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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 [itex]dx[/itex] are sometimes treated like ordinary scalars, but other times they are given special undefined properties like [itex](dx)^2=0[/itex], 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 [itex]dx[/itex] as something that tends to zero, but this way of thinking does not help very much in algebraic manipulations.