Definition:Distributional Partial Derivative
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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