Distributional Partial Derivatives Commute

Theorem

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

Then in the distributional sense:

$\dfrac {\partial^2 T} {\partial x_i \partial x_j} = \dfrac {\partial^2 T} {\partial x_j \partial x_i}$

where

$i, j \in \N : 1 \le i, j \le d$

Proof

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

 $\ds \map {\dfrac {\partial^2 T} {\partial x_i \partial x_j} } \phi$ $=$ $\ds -\map {\dfrac {\partial T} {\partial x_j} } {\dfrac {\partial \phi} {\partial x_i} }$ Definition of Distributional Partial Derivative $\ds$ $=$ $\ds \map T {\dfrac {\partial^2 \phi} {\partial x_j \partial x_i} }$ Definition of Distributional Partial Derivative $\ds$ $=$ $\ds \map T {\dfrac {\partial^2 \phi} {\partial x_i \partial x_j} }$ Partial Differentiation Operator is Commutative for Continuous Functions, Definition of Test Function $\ds$ $=$ $\ds - \map {\dfrac {\partial T} {\partial x_i} } {\dfrac {\partial \phi} {\partial x_j} }$ Definition of Distributional Partial Derivative $\ds$ $=$ $\ds \map {\dfrac {\partial^2 T} {\partial x_j \partial x_i} } \phi$ Definition of Distributional Partial Derivative