# Infinitesimal input, macroscopic output

I must admit that I never got well how physicists handle infinitesimal quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and Shubin's The Schrödinger Equation) destabilize me somewhat:

It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs. To appreciate what I'm hinting at consider [...] electronics, [...] transistors, [...] TV[...].

Can you provide me with some simple example of such a phenomenon? In my opinion this could not be possible, because as soon as you start introducing infinitesimal quantities, all subsequent equations need be truncated to first-order. So an equation like

$$\text{macroscopic output}=F(\text{macroscopic input} + \text{infinitesimal variation})$$

cannot be correct. I must be wrong, but why?

• Consider the freezing of water, i.e., a first order phase transition. There is a discontinuity in the order parameter and you cannot possibly Taylor-expand around that. – Lagerbaer Dec 1 '11 at 1:48