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I'm trying to integrate

$$\int_a^b \frac{d}{dt} \left[ \frac{du}{dx}\right]dx.$$

Assume $u$ is a sufficiently smooth function of both $t$ and $x$. Since the integral operator is in space only, can I simply factor out the time derivative operator and rewrite this integral as

$$\frac{d}{dt} \int_a^b \left[ \frac{du}{dx}\right]dx?$$

If so, what property allows me to do this? Linearity of the integral operator? Linearity of the derivative operator? Something else?

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up vote 1 down vote accepted

It relies on several things. First rewriting the derivative as limit:

$$ \int^b_a \lim_{\epsilon\to 0} \frac{\frac{\partial }{\partial x}u(x,t+\epsilon) - \frac{\partial }{\partial x}u(x,t) }{\epsilon} dx\tag{1} $$ The question is: whether above is the same as $$ \lim_{\epsilon\to 0}\frac{1}{\epsilon}\left(\int^b_a \frac{\partial }{\partial x}u(x,t+\epsilon) dx - \int^b_a\frac{\partial }{\partial x}u(x,t)dx\right) \tag{2}$$ From (1) to (2), we used the linearity of the integration, and more importantly, we interchanged the limit and the integration.

About interchanging the limit and the integration, proper assumption has to be made on $u$, for example, $\frac{\partial u}{\partial x}$ and $\frac{\partial^2 u}{\partial x \partial t}$ both being continuous will suffice. More weakly, the conditions of either dominated convergence theorem or monotone convergence theorem allow the interchanging. Therefore, the smoothness of $u$ is also used.

If the partial derivative of $u$ has some weird discontinuities, there are counterexamples that you can not interchange of the derivative and the integral sign.

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