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I am trying to compute the following partial derivative: $$ \frac{\partial}{\partial x}\iint_{\Omega(x)} f(x,\mathbf{w})d\mathbf{w} $$ with $f:\mathbb{R}\times \mathbb{R}^2\to \mathbb{R}$. In this article: Leibnitz Integral rule, the extension to high dimensional cases is only formulated for vector fields on time-variant surfaces, and I'm not sure how to translate that formulation for my case. How can I do that?

Nicola Lissandrini
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  • In your case, $x\in\mathbb R$ so you could just call it time and it would look the same. Or are you saying that you don't have a formula that expresses $\Omega(x)$ as the flow of some set $\Omega_0$? – Calvin Khor Feb 17 '21 at 11:37
  • I could do that, but then in the formula there are concepts of "velocity" and "surface" that I don't know how to translate in my case, especially since in my case I lose the intuition of "flow" and the related velocities – Nicola Lissandrini Feb 17 '21 at 12:50
  • As your question is currently written, I see no reason for you to lose the intuition. If your set is weird, then the associated flow will be weird and then the derivative is weird, but it will still be given by the formula in the link. If you think the details of your case are important, maybe add them to the question? – Calvin Khor Feb 17 '21 at 14:11
  • I'm not confident with the concept of "flow of a set". How is it defined exactly? Can you give some references? – Nicola Lissandrini Feb 17 '21 at 14:50
  • I don’t know if I was using precisely the right terms but I mean whatever it is that Wikipedia uses. Alternatively just follow the proof of the leibniz integral rule: change variables to a fixed domain and then differentiate normally under the integral sign – Calvin Khor Feb 17 '21 at 15:11
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    I typed this a while ago, maybe this is useful https://math.stackexchange.com/questions/3255547/derivative-of-a-double-integral-over-a-variable-circular-region/3255554#3255554 , but again, you have added no details to your question. if you want specific help,provide specific details – Calvin Khor Feb 17 '21 at 15:23
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    The question you linked should clarify everything I need, thanks a lot. – Nicola Lissandrini Feb 17 '21 at 16:29
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    Does this answer your question? Derivative of a double integral over a variable circular region – SMA.D Feb 27 '21 at 18:24

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