partial derivatives of Dirac functions

Question

I was reading my courses, and couldn't understand an exercise:

The question was: simplify in $D'(\Bbb R^n)$ $\sum_{i=1}^n x_i\frac{\partial \delta}{\partial{x_i}}$

on my correction, I had written:

$\varphi \in D(\Bbb R^n)$,

$<\sum_{i=1}^n x_i\frac{\partial \delta}{\partial{x_i}}, \varphi>=\sum_{i=1}^n <x_i\frac{\partial \delta}{\partial{x_i}},\varphi>$
$=\sum_{i=1}^n <\frac{\partial \delta}{\partial{x_i}},x_i\varphi>$
$=\sum_{i=1}^n -<\delta,\frac{\partial{(x_i\varphi)} }{\partial{x_i}}>$
$=\sum_{i=1}^n -<\delta,\varphi + x_i\frac{\partial{\varphi} }{\partial{x_i}}>$
$=\sum_{i=1}^n -<\delta,\varphi> -<\delta,x_i\frac{\partial{\varphi} }{\partial{x_i}}>$
$=\sum_{i=1}^n -<\delta,\varphi> - 0$
$=-n<\delta,\varphi>$

above, I don't understand why for each i, $<\delta,x_i\frac{\partial{\varphi} }{\partial{x_i}}>=0$

Answer 1

Remember that $\langle \delta,f\rangle = f(0)$. Because $x_i(0) = 0$, we're done.