Definition:Distributional Derivative

Definition

Let $\phi \in \map \DD \R$ be a test function.

Let $T \in \map {\DD'} \R$ be a distribution.

The distributional derivative $\ds \dfrac {\d T} {\d x} \in \map {\DD'} \R$ is defined by:

$\map {\dfrac {\d T} {\d x}} \phi := - \map T {\dfrac {\d \phi} {\d x}}$

Higher Derivatives

The $n$th distributional derivative of a distribution $T \in \map \DD \R$ is defined as:

$T^{\paren n} := \begin {cases} \paren {T^{\paren {n - 1}}}' & : n > 0 \\ T & : n = 0 \end {cases}$

Partial Derivative

Let $d \in \N$.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $T \in \map {\DD'} {\R^d}$ be a distribution.

Let $i \in \N : 1 \le i \le d$.

The distributional partial derivative $\ds \dfrac {\partial T} {\partial x_i} \in \map {\DD'} {\R^d}$ is defined by:

$\map {\dfrac {\partial T} {\partial x_i}} \phi := - \map T {\dfrac {\partial \phi} {\partial x_i}}$

Also see

• Results about distributional derivatives can be found here.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense