multivariable calculus - The meaning of Differentials in Integration

This is further to the questions discussed in a previous post

Here is an example of what I mean: Suppose that $C$ is a closed path in the plane and consider the line integral of $xy\,dx+x^2\,dy$ over $C$. Note that this can be written as $$\oint_C x\,d(xy)$$

Question: If we let $A$ denote the starting and ending point of the path, is it valid to use "Integration by Parts" to write $$\oint_C x\,d(xy)=x\cdot xy\bigg|_A^A -\oint_C xy\cdot dx=-\oint_C xy\cdot dx$$ Note that the claim is valid, as the sum of the two integrands is $2xy\,dx+x^2\,dy$ which is conservative, being the gradient of $f(x,y)=x^2y$. Therefore the sum of the two integrals in the last equation is zero, hence they are negatives of each other. (And obviously, the second integral will usually be more desirable to compute than the first.)

But what does this mean, and why is it true?

More generally, I think it would insightful to have a discussion about differentials in the context of integration and how/why the Leibniz notation can or cannot be used to intuitively understand calculations such as the above.

And how can we interpret differentials of general quantities which may involve several variables, as opposed to just single ones? Does the expression $d(xy)$ only make sense in the context of a path $C$, where we can view it as the change in $g(x,y)=xy$ over small segments of that curve, or does it admit a broader interpretation?

Answer

There is a satisfactory modern account of this in terms of differential forms, though it is not the only possible account. Here $d(xy)$ is a differential 1-form. It is shown in general theory that it safisfies the suitable version of the Leibniz rule. Therefore these manipulations are easily accounted for in that framework. In general, the viewpoint is that what one integrates when path integrals are concerned are not functions but differential 1-forms. One can then interpret the results in terms of functions, but the interpretation is less elegant and sometimes you run into problems, as the problem with the notorious change of sign when you switch $dx$ and $dy$ in multiple integrals.

See Writing Integrals using Differential Forms for a related discussion.