There is a discussion of this in Boyer, at p. He had a notation $o$ for an infinitesimal change in the independent variable, so that if $x$ depends on $t$, then what in Leibniz notation would be written as $dx$ would be notated in Newton's notation as $\dot$, omitting the $o$ when context made it clear that what was intended was an infinitesimal change in $x$. newest tensor calculus questions physics stack exchange june 1st, 2020 - tensor calculus tensor analysis is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold but which accounts for respective sub spaces their symmetries and their. Newton's notation was unclearly presented, and many people didn't understand how he intended it to be used. It works whether you want to think in terms of variables or functions, limits or infinitesimals. Unlike Newton's notation, it makes it easy to do dimensional analysis, and it works well when you have lots of different variables that you might be differentiating or integrating with respect to. Leibniz's notation has some objective advantages. For example, in physics, if you have a function of position and time, it's common to use dots for time derivatives and primes for spatial derivatives. For example, see Hutton, 1807,, which uses the dot notation and terms like "fluent." We still do use elements of Newton's notation in many fields. Many textbooks in English did use Newton's notation and terminology for a long time. Moreover, they generalise to the infinite-dimensional context seamlessly, unlike the usual calculus for which there are many differing techniques). (It's worth adding that doing differential geometry with intuitionistic logic allows the introduction of infinitesimals that is much closer to how Newton envisaged them, his fluxions, rather than the traditional epsilon-delta techniques of traditional analysis. Think of it this way: You’ve got some equations, you feed in some numbers, and after a bunch of work, you come up with x + n 3 + n. Not just intuitively, but mathematically. But infinity is always hard to deal with. In other words, the notation associated with their names reflected their interests. Answer (1 of 2): Well, physics needs infinity in some places. Had Newton been more interested in geometry than physics, and Liebniz more interested in physics than geometry it would have been likely we would have seen their notations being swapped. Thus Liebniz's notation is more natural here. This should indicate the degree, the dependent and independent variables. When one is interested in the calculus for its own sake, then a more comprehensive notation is necessary. Buy since we are only interested in the first two, 1 & 2, we don't have to indicate the degree by a numerical prefix (as they do in some notations), we can simply indicate it by a single or double dot. Hence he needed only to indicate the degree of the derivative. Since the dependent variable implicitly understood, there is no need for the notation to reflect this. As a physicist he was mostly interested in the first and second derivatives of time. This is also used for the Frechet and Gateaux derivative (which is implicitly used in the notation for tangent bundles in differential geometry). For example, there is also Heaviside's operational D. I do not expect this to be a good question, but still I am curious about mathematicians' point of view on that if they ever know this situation in physics.There are many notations for derivatives as the concept has been expanded in many different ways. I apologise for this very general question. Is this kind of "infinitesimal techniques" all rigorously defined, for example, some in the language of distributions? $$\int^$ is a fermion creation/annihilation operator. For example, when we define a following Fourier transform, we add a positive infinitesimal and let it go to zero in the end: As you may know that there are a bunch of heuristic techniques in physics to make integrals converge.
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