Regarding the infinitesimal form of the First Law of Thermodynamics
The first law of thermodynamics is a statement of energy conservation thus and defines the internal energy E as an extensive state function. In an infinitesimal transformation, the first then law reduces to
$$\mathrm{d}E = \mathrm{d}Q + \mathrm{d}W$$

where $\mathrm{d}E$ is a total (exact) differential for infinitesimal transformation. 

However, $\mathrm{d}Q$ and $\mathrm{d}W$ are not exact ($Q$ and $W$ are not state functions); $Q$ and $W$ in a thermodynamics transformation are process-dependent. All of these are properties of functions of more than one variables