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 [{\bb R}^{n}], i.e. an infinitely differentiable mapping from an open subset Ω of [{\bb R}^{n}] to Ω′ in [{\bb R}^{n}], whose Jacobian [J(\sigma) = \det \left[{\partial \sigma ({\bf x}) \over \partial {\bf x}}\right]] vanishes nowhere in Ω. By the implicit function theorem, the inverse mapping [\sigma^{-1}] from Ω′ to Ω is well defined.

If f is a locally summable function on Ω, then the function [\sigma^{\#} f] defined by [(\sigma^{\#} f)({\bf x}) = f[\sigma^{-1}({\bf x})]] is a locally summable function on Ω′, and for any [\varphi \in {\scr D}(\Omega')] we may write: [\eqalign{{\textstyle\int\limits_{\Omega'}} (\sigma^{\#} f) ({\bf x}) \varphi ({\bf x}) \;\hbox{d}^{n} {\bf x} &= {\textstyle\int\limits_{\Omega'}} f[\sigma^{-1} ({\bf x})] \varphi ({\bf x}) \;\hbox{d}^{n} {\bf x} \cr &= {\textstyle\int\limits_{\Omega'}} f({\bf y}) \varphi [\sigma ({\bf y})]|J(\sigma)| \;\hbox{d}^{n} {\bf y} \quad \hbox{by } {\bf x} = \sigma ({\bf y}).}] In terms of the associated distributions [\langle T_{\sigma^{\#} f}, \varphi \rangle = \langle T_{f}, |J(\sigma)|(\sigma^{-1})^{\#} \varphi \rangle.]

This operation can be extended to an arbitrary distribution T by defining its image [\sigma^{\#} T] under coordinate transformation σ through [\langle \sigma^{\#} T, \varphi \rangle = \langle T, |J(\sigma)|(\sigma^{-1})^{\#} \varphi \rangle,] which is well defined provided that σ is proper, i.e. that [\sigma^{-1}(K)] is compact whenever K is compact.

For instance, if [\sigma: {\bf x} \;\longmapsto\; {\bf x} + {\bf a}] is a translation by a vector a in [{\bb R}^{n}], then [|J(\sigma)| = 1]; [\sigma^{\#}] is denoted by [\tau_{\bf a}], and the translate [\tau_{\bf a} T] of a distribution T is defined by [\langle \tau_{\bf a} T, \varphi \rangle = \langle T, \tau_{-{\bf a}} \varphi \rangle.]

Let [A: {\bf x} \;\longmapsto\; {\bf Ax}] be a linear transformation defined by a non-singular matrix A. Then [J(A) = \det {\bf A}], and [\langle A^{\#} T, \varphi \rangle = |\det {\bf A}| \langle T, (A^{-1})^{\#} \varphi \rangle.] This formula will be shown later (Sections 1.3.2.6.5[link], 1.3.4.2.1.1[link]) to be the basis for the definition of the reciprocal lattice.

In particular, if [{\bf A} = -{\bf I}], where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting [A^{\#} \varphi] by [\breve{\varphi}] we have: [\langle \breve{T}, \varphi \rangle = \langle T, \breve{\varphi} \rangle.] T is called an even distribution if [\breve{T} = T], an odd distribution if [\breve{T} = -T].

If [{\bf A} = \lambda {\bf I}] with [\lambda \gt 0], A is called a dilation and [\langle A^{\#} T, \varphi \rangle = \lambda^{n} \langle T, (A^{-1})^{\#} \varphi \rangle.] Writing symbolically δ as [\delta ({\bf x})] and [A^{\#} \delta] as [\delta ({\bf x}/\lambda)], we have: [\delta ({\bf x}/\lambda) = \lambda^{n} \delta ({\bf x}).] If [n = 1] and f is a function with isolated simple zeros [x_{j}], then in the same symbolic notation [\delta [\;f(x)] = \sum\limits_{j} {1 \over |\;f'(x_{j})|} \delta (x_{j}),] where each [\lambda_{j} = 1/|\;f'(x_{j})|] is analogous to a `Lorentz factor' at zero [x_{j}].








































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