Can You Interchange Derivatives and Integrals in Different Variables?

[Hertz]

Hi, so this is just a quick question about taking a derivative of an integral. Assume that I have some function of position $A(x, y, z)$, then assume I am trying to simplify $$D_i\int{A dx_j}$$ where $i≠j$. So, I'm taking the partial derivative of the integral of A, but the derivative and integral are with respect to different variables.

Considering that the integral is similar to a summation, I intuitively believe that I can take this step:
$$D_i\int{A dx_j}=\int{D_i(A dx_j)}$$

This is where I am confused. Can I take this step?:
$$\int{D_i(A dx_j)}=\int{D_i(A) dx_j}$$

I believe that this is right, but I don't exactly know why. It's as if $dx_j$ is a constant. Is this the case? I can't really see why on my own because I've seen instances where differentials depend on other differentials and obviously I've seen cases where one differential over another differential changes with position, which implies the differentials themselves change with respect to each other. I don't know, I'm just confused about how to think about this. How can I justify taking the $dx_j$ out of the $D_i$?

Lastly, what about the reverse case, where I have $\int{D_i(A) dx_j}$ can I convert this too $D_i\int{A dx_j}$? Thanks!

 

[gopher_p]

The short answer is, yes - switching the order of differentiation/integration in the manner you have indicated is valid. The long answer is https://en.wikipedia.org/wiki/Leibniz_integral_rule.

Also be aware that the $dx$ in the integral notation, at least as far the Reimann integral is concerned, is not the same as the $dx$ of differentials. The integral version is more of a notational artifact that indicates the variable of integration. It's not uncommon to see the two versions mix and mingle, such as when studying differential equations or setting up physics problems and other applications problems. and most of the time it's okay because it works. But just be aware that there's a minor abuse of notation going on in those cases.