Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as $\Delta P/P=-\gamma \Delta V/V$?

Question

Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as

$$\frac{\Delta P}{P}=-\gamma \frac{\Delta V}{V},$$

where $P_2-\Delta P=P_1$ and $V_2+\Delta V=V_1$ as in this website

http://newt.phys.unsw.edu.au/jw/Helmholtz.html

Answer 1

It's only an approximation for small changes in pressure and volume. We can write your original equation as $$PV^{\gamma}=C$$ If we take the differential of this relationship using the product rule, we obtain:$$(dP)V^{\gamma}+\gamma PV^{\gamma-1}dV=0$$Dividing this by $PV^{\gamma}$ yields: $$\frac{dP}{P}+\gamma \frac{dV}{V}=0$$or, equivalently, $$\frac{dP}{P}=-\gamma \frac{dV}{V}$$