e one upper and one lower since they are dummy indices and hence the position

# E one upper and one lower since they are dummy

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, i.e. one upper and one lower, since they are dummy indices and hence the position of the index of the unbarred coordinate should be opposite to its position in the unbarred tensor, (ii) an upper index in the denominator is in lieu of a lower index in the numerator, and (iii) the order of the coordinates should match the order of the indices in the tensor: ¯ A ijkl = x n ¯ x i x p ¯ x j x q ¯ x k x r ¯ x l A npqr ¯ A ijkl = ¯ x i x n ¯ x j x p ¯ x k x q ¯ x l x r A npqr (78) ¯ A ij kl = ¯ x i x n ¯ x j x p x q ¯ x k x r ¯ x l A np qr We also replaced the “ \$ ” sign in the final set of equations with the strict equality sign “=” as the equations now are complete. The covariant and contravariant types of a tensor are linked through the metric tensor , as will be detailed later in the book (refer to § 6). As indicated before, for orthonormal Cartesian systems there is no difference between covariant and contravariant tensors, and hence the indices can be upper or lower although it is common to use lower indices in this case. A tensor of m contravariant indices and n covariant indices may be called type ( m, n ) tensor. When one or both variance types are absent, zero is used to refer to the absent variance type in this notation. Accordingly, A k ij is a type ( 1 , 2 ) tensor, B ik is a type ( 2 , 0 ) tensor, C m is a type ( 0 , 1 ) tensor, and D ts pqr is a type ( 2 , 3 ) tensor. The vectors providing the basis set for a coordinate system are of covariant type when they are tangent to the coordinate axes, and they are of contravariant type when they are perpendicular to the local surfaces of constant coordinates. These two sets, like the
2.6.1 Covariant and Contravariant Tensors 64 tensors themselves, are identical for orthonormal Cartesian systems. Formally, the covariant and contravariant basis vectors are given respectively by: E i = r ∂x i E i = x i (79) where r = x i e i is the position vector in Cartesian coordinates and x i is a general curvi- linear coordinate. As before, a superscript in the denominator of partial derivatives is equivalent to a subscript in the numerator . It should be remarked that in general the basis vectors (whether covariant or contravariant ) are not necessarily of unit length and/or mutually orthogonal although they may be so. [32] The two sets of covariant and contravariant basis vectors are reciprocal systems and hence they satisfy the following reciprocity relation : E i · E j = δ j i (80) where δ j i is the Kronecker delta (refer to § 4.1) which can be represented by the unity matrix (see § Special Matrices). The reciprocity of these two sets of basis vectors is illustrated schematically in Figure 14 for the case of a 2D space. A vector can be represented either by covariant components with contravariant coor- dinate basis vectors or by contravariant components with covariant coordinate basis vectors. For example, a vector A can be expressed as: A = A i E i or A = A i E i (81) where E i and E i are the contravariant and covariant basis vectors respectively. This is illustrated graphically in Figure 15 for a vector A in a 2D space. The use of the covariant

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

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