-1: This answer is no good. Fermi doesn't mean differentials, he means infinitesimals. The two concepts are different, only coinciding to first order, and the mathematical obscurantism regarding this is intolerable.

@RonMaimon: Please elaborate on that statement. Is there anything you'd not get from the formalism involving $dV$ viewed as the exterious differential acting on a function $V$? What's the point of insisting on infinitesimals here?

@Nikolaj: I shouldn't have downvoted, the answer is not wrong, but I don't like that to understand something as obvious as an infinitesimal relation, the student is diverted to the superficially intimidating ideas of duals of tangent spaces. The two concepts are only obliquely related. For an example of something infinitesimal which is not interpretable as a differential which arises in gasses, consider the number fluctuations in a finite region over an infinitesimal time dt, these scale as $\sqrt{dt}$. You can't express this infinitesimal relation using differentials.

@RonMaimon: Sounds interesting, do you have any reference to the specific problem?

I could second that request for more references. I appreciate both answers, but the second one actually left more confuse. I am trying to find the time read this, but other than that, I have found no more books about this. Most calculus books I've seen never go beyond $\epsilon - \delta$ land...

@wmnorth: The idea of an infinitesimal is just a tiny number which is so small, it's square is negligible. Then you ask, what is the change in, say, (x^2) when you go to (x+dx)^2, and this is 2x dx, which gives the definition of derivative. The infinitesimals are the main idea in calculus, and I don't understand how a book can omit them. If you look at old enough calculus books (the best books are around the mid to late 19th century), you can find a discussion. Feynman also discusses infinitesimals in vol 1. They are commonplace.

@Ron: OK, I'll look it up (if you have a suggestion for a couple of good ones, I'm listening ;-) )

@Nikolaj: In the last comment adressed at you, I should have said "the displacement in a time $dt$ is $\sqrt{dt}$". The mathematician's version of this omits the differential square-roots from the notation, and is called "Ito's lemma". The Feynman path integral is the same as this. The analogs for Levy flights are still not worked out anywhere, because of the avoidance of fractional roots of infinitesimals.

@wmnorth: Well, there is also this. And RonMaimon: I'm afraid I still don't understand what the problem really is. You said "number fluctuations", is that a typo, or what numbers are we talking about here? And do we have a situation with $dn=\sqrt{t}dt$ or $dn=\sqrt{dt}$ or something else? (just saw the new comment, i.e. you probably mean $\sqrt{dt}$). And so you give out a source for "trivial" stuff after all. :D

@wmnorth: I learned calculus in a tradition limits way from Serge Lang's calculus book. I can't honestly recommend another book, because the material is too elementary, so I wouldn't know what makes one book better than another anymore, they all look the same now. But I do remember being annoyed by Lang that I was forced to rediscover the infinitesimal interpretation for myself. Lang has an epsilon-delta appendix. I don't think a nonstandard analysis textbook is what you want, they tend to introduce unnecessary complications, like "standard part" which are only useful for logicians.

@Nikolaj: The right form is $dn= \eta(t) dt$, where $\eta(t)$ is a random quantity which has the defining property that the integral from a to b of $\eta$ is a Gaussian random number of variance $a-b$. The need to deal with square-root infinitesimals comes when you need to show that $n(t+dt)*{dn\over dt}|_t - n(t-dt){dn\over dt}|_t $ is not zero, but constant for infinitesimal dt. This is standard stochastic calculus, but it is considered "advanced" because of the presence of square roots of infinitesimals (although they don't call them that in formal mathematics, of course).

@Ron, your criticism of this answer is completely incoherent gibberish. "Differential" (noun) is the correct word for ${\rm d}f$ which, in calculus (and physics), always means exactly the same thing that you terminologically incorrectly represent by the (noun?) "infinitesimal". See the first line under "mathematics" at en.wikipedia.org/wiki/Differential - All these differentials (and this is the only correct word!) are always and by definition infinitesimal (adjective) quantities so it's nonsensical to say that the agreement holds only to the leading order: there's no other order.

@Lubos: You are just wrong about this. The notion of infinitsimal does not coincide with the notion of differential, as can be seen from the case in my comment above, where $dn^2 = dt$. How do you interpret a second-order infinitesimal coinciding with a first order? You just never studied Ito calculus. It is important to learn this, because it can be interpreted as the first rigorous path integral construction.

@Lubos: Perhaps I misunderstood your comment. Are you saying that differentials are always infinitesimal? That is not so. The answer above gives a conceptually different definition, namely as the dual space of the space of tangent vectors. There is nothing infinitesimal, nor displaced, about duals to vectors. Dual vectors are a normal workaday finite mathematical object, with no limiting connetations. The surprise is that one can partially identify these objects with the infinitesimal objects of calculus. It requires explanantion. The infinitsimals come first, esp. in Ito, see below.

@Ron (Finally got time to read all this) Don't you see the irony here? You downvoted this answer despite it being correct also by your own judgement and only because you disliked the mathematical formalism (your word: "obscurantism") and now you come brandishing Itō calculus. In an attempt to elucidate what this answer has so unbearably obscured, I suppose?

@Adam: The point of the Ito calculus is not obscurantism, just the opposite! The Ito calculus makes it clear that you need to use infinitesimal increments, and not falsely pretend that the concept is captured by linear algebra on tangent spaces. The Ito calculus, expressed correctly, is as transparent as ordinary calculus, it just requires you to note that $dn^2$ is an infinitesimal of the same order of magnitude as $dt$, and this doesn't happen in the realm of differentiable stuff. I wish there were a differentiable example of cases where differential orders don't match. But there can't be.

@Adam: I hope you will come to understand that the interpretation of "dV" and "dP" as "coordinate projecting one forms" is 20th century dipshits taking a dump on the great classical work of Cavalieri and Leibnitz, that it is motivated by snobbery and stupidity (because Cavalieri was not an academic) and that absent the genius of Abraham Robinson, the issue could have festered indefinitely. I hope you take the time to learn Ito calculus to the point where you will see that my example is apropos, and this is not smoke and mirrors.