From Multivariate integration of a derivative w.r.t. a single variable, from words_that_end_in_GRY's answer.
From elementary calculus, I always thought $\int^{}$ and $dx$ were just a kind of "command" button to integrate something between the two, rather than a genuine mathematical object.
(I don't mean this is widely accepted definition of the terms, but by "mathematical object" I mean anything that can be subject to mathematical operation. By "command button" I mean symbol for operation of some kind. In "2+2=4" 2 and 4 will be mathematical objects and + and = will be command button.)
That understanding caused no trouble in elementary calculus like
$\int^{} \mathbf f(x) \mathbf {dx}$.
But it starts to confuse me when things like
$g(\mathbf r) = \int^{\mathbf r} \mathbf F(\mathbf r')\cdot \mathbf {dr'}+c$.
comes up. dot product! $dr'$ is being dot producted! Then clearly it sounds like $$dr'$ is some genuine mathematical object.
I remember I felt similar confusion in differential equation course and many others.
In short, what's the true rigorous meaning of dx symbol in general (possibly multivariate) setting?
