- #106
Dale
Mentor
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I don't see any question of yours which has been partially answered.Mr Real said:
1) You wanted to know why the integral of x.dF did not also give work just like F.dx. The complete answer was that it's a calculus error to switch the quantities like that. The area under a curve is not the same as the area to the left of the curve. So whatever x.dF might represent it is not generally work.
2. You wanted to know why textbooks don't spend time discussing this. The complete answer is that there are an infinite number of mistakes you could make, and this is a very rare one. Authors focus primarily on the correct ways, not one of the infinite incorrect ways. When they do address mistakes, it is only the common ones.
3. You wanted to know why the integral of x.dF is not useful in its own right. The complete answer is twofold, first, it is just an odd expression. It is not common to express x as a function of F, and it is often not even possible to do so, and the quantity dF is a weird quantity without physical use elsewhere. Second, there is no known relationship of that quantity to other known physical quantities of interest, nor does it posses known properties that make it a primary quantity of interest.
What could possibly be considered incomplete about this discussion? Not only has this community correctly and completely answered your question, we have been exceptionally patient with you. This discussion should have ended on the first page as soon as it was pointed out that it is a mathematical mistake to switch the terms that way. So your answer has been correct, complete, and exceptionally supportive.
It is highly frustrating to have such thoughtful and helpful replies from the community regarded as insufficient.
At this point I am going to close this thread. I only hope that in the future you can recognize when you have received such high quality responses, and be a little more encouraging to the community that provided them.