# In physical calculations, is the elimination of higher-order small quantities an approximation or a strict equality in mathematics?

Physics sometimes uses a technique called the method of differentials, which seems magical and not very systematic. This makes me unsure which variable I should take the differential of, and sometimes when I choose different differentials, I get the wrong result in my calculations (which may be my own fault). So I'm wondering, in physics calculations, is the elimination of higher-order differentials an approximation at the physical level or is it strictly equal at the mathematical level? (Physics derivations always make me uncomfortable... it's not as satisfying as mathematics.)

• Mathematicians are using "small quantities" all the time in the epsilon/delta definition of limits, which allows them to develop a large fraction of calculus. There are cases where one has to go to second and higher order, which are e.g. covered by L'Hopital's rule. If it's good enough for mathematics (it certainly was a couple centuries ago), then it's probably good enough for physics. Having said that, most of the problems that you will solve with these methods are fairly entry-level. Apr 2 at 3:14
• Yes it usually they agree with each other. Perhaps it would be easier to discuss the matter around a specific example. Note that in some cases, the mathematical error resulting from an approximate calculation is sometimes dwarfed by the error due to the physical limitations of the original equations.
– LPZ
Apr 2 at 10:43

The traditional non-infinitesimal approach to analysis and the approach of infinitesimal analysis in the manner of Robinson both use classical logic and are mathematically equivalent. Both use the notion of limit. While in the non-infinitesimal approach, limits are defined by means of epsilon-delta alternating-quantifier definitions, in the approach using infinitesimals one defines the limit of, say, $$f(x)$$ as $$x$$ tends to $$0$$ as the standard part, or shadow, of $$\frac{f(x+\alpha)-f(x)}{\alpha}$$ where $$\alpha$$ is a nonzero infinitesimal. Here an infinitesimal is a number smaller in absolute value than every standard positive real number. The shadow of a limited number is the standard number infinitely close to it.