You are probably right, I underestimated the complexity of the equation given it's age, ignoring the fact that mathematicians then had access to very complex tools like differential equations and many other things.
However I had been lead to a possible right direction to manage to define this...
Well that is a very interesting read, and one that opens a whole other can of worms for me to study. while I am familiar with many of the concepts used, and could (midly) understand the read, I'm not ready to be comfortable with the use of differential forms, less for such an examination of line...
Thank you for your reply! I was anxious for one hahaha, been stuck in this topic for a couple of days now.
As I said, I have understood the "work" conception of the line integral, but such conception is of course, limited to that specific context (line integral as a mean of obtaining work)...
So, as i understand, the geometrical meaning of this type of integral should still be the area under the curve, however, I really do not see how you can obtain each infinitesimal rectangle from the dot product.
I have understood the typical work example, that is, the line integral as the sum...