International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 33   | 1 | 2 |

## Section 1.3.2.3.9.5. Transformation of coordinates

G. Bricognea

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#### 1.3.2.3.9.5. Transformation of coordinates

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Let σ be a smooth nonsingular change of variables in , i.e. an infinitely differentiable mapping from an open subset Ω of to Ω′ in , whose Jacobianvanishes nowhere in Ω. By the implicit function theorem, the inverse mapping from Ω′ to Ω is well defined.

If f is a locally summable function on Ω, then the function defined byis a locally summable function on Ω′, and for any we may write:In terms of the associated distributions

This operation can be extended to an arbitrary distribution T by defining its image under coordinate transformation σ throughwhich is well defined provided that σ is proper, i.e. that is compact whenever K is compact.

For instance, if is a translation by a vector a in , then ; is denoted by , and the translate of a distribution T is defined by

Let be a linear transformation defined by a nonsingular matrix A. Then , andThis formula will be shown later (Sections 1.3.2.6.5, 1.3.4.2.1.1) to be the basis for the definition of the reciprocal lattice.

In particular, if , where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting by we have:T is called an even distribution if , an odd distribution if .

If with , A is called a dilation andWriting symbolically δ as and as , we have:If and f is a function with isolated simple zeros , then in the same symbolic notationwhere each is analogous to a `Lorentz factor' at zero .