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
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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 ]
...

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 ]
...