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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 26   | 1 | 2 |

Section 1.3.2.2.3. Multi-index notation

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.2.3. Multi-index notation

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When dealing with functions in n variables and their derivatives, considerable abbreviation of notation can be obtained through the use of multi-indices.

A multi-index [{\bf p} \in {\bb N}^{n}] is an n-tuple of natural integers: [{\bf p} = (p_{1}, \ldots, p_{n})]. The length of p is defined as[|{\bf p}| = {\textstyle\sum\limits_{i = 1}^{n}}\, p_{i},]and the following abbreviations will be used:[\displaylines{\quad (\hbox{i})\qquad\,\,{\bf x}^{{\bf p}} = x_{1}^{p_{1}} \ldots x_{n}^{p_{n}}\hfill\cr \quad (\hbox{ii})\,\qquad D_{i\,} f = {\partial f \over \partial x_{i}} = \partial_{i}\, f\hfill\cr \quad (\hbox{iii})\qquad D^{{\bf p}} f = D_{1}^{p_{1}} \ldots D_{n}^{p_{n}} f = {\partial^{|{\bf p}|} f \over \partial x_{1}^{p_{1}} \ldots \partial x_{n}^{p_{n}}}\hfill\cr \quad (\hbox{iv})\qquad {\bf q} \leq {\bf p} \hbox{ if and only if } q_{i} \leq p_{i} \hbox{ for all } i = 1, \ldots, n\hfill\cr \quad (\hbox{v})\qquad\,{\bf p} - {\bf q} = (p_{1} - q_{1}, \ldots, p_{n} - q_{n})\hfill\cr \quad (\hbox{vi})\qquad {\bf p}! = p_{1}! \times \ldots \times p_{n}!\hfill\cr \quad (\hbox{vii})\qquad\!\! \pmatrix{{\bf p}\cr {\bf q}\cr} = \pmatrix{p_{1}\cr q_{1}\cr} \times \ldots \times \pmatrix{p_{n}\cr q_{n}\cr}.\hfill}]

Leibniz's formula for the repeated differentiation of products then assumes the concise form[D^{\bf p} (fg) = \sum\limits_{{\bf q} \leq {\bf p}} \pmatrix{{\bf p}\cr {\bf q}\cr} D^{{\bf p} - {\bf q}} f D^{\bf q} g,]while the Taylor expansion of f to order m about [{\bf x} = {\bf a}] reads[f({\bf x}) = \sum\limits_{|{\bf p}| \leq m} {1 \over {\bf p}!} [D^{\bf p} f ({\bf a})] ({\bf x} - {\bf a})^{\bf p} + o (\|{\bf x} - {\bf a}\|^{m}).]

In certain sections the notation [\nabla f] will be used for the gradient vector of f, and the notation [(\nabla \nabla^{T})f] for the Hessian matrix of its mixed second-order partial derivatives:[\displaylines{\nabla = \pmatrix{\displaystyle{\partial \over \partial x_{1}}\cr \vdots\cr\noalign{\vskip6pt} {\displaystyle{\partial \over \partial x_{n}}}\cr}, \quad \nabla f = \pmatrix{\displaystyle{\partial f \over \partial x_{1}}\cr \vdots\cr\noalign{\vskip6pt} {\displaystyle{\partial f \over \partial x_{n}}}\cr},\cr (\nabla \nabla^{T}) f = \pmatrix{\displaystyle{\partial^{2} f \over \partial x_{1}^{2}} &\ldots &{\displaystyle{\partial^{2} f \over \partial x_{1} \partial x_{n}}}\cr \vdots &\ddots &\vdots\cr\noalign{\vskip6pt} {\displaystyle{\partial^{2} f \over \partial x_{n} \partial x_{1}}} &\ldots &{\displaystyle{\partial^{2} f \over \partial x_{n}^{2}}}\cr}.}]








































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