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Covariant coordinates – dual or reciprocal space
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.4, p. [ doi:10.1107/97809553602060000900 ]
: 1.1.2.4.2. Reciprocal space | top | pdf | Let us take the scalar products of a covariant vector and a contravariant vector : [using expressions (1.1.2.5), (1.1.2.11) and (1.1.2.13) ]. The relation we obtain ...

Change of basis
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.1, p. [ doi:10.1107/97809553602060000900 ]
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Properties of the metric tensor
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.4.3, p. [ doi:10.1107/97809553602060000900 ]
to the properties of tensors In a change of basis, following (1.1.2.3) and (1.1.2.5), the 's transform according to Let us now consider the scalar products,, of two contravariant basis vectors. Using (1.1.2.11 ...

Metric tensor
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.2, p. [ doi:10.1107/97809553602060000900 ]
: This important property will be used in Section 1.1.2.4.1 . References International Tables for Crystallography (2013). Vol. D. ch. 1.1, p. 5 © International Union of Crystallography 2013 ...

Reciprocal space
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.4.2, p. [ doi:10.1107/97809553602060000900 ]
to the properties of tensors Let us take the scalar products of a covariant vector and a contravariant vector : [using expressions (1.1.2.5), (1.1.2.11) and (1.1.2.13) ]. The relation we obtain,, is identical ...

Basic properties of vector spaces
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2, p. [ doi:10.1107/97809553602060000900 ]
. This set of n vectors forms a basis since (1.1.2.12) can be written with the aid of (1.1.2.13) as The 's are the components of x in the basis . This basis is called the dual basis . By using (1.1.2.11) and (1.1.2.13 ...

Covariant coordinates
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.4.1, p. [ doi:10.1107/97809553602060000900 ]
to the properties of tensors Using the developments (1.1.2.1) and (1.1.2.5), the scalar products of a vector x and of the basis vectors can be written The n quantities are called covariant components, and we ...

Orthonormal frames of coordinates – rotation matrix
Authier, A., International Tables for Crystallography (2013). Vol. D, Section 1.1.2.3, p. [ doi:10.1107/97809553602060000900 ]
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