(*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 ,

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.