What are equations such as $dz = dx + dy$ called? These look like differential equations:
$$dz = dx + dy,$$
$$dU = TdS - PdV,$$
and many other equations in physics, but they aren't ordinary differential equations or partial differential equations. What are they then?
 A: I don't know what physicists call them. What they are (in physics) are equations that relate small (infinitesimal) changes in physical quantities. For example, 
$$
ds = v \ dt
$$
says that a small change in time causes a small change in position that's proportional to the velocity.
Your second example looks like one from thermodynamics. $T$ is temperature, $P$ pressure and $V$ volume.  (I don't remember enough physics to identify $S$ and $U$.)
Often these equations lead to differential equations - ordinary or partial.
In some formal mathematical physics they are equations that relate quantities defined for manifolds.
A: In mathematics -- especially multivariable calculus and differential geometry -- quantities like $y\,dx$ or $T\,dS - P\,dV$, etc., are called differential $1$-forms.
An equation of the form $df = 0$, where $f$ is a function, is called an exact equation.
More generally, an equation of the form $\alpha = 0$, where $\alpha$ is any differential $1$-form, is called a Pfaffian equation.
