In this demonstration we use rank 4 tensors as examples since this is

In this demonstration we use rank 4 tensors as

This preview shows page 61 - 65 out of 171 pages.

In this demonstration we use rank-4 tensors as examples since this is sufficiently general
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2.6.1 Covariant and Contravariant Tensors 61 and hence adequate to elucidate the rules for transforming tensors of any rank. The demonstration is based on the assumption that the transformation is taking place from the unbarred system to the barred system; the same rules should apply for the opposite transformation from the barred system to the unbarred system. We use the sign “ $ ” for the equality in the transitional steps to indicate that the equalities are under construction and are not complete. We start with the very generic equations between the barred tensor ¯ A and the unbarred tensor A for the three types: ¯ A $ A (covariant) ¯ A $ A (contravariant) (73) ¯ A $ A (mixed) We assume that the barred tensor and its coordinates are indexed with ijkl and the unbarred are indexed with npqr , so we add these indices in their presumed order and position (lower or upper) paying particular attention to the order in the mixed type: ¯ A ijkl $ A npqr ¯ A ijkl $ A npqr (74) ¯ A ij kl $ A np qr Since the barred and unbarred tensors are of the same type, as they represent the same tensor in two coordinate systems, [30] the indices on the two sides of the equalities should match in their position and order. We then insert a number of partial differential operators on the right hand side of the equations equal to the rank of these tensors, which is 4 in our [30] Similar basis vectors are assumed.
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2.6.1 Covariant and Contravariant Tensors 62 example. These operators represent the transformation rules for each pair of corresponding coordinates, one from the barred and one from the unbarred: ¯ A ijkl $ A npqr ¯ A ijkl $ A npqr (75) ¯ A ij kl $ A np qr Now we insert the coordinates of the barred system into the partial differential operators noting that (i) the positions of any index on the two sides should match, i.e. both upper or both lower, since they are free indices in different terms of tensor equalities, (ii) a superscript index in the denominator of a partial derivative is in lieu of a covariant index in the numerator , [31] and (iii) the order of the coordinates should match the order of the indices in the tensor, that is: ¯ A ijkl $ x i x j x k x l A npqr ¯ A ijkl $ x i x j x k x l A npqr (76) ¯ A ij kl $ x i x j x k x l A np qr For consistency, these coordinates should be barred as they belong to the barred tensor; hence we add bars: ¯ A ijkl $ ¯ x i ¯ x j ¯ x k ¯ x l A npqr ¯ A ijkl $ ¯ x i ¯ x j ¯ x k ¯ x l A npqr (77) ¯ A ij kl $ ¯ x i ¯ x j ¯ x k ¯ x l A np qr [31] The use of upper indices in the symbols of general coordinates is to indicate the fact that the coordinates and their differentials transform contravariantly.
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2.6.1 Covariant and Contravariant Tensors 63 Finally, we insert the coordinates of the unbarred system into the partial differential operators noting that (i) the positions of the repeated indices on the same side should be opposite , i.e. one upper and one lower, since they are dummy indices and hence the
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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