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.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Derivative of an Integral

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