Where Can I Find the Rules for Manipulating Infinitesimals?

[mnb96]

Hello,
I have been thought differential-calculus ages ago, but now when started reading some physics books (where infinitesimal quantities are used again and again) I realized I know nothing about calculus.

I am unable to specify where exactly my problem lies, but I guess it lies in how to perform algebraic manipulations with quantities like dx, dy and so on...
I often get the feeling that infinitesimal like $dx$ are sometimes treated like ordinary scalars, but other times they are given special undefined properties like $(dx)^2=0$, and many others.

So where are these rules/properties written or deduced from? They cannot certainly be arbitrary.
Are there strict definitions that avoid run into silly mistakes?
At the moment I naively make the mental association of $dx$ as something that tends to zero, but this way of thinking does not help very much in algebraic manipulations.

 

[HallsofIvy]

Actually, a rigorous development of Calculus with infinitesmals is called "non-standard analysis" and requires some very deep results from logic to get the "infinitesmals" themselves. It is much easier to use limits to get Calculus results and use the "differentials" as a mnemonic device.

 

[mnb96]

Let´s say that I do accept the intuition behind infinitesimals because I know it has recently been given a solid and rigorous grounding (in Non-standard Analysis); but sometimes I cannot use this intuition to manipulate expressions involving differentials.

I just found "www.unco.edu/NHS/mathsci/facstaff/parker/math/Infinitesimal_Paper.pdf"[/URL] a short overview on the history of development of such concepts (from Newton to Non-standard Analysis), that exposes (at page 4) some of the ideas initially adopted by Leibniz:

1) [I]dx[/I] is indistinguishable from 0
2) [I]dx[/I] is neither equal to, nor not-equal to 0
3) $$\mathbf{(dx)^2}$$ [B]is equal to zero[/B]
4) [I]dx[/I] becomes vanishingly small

Apparently there are some rules to corectly use infinitesimal.
My "intuition" on infinitesimal was mostly based on points (1) and (4).
Nobody (nor my textbook) ever mentioned or stressed enough the properties (2) and (3), especially [B](3)[/B].-- Is this really everything I should know to get along with algebraic manipulation and/or proofs involving differential quantities?

-- I also don´t quite understand what do you mean by "using the limits". How would you use limits to prove $(dx)^2=0$ ?

-- I also bet we are not allowed to make the association $$dx = \lim_{\Delta x \to 0} \Delta x$$, because we´d get [I]dx=0[/I]