Heat Current (Thermal Conduction) Why isn’t heat current written as $\textrm{d}Q/\textrm{d}t$ and why only as $\Delta Q/ \Delta t$?
 A: There are a number of historical reasons for this, but the reasons the notation has remained is an important mathematical.  Writing $\frac{dQ}{dt}$ would imply that there was a state function $Q$, of which that expression was the time derivative.  However, such a $Q$ does not exist.  Another way to describe this is that the infinitesimal heat transfer $dQ$ is "not an exact differential."  Similar, the infinitesimal work done by a system, $d$ is also not an exact differential; only by taking the sum do we obtain an exact differential $dE$, where the internal energy $E$ is a true state function.
Since $Q$ and $W$ are not state functions, it is often useful use notation, like $\frac{\Delta Q}{\Delta t}$ that emphasizes that this ratio is not the derivative of a function.  Another notation that is sometimes used instead of $dQ$ is $\not dQ$, for the same reason.
The use of a $\Delta$ in an expression like $\Delta Q$ or $\Delta W$ does not indicate that $\Delta W$ is the difference between two values of $W$, which would be impossible, since $W$ is not a state function.  Instead, $\Delta W$ is just defined as an integral of $\not dW$.  Another similar notation that is frequently used in engineering is $\dot{Q}$, meaning heat transfer per unit time, even though it is not the time derivative of anything.
