TITLE: What are equations such as $dz = dx + dy$ called?
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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.