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Math Help - not sure why this is smooth

  1. #1
    Senior Member slevvio's Avatar
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    not sure why this is smooth

    Define g: V \rightarrow \mathbb{R} smooth where V \subseteq \mathbb{R}^n is open. Then define a map, for a fixed x \in V

    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
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  2. #2
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    Re: not sure why this is smooth

    \frac{\partial{h_i}}{\partial{x^j}} = \int_0^1 t\frac{\partial^2{g}}{\partial{x^ix^j}}(t \mathbf{x}) dt
    Last edited by xxp9; October 11th 2011 at 11:25 AM.
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  3. #3
    Senior Member slevvio's Avatar
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    Re: not sure why this is smooth

    hmmm why is this the case?
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