Distributional derivative - Imported Knowlpedia Content

Definition

Let $T$ be a distribution on an open set $\Omega\subseteq\mathbb R^n$ . For a multi-index $\alpha$ , its distributional derivative $\partial^\alpha T$ is the distribution defined by

\langle\partial^\alpha T,\varphi\rangle =(-1)^{|\alpha|}\langle T,\partial^\alpha\varphi\rangle

for every $\varphi$ in the test-function space. The sign is the one forced by repeated integration by parts. Continuity of $\varphi\mapsto\partial^\alpha\varphi$ on the test-function space ensures that $\partial^\alpha T$ is again a distribution. For $|\alpha|=1$ , this extends the classical partial derivative.

Agreement with classical differentiation

If $f$ is continuously differentiable and $T_f$ is the distribution induced by $f$ , integration by parts gives

\partial_jT_f=T_{\partial_jf}.

No boundary term appears because test functions have compact support inside $\Omega$ . Hence the generalized definition does not alter classical derivatives where those derivatives exist.

Singular derivatives

Distributional differentiation can detect jumps. On $\mathbb R$ , the Heaviside function $H$ defines a regular distribution and satisfies $H'=\delta_0$ . More generally, differentiating a piecewise smooth function produces its ordinary derivative away from jumps together with delta terms whose coefficients are the jump sizes.

Algebraic properties

Distributional derivatives are linear, commute with one another, and exist to every order for every distribution. If $a$ is smooth, the product rule

\partial_j(aT)=(\partial_ja)T+a\,\partial_jT

holds. These properties let differential operators with smooth coefficients act on distributions without requiring pointwise differentiability Hörmander, §2.1.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: §2.1 on differentiation of distributions.
F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, 2nd ed., Cambridge University Press, 1998. Publisher record. Relevant: Chapter 2 on operations on distributions.