What's the difference between $\frac{\delta}{dt}$ and $\frac{d}{dt}$? I have read the few questions on calculi notation, particularly the notations on partial and total derivatives. My question seems to have not been answered, or at least not brought to my attention. If there has been a similar thread, please direct it to me. Thanks :)
Anyways, my question is simple: what's the difference between $\frac{\delta}{dt}$ and $\frac{d}{dt}$? In other words:
$$
\frac{\delta x}{\delta y} \text{ is } \frac{dx}{dy} \text{?}
$$
 A: In thermodynamics we conventionnally use $\delta Q$ insead of $\partial Q$ or $dQ$ to make it clear that the heat $Q$ is not a thermodynamical-state unlike the temperature T, entropy S, enthalpy H, total_energy U, Pressure P. For a thermodynamical state $X$ you can write $dX$ or $\partial X$, but for a quantity which is not a thermo-state we dont use the conventional differential signs $\partial$ or $d$.
In other words in thermodynamics writing $$\frac{\delta Q}{dt}$$ means that you are calculating the total time derivative of a quantitiy which is not as thermo-state <=> Q doesn't respect Schwarz's equality.
In analytical mechanics $\delta$ represents the differential of virtual quantities like virtual displacement $\delta \vec{r}$, virtual work $\delta W$, etc...
But of cours those notations are conventions between physicists.
A: We have a very good explanation on the $\delta$ notation in the Chapter 6 of the book "Classical Dynamics" by Thornton and Marion. Despite the fact that it is a book for physicists you will see the explanation of such a notation by means of the calculus of variations. It's difference from the other simbols for derivatives is also covered there.
