This was a little too long to insert into a comment, so I write it as an answer here.
Here's an answer I wrote on MSE How does the idea of a differential dx work if derivatives are not fractions?. There I explain the idea of interpreting them as 1-forms (this is just fancy vocabulary for a simple idea).
Also, if you want to see how some of these basic differential geometric ideas are used in physics, I'd strongly recommend you read Bamberg and Sternberg's A Course of Mathematics for Students of Physics, which is a very readable text. Check out both volumes 1 and 2 (there's stuff on electrostatics, magnetostatics, optics, Maxwell's equations). Thermodynamics specifically is covered in Chapter 22 (the last chapter in Volume 2), and they follow the geometric approach of Caratheodory. To understand this, you don't need to read every single chapter beforehand; you only need to read chapter 5 (basic differential calculus in several variables) and occasionally you'll need the material of chapter 15 (on exterior derivatives, and closed and exact forms). In this language the first law says that if we take the work 1-form $\omega$ and the "heat" $1$-form $\alpha$, then their sum $\alpha+\omega$ is closed (i.e $d(\alpha+\omega)=0$) and thus can be locally written as $dU=\alpha+\omega$ for some smooth function $U$.
So, it is not $\omega$ nor $\alpha$ alone which is closed, but rather their sums. Typically, to reflect this assertion, the classical notation writes it as $\delta Q$ or $\delta W$, or even $dQ, dW$ with a little line crossed over the $d$ (not too sure how to write this in mathjax). Another book (more advanced) which develops the exterior calculus and shows its applications to physics is Applied Exterior Calculus, by Dominic G.B. Edelen.