not sure why this is smooth

Oct 2007
347
17
Define \(\displaystyle g: V \rightarrow \mathbb{R}\) smooth where \(\displaystyle V \subseteq \mathbb{R}^n\) is open. Then define a map, for a fixed \(\displaystyle x \in V\)

\(\displaystyle h_i(x) = \int_0^1 \frac{\partial g}{\partial x_i}(tx) dt\)

(we may assume tx is in V)

Can anyone explain to me why this function is smooth?

Thanks for any help
 
Mar 2010
293
91
Beijing, China
\(\displaystyle \frac{\partial{h_i}}{\partial{x^j}} = \int_0^1 t\frac{\partial^2{g}}{\partial{x^ix^j}}(t \mathbf{x}) dt\)
 
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