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(i) Notation for direct and reciprocal basis vectors: ![[\displaylines{ {\bf a} = {\bf a}_{1}, {\bf b} = {\bf a}_{2}, {\bf c} = {\bf a}_{3}\cr {\bf a^{*}} = {\bf a}^{1}, {\bf b^{*}} = {\bf a}^{2}, {\bf c^{*}} = {\bf a}^{3}.}]](/teximages/bach1o1/bach1o1fd37.svg) Subscripted quantities are associated in tensor algebra with covariant, and superscripted with contravariant transformation properties. Thus the basis vectors of the direct lattice are represented as covariant quantities and those of the reciprocal lattice as contravariant ones.
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(ii) Summation convention: if an index appears twice in an expression, once as subscript and once as superscript, a summation over this index is thereby implied and the summation sign is omitted. For example, ![[{\textstyle\sum\limits_{i} \sum\limits_{j}} x^{i} T_{ij} x\hskip 2pt^{j} \hbox{ will be written } x^{i} T_{ij} x\hskip 2pt^{j}]](/teximages/bach1o1/bach1o1fd38.svg) since both i and j conform to the convention. Such repeating indices are often called dummy indices. The implied summation over repeating indices is also often used even when the indices are at the same level and the coordinate system is Cartesian; there is no distinction between contravariant and covariant quantities in Cartesian frames of reference (see Chapter 3.3![[link]](/graphics/greenarr.gif)
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(iii) Components (coordinates) of vectors referred to the covariant basis are written as contravariant quantities, and vice versa. For example, ![[\eqalign{ {\bf r} &= x{\bf a} + y{\bf b} + z{\bf c} = x^{1} {\bf a}_{1} + x^{2} {\bf a}_{2} + x^{3} {\bf a}_{3} = x^{i} {\bf a}_{i}\cr {\bf h} &= h{\bf a^{*}} + k{\bf b^{*}} + l{\bf c^{*}} = h_{1} {{\bf a}^{1}} + h_{2} {{\bf a}^{2}} + h_{3} {{\bf a}^{3}} = h_{i} {\bf a}^{i}.}]](/teximages/bach1o1/bach1o1fd39.svg)
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