Probably most of you know a short article by Terence Tao on differential forms (http://www.math.ucla.edu/~tao/preprints/forms.pdf). He talks about three integration concepts, two of which are the definite integrals. The integration of functions on intervals is called the "unsigned definite integral":
$$ \int\limits_{\left[a,b\right]}f $$
This concept I understand quite well. It is basically the the Riemann (or Darboux) integrals or functions on some intervals. This theory I understand pretty well. The second concept he introduces is the "signed definite integral" which we can write as:
$$ \int\limits_{a}^{b}f \ \ \ \ \text{or rather (see issues below)} \ \ \ \ \int\limits_{a}^{b}f(x) \ \mathrm{d}x $$
This integral is actually very much different (in my opinion) from the Riemann integral, even though there is a close relationship between them. I would like to understand these relations and concepts further, but I have some problems in tracing down the origins of the "signed" integral. By origins I mean not the construction but rather the issues like:
What is being integrated? Is it still just a function, or already a differential form? What would be main difference in 1-D case?.
What we integrate over? In higher dimensions we talk about curves, paths etc. But what would be the simplest case in 1-D? What kind of basic definitions should I use to introduce something like "oriented interval"?
If something is unclear I will obviously try to fill in any gaps in these questions.
Many thanks,
Radek
