For a few weeks now I've been trying to reconcile my understanding of differential forms, measure theory, integration and differential equations with the use of differentials in my engineering course (but this is also applicable to many expositions of physics I have seen).
I have come across a few different definitions of integrals, i.e. the Riemann integral, Lebesgue integral and also integration of forms on manifolds.
I understand this may be a rather open-ended question, but please bear with me (as I am only a 2nd year undergraduate). I've been having a very tough time satisfying my intellectual need from the results of my searches alone. So, as a last resort I am posting a question on here. View this meta question for more information on this stance I am taking.
Question: How does one rigorously move from a drawing, or a geometric representation of a "volume element" (in any dimension ideally, but specifically in 3-dimensions) to an integral. From my understanding of how people perceive dV, it is sometimes treated in my engineering courses as a very small volume (whatever that means) or part-of-a-whole, which seems hardly rigorous by any account.
"dV" is then just thrown into this integral below via what looks to be intuition (but what is dV, is it a differential form, if so, how does that come from this "very small" volume element)?
Below is the link to the page where this "conversion" is made:
Fluid_Mechanics_for_Mechanical_Engineers/Differential_Analysis_of_Fluid_Flow