>>11158701I considered how going from V to V+dV causes dV to be stuck in the small funnel so dV would press against V in such a way as to make volume expansion linear rather than cubic since any additional increase in dV would cause the pressure to increase by dp but even in the ideal case it would make thermal expansion quadratic rather than linear but there is a better explanation.
Imagine a cube with size length x. Its volume is x^3. Increase length by dx. Its volume now is (x+dx)^3 = x^3+3x^2dx+3xdx^2+dx^3.
We assume that the increase in length dx is so small that dx^2 and dx^3 can be ignored. We end up with an increase of 3x^2dx.
The next step, going from length (x+dx) to (x+dx)^2 would means again that the volume will approximately increase by 3(x+dx)^2dx = 3x^2dx+6xdx^2+3dx^3. Here again we can ignore dx^2 and dx^3 for sufficiently small dx and get 3x^2dx. We now see for any sufficiently small dx, we can calculate the increase in volume solely based on a fixed value 3x^2 where x is the initial length. This argumentation is based on the second rather than the first differential.
