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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?

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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.

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Mikhail Katz
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    Would you say that differential forms are true to the original spirit of Leibniz' differentials? Or are they essentially unrelated? As I said I haven't learned about them yet, and now I'm positively crashing books over my head to get to that section :) Maybe I should wait to continue these questions until I have covered that... –  May 06 '14 at 16:49
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    @NotNotLogical: I have somewhat the "opposite" problem. I am relatively familiar with Leibniz's differentials, but I have little experince concerning path integrals. However, I do know of problems regarding the fraction-like behaviour of infinitesimals in occasions like differential equations of the form $$\frac{dy}{dx}=y'(x)=\frac{g(x)}{h(y)}$$ because whereas the relation corresponds to taking limits pointwise it is not clear a priori, though, how it should be interpreted as a limit of sums in the integral expression $$\int h(y)\ dy=\int g(x)\ dx$$ – String May 07 '14 at 09:21
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    @NotNotLogical, differential forms are an efficient tool in formalizing some of these calculations, but they stay within the realm of the real numbers and associated structures such as tangent bundles, tensor products, etc. In the sense that no infinitesimals appear in this picture I would say that differential forms are not in the spirit of Leibniz's approach. The same could be said about vector fields which are dual to differential 1-forms. It so happens that we have just developed an approach to vector fields in the spirit of Leibniz and Euler; see this. – Mikhail Katz May 07 '14 at 13:16