![]() The definition you mention from your textbook doesn't really make sense when taken literally, at least when taken out of context like this. Being able to say that requires that we consider $0$ to be infinitesimal if we did not, then we would have to say something more awkward, like "$f(x)$ is either infinitesimal or zero whenever $x$ is infinitesimal". It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. in nonstandard analysis, if $f$ is a standard, continuous function with $f(0) = 0$, then we would like to say "$f(x)$ is infinitesimal whenever $x$ is infinitesimal". One exception is a recent reconstruction of infinitesimals positive numbers smaller than every real number devised by the logician Abraham Robinson and developed further by H. Modern application of infinitesimals In mathematics, nonstandard calculusis the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. The typical mathematical usage of infinitesimal is in a sense where 0 would be included e.g. of, pertaining to, or involving infinitesimals. immeasurably small less than an assignable quantity: to an infinitesimal degree. However, that makes for a bad mathematical definition. indefinitely or exceedingly small minute. And ds is actually not a differential form (as far as I can tell, since it is not linear), it seems a mess. In his Arithmetica Infinitorum (The Arithmetic of Infinitesimals) of 1655, the result of his interest in Torricelli’s work, Wallis extended Cavalieri’s law of quadrature by devising a way to include negative and fractional exponents thus he did not follow Cavalieri’s geometric approach and instead assigned numerical. With that in mind, I am not surprised to find that the English meaning of infinitesimal excludes zero. Apart from the square of an infinitesimal, there is also the square of an infinitesimal (which basically is a matrix, think of ds2 in Landau-Lifshitz Volume II). There are all sorts of conventions like if if you ever hear someone talk about a "small number", you're supposed to assume there's a good reason for using that phrase instead of "zero", and thus should assume that the number is, in fact, nonzero, despite the fact zero is a small number.įor a nonnumeric example of this phenomenon, if I told you I lived near Paris, you would infer that I do not live in Paris. Natural language is a bad reference for mathematical definitions it's 'optimized' for quickly conveying meaning in 'natural' settings, not for expressing things precisely.
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