I was lately wondering about the use of differentials in physics. I mean, usually $dx$ is thought of as a small increment in $x$, but does this have any rigorous meaning mathematically.
Doubts started to appear when I saw the first law of thermodynamics:
$$dU = dQ + dW.$$
What does this even mean? (not thermodynamically, but more mathematically). Again, does this have a somehow rigorous meaning? If it isn't rigorous, how are we able to manipulate them?
It's a shorthand for$$\frac{dU}{dp}=\frac{dQ}{dp}+\frac{dW}{dp}$$for all choices of a parameter $p$ that tracks the system's evolution. The special case where $p$ is the time elapsed is equivalent to the general case by the chain rule.
dQ + dW - dU is path independent when taken together.
– Aron
Sep 07 '21 at 07:53
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.
Thermodynamics deals with real mathematical differentials for the thermodynamic state functions. State functions are real functions of several variables, and different state functions are related via the Legendre transformation (see for its application in Thermodynamics).
As an example, the internal energy can be written as $$ dU = TdS - PdV + \sum_i\mu_idN_i, $$ and interpretation of the coefficients in the differential expansion as the partial derivatives allows to obtain the Maxwell relations, which in mathematical terms are nothing by the expression of the existence of the total differential).
The last term in the equation above however already shows that physicists tend to take liberties with mathematical notation, as $N_i$ in the last term is the number of particle,s i.e., a discrete variable. Similarly, one often stresses that $dQ$ and $dW$ are not real differentials, since they depend on the path that one chooses between the two points, and only their sum is not ambiguous - just as it is the case in math. However, as @J.G. has pointed out, the path is usually implied, and $Q,W$ can be thought as functions of a parameter along this path. Some books specifically use in this case symbols with crossed $d:$ đQ, đW.
I could be wrong, but the first beginning of thermodynamics in the general case is not expressed in complete differentials. As correctly noted above (@Roger Vadim), even many authors specifically introduce other designations for these values. Both expressions (and $dU$ and $\delta Q$) are physically infinitesimal increments, but the latter is not a differential. If you remember the Newton-Leibniz formula $$ \int\limits_{a}^{b}{dF} = F(b) - F(a), $$ then the integral (otherwise: the algebraic sum over the whole "trajectory" of summation) will depend only on the initial and final states of the system, and does not depend on how the system got from state (a) to state (b). At the same time, there are quantities that clearly depend on the shape of the trajectory of motion from (a) to (b).
In this regard, the answer to your question should sound approximately as follows. Some physical functions are total differentials in the sense of physically infinitesimal quantities. "Physically" in this case implies that the incremental magnitude itself is much smaller than some reference dimension, such as the size of a system or a possible quantum of measurement of some instrument. Only when the basic assumptions are made can the corresponding properties be used, and then you should always keep an eye on the meaning of this or that value, such as the same elementary work: $\delta W = \int\limits_{a}^{b}{\left( \vec{F} \cdot d\vec{r} \right)}$.
Strictly speaking, it's probably wrong to write your above equation as you have done so as a differential in one variable has to be referenced to another independent variable.
But most people know what is meant by it, i.e. the corresponding equation written in increments:
ΔU = ΔQ + ΔW
Written like that we can manipulate the equation w.r.t. other variables and convert to differential form by division by another increment and applying a 'tending towards zero' limit.
A derivative can be thought of as a ratio between differentials. For instance, $\frac {dy}{dx}=2$ can be interpreted as saying $dy =2dx$. The equation $dU=dQ+dW$ is just a version of such an equation with three terms. If you'd like, you can divide both sides by $dU$ to get $1 = \frac{dQ}{dU}+\frac{dW}{dU}$. And according to the chain rule, you can divide by any differential; as J.G. said, it's equivalent to $\frac{dU}{dp}=\frac{dQ}{dp}+\frac{dW}{dp}$ for any parameter $p$.
Saying that two expressions of infinitesimals are equal can be considered to be a claim that they are equal "in the limit". Stated rigorously, we can say $\Delta U = \Delta Q+\Delta W+\epsilon$ where $\lim_{\Delta U \rightarrow 0}\frac {\Delta \epsilon}{\Delta U}=0$ (of course, since the relationship between terms is constant, it turns out that $\epsilon$ is identically zero).
