When you look at the original and rigorous definition of the Riemann integral you found out that it does not actually speak about the 'orientation' of the set over you integrate on. Thus, I my opinion concepts used in computation of, for example, work done by moving a charge in the electric field, are quite different. Terence Tao gives some information about it in his short article about differential forms, but still without much explanation (he distinguishes between signed and unsigned integrals). Thus, my question is what mathematical structure we would need to impose onto the space (in 1D for simplicity) to get the proper 'signed' integrals to appear when analysing physical situations?
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$\begingroup$ wiki page for the line integral $\endgroup$– By SymmetrySep 25, 2016 at 23:44
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$\begingroup$ Thanks. I should have missed that. My question steams from on how to describe such issues to 1st year students of chemistry. They must use such integrals in the physics course, and their maths course is just about Riemman integral, which is not convergent with the previous approach... $\endgroup$– rk85Sep 26, 2016 at 19:27
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