 75, 87 Main diagonal, 35, 36, 38, 44 Mapping of spaces, 51 Matrix, 34, 35, 49 algebra, 34, 87, 94, 116, 117 notation, 37, 86, 117, 118 of cofactors, 39 Metric tensor, 54, 63, 149–154, 157, 158 Mixed basis tensor, 65, 79 Kronecker δ , 112 metric tensor, 150, 153, 157 tensor, 50, 57, 60, 61, 65, 74, 77, 79, 87, 150, 151 Modulus, 13, 23 Multiplication of matrices, 36, 37, 45, 46, 86, 89, 94, 117, 118, 143 of matrix by vector, 89, 117, 143 of tensor by scalar, 83–85, 87, 92, 93 of tensors, 69, 83, 85–88, 92, 93, 158 Mutually orthogonal, 16, 22, 64, 78, 103 perpendicular, 12, 17, 20 nabla operator, 7, 13, 29, 31, 42, 120, 121, 124–126, 144, 146, 147 Non- repetitive permutation, 45, 82, 104, 113 scalar tensor, 15, 86 singular matrix, 39 Normal vector, 30, 33 Numerical tensor, 96, 112 Oblique coordinate system, 12, 16 Odd permutation, 52, 75, 98, 99, 101, 113 Opposite cyclic permutation, 28 transformation, 18, 21, 61 Order of tensor, 50, 55, 74 166
Origin of coordinates, 16, 19, 52, 66, 67 Original space, 51 Orthogonal transformation, 66, 101 Orthonormal basis vectors, 103, 105, 113, 114, 124, 125 Cartesian system, 12, 15–17, 23, 41, 54, 55, 63, 64, 103, 149, 152 Outer product, 11, 37, 80, 86, 88–91, 93, 94, 121, 158 Parallelepiped, 27, 28 Parallelogram, 24, 26 Parenthesis notation, 26, 42, 53, 54 Parity, 53, 67, 75 Partial anti-symmetry, 82 derivative, 13–15, 62, 64, 78, 156–158, 160, 161 derivative operator, 158 differentiation, 14, 158 symmetry, 82 Passive transformation, 52 Permutation, 52, 53, 75, 104, 137 of tensor, 83, 91, 92, 95 symbol, 68 tensor, 67, 68, 96–101, 104–106, 109, 111– 114 Physical component, 121 Plane polar coordinate system, 19, 20, 41, 125, 147 Polar tensor, 65 vector, 67, 68, 79 Polyad, 12 Principle of invariance, 48, 74 Product rule of differentiation, 158 Projection, 17, 23, 24, 27 Proper rotation, 52, 70 tensor, 66, 68 transformation, 51, 52, 75, 97 vector, 68 Pseudo scalar, 68, 80 tensor, 65–69, 80, 84, 101, 113 vector, 66–68, 79, 80, 119 Quotient rule, 91, 92, 95 Range of coordinate, 16, 19 of index, 35, 47, 48, 53–55, 86, 133 Rank of tensor, 11–13, 40, 47–50, 55, 57, 58, 74, 85 Reciprocal metric tensor, 151, 154 system, 64 Reciprocity relation, 64, 65, 79 Rectangular Cartesian system, 16, 20, 29, 52, 58, 59 Relative tensor, 68, 69, 80, 88, 94, 101 Repetitive permutation, 45, 82, 104 Ricci, 9, 10 calculus, 9, 10 curvature tensor, 10 Riemann, 9 -Christoffel curvature tensor, 9 Riemannian manifold, 9 metric tensor, 9 167
Right hand rule, 16, 24, 25 twist rule, 33, 34 Rotation, 31, 52, 70, 101 Round bracket, 71, 81 Row of matrix, 26, 28, 35–38, 108, 114, 117 Rules of covariant differentiation, 158–161 of ordinary differentiation, 158–161 Scalar, 11, 12, 35, 40, 47, 49, 50, 56, 57, 70, 74, 76, 77, 156 invariant of tensor, 119, 144 operator, 32, 123, 124 product, 23 triple product, 26–28, 42, 68, 105, 118, 144 Semicolon notation, 14, 160 Sign function, 100 Singular matrix, 36, 39, 45, 46, 144 Skew-symmetric tensor, 71 Solenoidal, 31, 43, 132, 147 Spherical coordinate system, 7, 19–21, 41, 125, 147, 152, 154, 156 Square bracket, 13, 71, 81 matrix, 35, 36, 38, 39, 44 Stokes, 116 theorem, 33, 34, 44, 133, 134, 147 Subscript index, 13, 14, 47, 50, 51, 59, 60, 64, 157 Substitution operator, 102 Summation convention, 53, 54, 75, 102, 124, 125 Super factorial function, 100 Superscript index, 47, 50, 51, 60, 62, 64, 78, 157 Surface element, 33, 133 Symbolic notation, 13, 37, 49, 86, 88, 117, 122 Symmetric tensor, 59, 70–73, 81, 82, 97, 104, 114, 150 Symmetry of tensor, 73, 81, 105, 114, 153 Tensor, 8, 11–13, 49 algebra, 2, 11, 15, 83, 117 analysis, 9 calculus, 1, 2, 8–11, 14, 15, 23, 40, 49, 50, 53, 54, 58, 69, 74, 77, 96, 119, 134, 148 density, 65, 69 equality, 50, 53, 55, 56, 62, 66, 74, 76, 80 expression, 50, 53, 55, 56, 66, 74–76, 80 identity, 50 notation, 29, 47, 49, 58, 78, 105, 116, 119–121, 127, 132–134, 143–147 operation, 83, 92 term, 50, 53, 55, 56, 66, 67, 74, 76, 80, 158

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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