The covariant and contravariant metric tensors are inverses of each other that

The covariant and contravariant metric tensors are

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The covariant and contravariant metric tensors are inverses of each other, that is: [ g ij ] = g ij - 1 g ij = [ g ij ] - 1 (249) Hence: g ik g kj = δ i j g ik g kj = δ j i (250) It is common to reserve the term metric tensor to the covariant form and call the con- travariant form, which is its inverse, the associate or conjugate or reciprocal metric tensor . As a tensor, the metric has a significance regardless of any coordinate system although it requires a coordinate system to be represented in a specific form. For orthog-
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6 METRIC TENSOR 152 onal coordinate systems the metric tensor is diagonal , i.e. g ij = g ij = 0 for i 6 = j . As indicated before, for orthonormal Cartesian coordinate systems in a 3D space, the metric tensor is given in its covariant and contravariant forms by the 3 × 3 unit matrix, that is: [ g ij ] = [ δ ij ] = 1 0 0 0 1 0 0 0 1 = δ ij = g ij (251) For cylindrical coordinate systems with coordinates ( ρ, φ, z ), the metric tensor is given in its covariant and contravariant forms by: [ g ij ] = 1 0 0 0 ρ 2 0 0 0 1 g ij = 1 0 0 0 1 ρ 2 0 0 0 1 (252) while for spherical coordinate systems with coordinates ( r, θ, φ ), the metric tensor is given in its covariant and contravariant forms by: [ g ij ] = 1 0 0 0 r 2 0 0 0 r 2 sin 2 θ g ij = 1 0 0 0 1 r 2 0 0 0 1 r 2 sin 2 θ (253) As seen, all these metric tensors are diagonal since all these coordinate systems are orthogonal. We also notice that all the corresponding diagonal elements of the covariant and contravariant types are reciprocals of each other. This can be easily explained by the fact that these two types are inverses of each other, plus the fact that the inverse of an invertible diagonal matrix is a diagonal matrix obtained by taking the reciprocal of the corresponding diagonal elements of the original matrix, as stated in § Inverse of Matrix.
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6.1 Exercises 153 6.1 Exercises 6.1 Describe in details, using mathematical tensor language when necessary, the metric tensor discussing its rank, purpose, designations, variance types, symmetry, its role in the definition of distance, and its relation to the covariant and contravariant basis vectors. 6.2 What is the relation between the covariant and contravariant types of the metric tensor? Express this relation mathematically. Also define mathematically the mixed type metric tensor. 6.3 Correct, if necessary, the following equations: g i j = δ i i g ij = E i · E j ( ds ) = g ij dx i dx j g ij = E i · E j E i · E j = δ j i 6.4 What “flat metric” means? Give an example of a coordinate system with a flat metric. 6.5 Describe the index-shifting (raising/lowering) operators and their relation to the met- ric tensor. How these operators facilitate the transformation between the covariant, contravariant and mixed types of a given tensor? 6.6 Find from the Index all the names and labels of the metric tensor in its covariant, contravariant and mixed types.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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