conditions for $\int \frac{df}{dt}dx = \frac{d}{dt} \int fdx$

Asked Mar 18 '21 at 21:07

Active Mar 18 '21 at 21:07

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In several literature of partial differential equations I read some arguments as the following:

$$ \int_\Omega \frac{\partial f(x,t)}{\partial t} dx = \frac{\partial }{\partial t} \int_\Omega f(x,t) dx, $$ where $f:\mathbb{R}^2\to \mathbb{R}$ is a function. My question is: what are the sufficient conditions for that the last equality follows? is it always follows?

asked Mar 18 '21 at 21:07

Newinmaths

Boundary being fixed is one – Clemens Bartholdy Mar 18 '21 at 21:13
Leibniz's Rule. – Sean Nemetz Mar 18 '21 at 21:52
3

I'd suggest reading this : https://math.stackexchange.com/questions/12909/will-moving-differentiation-from-inside-to-outside-an-integral-change-the-resu and this for a counter-example: https://math.stackexchange.com/questions/8711/derivative-commuting-over-integral/8712 – Flewer47 Mar 18 '21 at 22:45

conditions for $\int \frac{df}{dt}dx = \frac{d}{dt} \int fdx$

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