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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 31-32   | 1 | 2 |

## Section 1.3.2.3.9.1. Differentiation

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

#### 1.3.2.3.9.1. Differentiation

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• (a) Definition and elementary properties

If T is a distribution on , its partial derivative with respect to is defined byfor all . This does define a distribution, because the partial differentiations are continuous for the topology of .

Suppose that with f a locally integrable function such that exists and is almost everywhere continuous. Then integration by parts along the axis givesthe integrated term vanishes, since ϕ has compact support, showing that .

The test functions are infinitely differentiable. Therefore, transpositions like that used to define may be repeated, so that any distribution is infinitely differentiable. For instance,where Δ is the Laplacian operator. The derivatives of Dirac's δ distribution are

It is remarkable that differentiation is a continuous operation for the topology on : if a sequence of distributions converges to distribution T, then the sequence of derivatives converges to for any multi-index p, since as An analogous statement holds for series: any convergent series of distributions may be differentiated termwise to all orders. This illustrates how robust' the constructs of distribution theory are in comparison with those of ordinary function theory, where similar statements are notoriously untrue.

• (b) Differentiation under the duality bracket

Limiting processes and differentiation may also be carried out under the duality bracket as under the integral sign with ordinary functions. Let the function depend on a parameter and a vector in such a way that all functionsbe in for all . Let be a distribution, letand let be given parameter value. Suppose that, as λ runs through a small enough neighbourhood of ,

 (i) all the have their supports in a fixed compact subset K of ; (ii) all the derivatives have a partial derivative with respect to λ which is continuous with respect to x and λ.

Under these hypotheses, is differentiable (in the usual sense) with respect to λ near , and its derivative may be obtained by differentiation under the sign':

• (c) Effect of discontinuities

When a function f or its derivatives are no longer continuous, the derivatives of the associated distribution may no longer coincide with the distributions associated to the functions .

In dimension 1, the simplest example is Heaviside's unit step function :Hence , a result long used heuristically' by electrical engineers [see also Dirac (1958)].

Let f be infinitely differentiable for and but have discontinuous derivatives at [ being f itself] with jumps . Consider the functions:The are continuous, their derivatives are continuous almost everywhere [which implies that and almost everywhere]. This yields immediately:Thus the distributional derivatives' differ from the usual functional derivatives by singular terms associated with discontinuities.

In dimension n, let f be infinitely differentiable everywhere except on a smooth hypersurface S, across which its partial derivatives show discontinuities. Let and denote the discontinuities of f and its normal derivative across S (both and are functions of position on S), and let and be defined byIntegration by parts shows thatwhere is the angle between the axis and the normal to S along which the jump occurs, and that the Laplacian of is given byThe latter result is a statement of Green's theorem in terms of distributions. It will be used in Section 1.3.4.4.3.5 to calculate the Fourier transform of the indicator function of a molecular envelope.

### References

Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th ed. Oxford: Clarendon Press.