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.)
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$\begingroup$ 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. $\endgroup$– FlatterMannApr 2 at 3:14
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2$\begingroup$ 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. $\endgroup$– LPZApr 2 at 10:43
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2 Answers
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is the elimination of higher-order differentials an approximation at the physical level or is it strictly equal at the mathematical level?
Unfortunately, the answer is that it depends. Differentials can be found in several different mathematical contexts.
With standard analysis calculus can be expressed in terms of limits of functions of real numbers. In that context in an expression with differentials of several orders it is indeed possible that the higher order terms go to zero in the limit while the lower order terms do not. This is the approach with standard analysis which, while not being the most intuitive, is (as the name suggests) the standard approach.
However, in non-standard analysis there are many different number systems. In the hyperreal number system, higher order infinitesimals are still non-zero. In contrast, in the dual number system, higher order infinitesimals are exactly zero by definition. These number systems, which include infinitesimal numbers as legitimate numbers, are more intuitive than standard analysis but are less well-known. They do not require limits.
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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.
Leibniz used to discard higher-order infinitesimals in his calculations, and emphasized that he worked with a generalized notion of equality (up to negligible terms), rather than strict equality. This is formalized in modern infinitesimal analysis in terms of the concepts outlined above.