Question: In my chemistry book, the formula for change in entropy is given as :

$$\int{dS} = \int{\frac{δq\_{rev}}{T}}$$

What is the meaning of $δq\_{rev}$? I know that it is the heat exchanged in a reversible process. But why $δq\_{rev}$? Why not $dq\_{rev}$? What is the difference between $δq\_{rev}$ and $dq\_{rev}$? What exactly is the meaning of $δ$ here and how is it different from $d$?

Answer: 
$\delta Q$ represents the inexact differential that means a change in heat from one state to another depends on the path.
$$\Delta Q=\int\_{\mathcal{P}} \delta Q \not= Q(\text{Final})-Q(\text{Initial})$$
It’s an inexact differential because, unlike properties like internal energy, pressure, temperature etc. heat (like work) is not a property so heat doesn’t “change” . The $\delta$ means *amount* of energy transferred in the form of heat. The amount transferred may then result in a *change* in the properties ($dT$, $dp$, $dU$, etc.) of the entities between which energy is transferred.

Hope this helps