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Will moving differentiation from inside, to outside an integral, change the result?

In analysis there is such

theorem: Let $$f:[0,1]\times\mathbb{R}\to\mathbb{R}$$ $$(x,\lambda)\mapsto f(x,\lambda)$$ such continuous function of independent variable $x$ and real parameter $\lambda$ that all partial derivatives exist and continuous everywhere. And let $$I(\lambda)=\int_{0}^{1}f(x,\lambda)dx$$ Then $$\frac{d}{d\lambda}I(\lambda)=\int_{0}^{1}\frac{\partial f(x,\lambda)}{\partial\lambda}dx.$$

Is there some analogous statement for multiple integrals?

Thanks

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    This looks similar to differentiating under the integral sign. Did you check (http://en.wikipedia.org/wiki/Leibniz_integral_rule) and (http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign)? there's a couple multi-dimensional examples.2012-03-06
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    Yes, there is a very general statement where you can replace $[0, 1]$ with any measure space (but you have to place some mild growth conditions on $f$). I posted it at some point on math.SE but I can't find it at the moment.2012-03-06
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    @QiaochuYuan: can you try to find your post, please2012-03-06
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    @QiaochuYuan and Aspirin: http://math.stackexchange.com/questions/12909/2012-04-01

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