Differentials
We often write \(\frac{dy}{dx} = f'(x)\). The term \(dy\) is called a Differential. It represents an infinitely small change in \(y\) caused by an infinitely small change \(dx\) in \(x\).
\[dy = f'(x) dx\]
This allows us to treat "small changes" as algebraic objects.
Worked Examples
Example 1: Approximating Volume Change
If the side of a cube \(s\) increases by a small amount \(ds\), how much does the volume \(V = s^3\) change?
- \(dV = \frac{dV}{ds} ds = 3s^2 ds\).
- Notice that \(3s^2\) is the surface area of three faces of the cube. The change in volume is essentially adding a thin layer of "paint" to the surface.
The Bridge to Quantum Mechanics
Differentials are the language of Phase Space. In Quantum Mechanics, we talk about the volume element \(dx dp\) in phase space. The Uncertainty Principle tells us that there is a minimum size for this differential: \(\Delta x \Delta p \geq \hbar/2\). We cannot make the "boxes" of space and momentum infinitely small. This fundamental "granularity" of the differential is why the world is quantum and not classical.