imply summation over their range eg for an n D space we have A i B i n X i 1 A

Imply summation over their range eg for an n d space

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imply summation over their range, e.g. for an n D space we have: A i B i = n X i =1 A i B i = A 1 B 1 + A 2 B 2 + . . . + A n B n (60) δ ij A ij = n X i =1 n X j =1 δ ij A ij (61) ijk A ij B k = n X i =1 n X j =1 n X k =1 ijk A ij B k (62) When dummy indices do not imply summation, the situation must be clarified by enclosing such indices in parentheses or by underscoring or by using upper case letters (with declaration of these conventions) or by adding a clarifying comment like “no summation on repeated indices”. [25] Each tensor index should conform to one of the forthcoming variance transformation rules as given by Eqs. 70 and 71, i.e. it is either covariant or contravariant . For orthonormal Cartesian coordinate systems, the two variance types (i.e. covariant and contravariant) do not differ because the metric tensor is given by the Kronecker delta (refer to § 4.1 and 6) and hence any index can be upper or lower although it is common to use lower indices in such cases. For tensor invariance, a pair of dummy indices should in general be complementary in their variance type, i.e. one covariant and the other contravariant. However, for or- thonormal Cartesian systems the two are the same and hence when both dummy indices are covariant or both are contravariant it should be understood as an indication that the underlying coordinate system is orthonormal Cartesian if the possibility of an error is [25] These precautions are obviously needed if the summation convention is adopted in general but it does not apply in some exceptional cases where repeated indices are needed in the notation with no intention of summation.
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2.3 General Rules 55 excluded. As indicated earlier, tensor order is equal to the number of its indices while tensor rank is equal to the number of its free indices; hence vectors (terms, expressions and equalities) are represented by a single free index and rank-2 tensors are represented by two free indices. The dimension of a tensor is determined by the range taken by its indices. The rank of all terms in legitimate tensor expressions and equalities must be the same. Moreover, each term in valid tensor expressions and equalities must have the same set of free indices (e.g. i, j, k ). Also, a free index should keep its variance type in every term in valid tensor expressions and equations, i.e. it must be covariant in all terms or contravariant in all terms. While free indices should be named uniformly in all terms of tensor expressions and equalities, dummy indices can be named in each term independently , e.g. A i ik + B j jk + C lm lmk D j i = E jk ik + F jm im (63) A free index in an expression or equality can be renamed uniformly using a different symbol, as long as this symbol is not already in use, assuming that both symbols vary over the same range, i.e. have the same dimension.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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