Definition:Distributional Partial Derivative

Definition

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 partial 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