TITLE: Justification of algebraic manipulation of infinitesimals
QUESTION [16 upvotes]: As an engineering student, I regularly see people making arguments like this:
Consider a rectangle of dimensions $x\times 4x$. If we make $x$ bigger by a small quantity $dx$ then this will make $4x$ bigger by $4\cdot dx$ so the area of that $x \times 4x$ rectangle will change from $4x^2$ to $$(x+dx)(4x+4dx)=4(x^2+2x\cdot dx+(dx)^2)\approx4x^2+8x\cdot dx$$
with the final step justified because $dx$ is a 'small' quantity so $(dx)^2$ will be so small as to be ignorable in some mathematically rigorous way. Thus the change in area $dA$ would be $8x\cdot dx$.
Arguments like this are very common. Another random example would be in Wikipedia's proof of the brachistochrone problem which starts with the statement 
$$ds^2=dx^2+dy^2$$
and proceeds to manipulate these infinitesimals as if they were ordinary constants or variables. 
I'm wondering if there's a simple, analytically rigorous justification for all of this manipulation. While I feel perfectly comfortable with the idea of the derivative of a function (considered as a limit), I've never seen a similar, rigorous justification for the algebraic manipulation of infinitesimals and the cancellation of 'small' terms (like $(dx)^2$). Any thoughts or help would be appreciated.
Thankyou

REPLY [3 votes]: There are various ways of answering your question. Tim van Beek has given one in terms of differential forms. Another nice way of looking at it is using nonstandard analysis (NSA). The nice thing about NSA, to my taste, is that it allows you say things the way that Leibniz and Gauss and Euler found it so useful to say them (i.e., with dx's and dy's), and it also allows you to be certain that what you're doing is logically rigorous. NSA doesn't have to be hard and scary. Jerome Keisler wrote a very nice freshman calculus book using NSA, now available online for free: http://www.math.wisc.edu/~keisler/calc.html My own book, using a similar approach, is here: http://www.lightandmatter.com/calc/ This is also a very nice treatment: http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm
The basic idea of NSA is that just as we expanded the integers to the rationals, and the rationals to the reals, we go one more step and expand the reals to the hyperreals. The hyperreals are a number system that includes the reals, but that also includes infinitesimals. The way you know whether you're doing something logically correct is that if you write down all of the elementary axioms of the real number system (x+y=y+x, etc.), then all of those are true in the hyperreals. "Elementary" means axioms that only say things like "for every number...," not ones that say "for every set of numbers..."