Why use differential forms for volume elements and stress components

[An1MuS]

1) In a lot of instances i see distances and volumes written in the differential form. For instance
$$dV = dxdydz$$ Why not just write it as V = xyz (or any other letters, but not in the differential form)?

In the image below, dx seems to be the inital length in x axis, and dy in the y axis. Why not just name them x and y ?

2) In trying to have the static equilibrium of a solid, we sum up the tensions and external forces acting on it. I understand from the image below for the o-x direction, $$\sigma_{xx} -\sigma_{xx} + F_x=0$$

but i don't understand the meaning of the differential part.

 

[AlephZero]

I think your two questions are really the same. If you understood "the differential part" of (2), you would also understand (1).

What Calculus courses have you done? To answer these questions, I think we need to know what you know already.

 

[An1MuS]

Where i live my calculus courses are divided into 3 parts, Mathematical Analysis I, II and III. I have done I and II, which means i studied up until triple integrals, higher derivatives, and a lot of other related topics and probably should already know the meaning of this. However in my courses when studying derivatives we focused mainly on the analytical part and not on the meaning of them, and the graphical interpretation. At least that's what i remember, since it was some years ago.