Why do we treat differentials as infinitesimals, even when it's not rigorous

Solution 1:

As you noted, learning about differential forms will give you some answers to your questions.

Treating a differential as an infinitesimal isn't always guaranteed to make sense. For instance if I'm integrating over some smooth manifold other than $\mathbb{R}$, the concept of an infinitesimal might not be well-defined. In essence, differential forms were constructed in a particular way to answer some of your questions.

Given some smooth manifold $X$, a differential form $d\omega$ is a function $X \to \Lambda(T_{x}(X)^{*})$. That is to say at each point $x$, $d\omega(x)$ gives us an alternating tensor on the tangent space at $x$. If you squint hard enough, you'll see that this amounts to measuring how much the derivative changes at a point, somewhat similar to the familiar $dx$ in $\mathbb{R}$.

Now differential forms are also anticommunative, $dx \wedge dy = -dy \wedge dx$, which implies $dx \wedge dx = 0$.

If you play around with this anticommunative property a little bit, you'll note that it automatically accounts for the determinant of the Jacobian, which is super useful! Essentially, you get the change of volume when you move your integral around for free by using differential forms.

So it's not fair to treat $dx$ as an infintesimal as that won't always make sense, and it doesn't generalize well. This kind of went off on a tangent, but differential forms are pretty cool and if you're interested I'd recommend the last chapter of Guillemin and Pollack (and if you're rusty on the prerequisites, chapters 1 and 2 of the same book). I'd say it's worth the price of admission to prove the generalized Stokes theorem, which is in my opinion on of the most elegant theorems around.