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

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 33   | 1 | 2 |

Section 1.3.2.3.9.5. Transformation of coordinates

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.3.9.5. Transformation of coordinates

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Let σ be a smooth non-singular change of variables in , i.e. an infinitely differentiable mapping from an open subset Ω of to Ω′ in , whose Jacobian vanishes 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 by is 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 through which 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 non-singular matrix A. Then , and This 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 and Writing symbolically δ as and as , we have: If and f is a function with isolated simple zeros , then in the same symbolic notation where each is analogous to a `Lorentz factor' at zero .