Infinitesimal and approximations in physics

Question

I'm a first year student studying physics. Solutions of many physics problems, which I've seen so far, are achived through solving this problem for infinitesimal part of problem's subject (some curve, rod, medium, etc.) and, during this step, we use many approximations: like counting infinitesimal part of curve as a line segment, or approximating eg. trigonometric functions. So, why is it even possible to use these approximations especially in physics? Why do we get correct answer after integrating all these approximated infinitesimal parts?

Answer 1

The main reason people are puzzled by the typical way of dealing with infinitesimal parts of a problem in Physics is the remnant of old and not more correct definitions of infinitesimal quantities. Historically, calculus was born with the idea that they are "quantities that are closer to zero than any real number, but that are not zero" (rephrasing the Wikipedia definition linked in the question). Derivatives were defined as the ratio and integrals as (infinite) sums of infinitesimals.

These original ideas had an important heuristic role. However, they were not consistent from the mathematical point of view. At least from the first part of the nineteenth century, they were abandoned by mathematicians in favor of more sound definitions based on the concept of limit. In the second part of the twentieth century, with the introduction of non-standard analysis was possible to put the original ideas on a sound basis.

Still, physicists apparently ignored (and keep ignoring) this sequence of historical facts and are used to build sound intuition and theories by reasoning on infinitesimals more or less like their eighteenth-century colleagues would have done.

The reason is that it is possible to translate whatever can be said using the word infinitesimal into sound mathematical statements based either on the limit concept or on non-standard analysis. The essential step is the rigorous definition of differential of a function $f(x)$ at a point $x_0$, as the best linear approximation to the local variation of the function: $$ df = f'(x_0)(x-x_0), $$ where $f'$ denotes the first derivative of $f$. It turns out that for smooth functions, the difference $\Delta f=f(x)-f(x_0)$ is well approximated by $df$ in a neighborhood of $x_0$, with an error going to zero at least like $(x-x_0)^2$.

When a physicist speaks about an infinitesimal variation of a function means a finite variation (expressed by a real number) so small that the first-order approximation provided by the differential is good enough. Of course, this remains an approximation. However, since it is a controlled approximation, it is easy to convert it into an exact formula referring to the derivative the differential is based on. In some cases, one deals with a sum of differences. When the sum contains many small differences, it can be approximated by a finite sum of differentials, which becomes an integral when the proper limit has been taken.

To summarize, the correct interpretation of the infinitesimal quantities in Physics is as differentials, i.e., linear approximations to the local variations, used in neighborhoods so small that deviations from linearity can be safely neglected.