Questions tagged [infinitesimals]
For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).
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My proof of existence of nilsquare infinitesimals (synthetic differential geometry) [closed]
I tried to prove that nilsquare infinitesimals $\varepsilon$ exist and $\varepsilon$ is indistinguishable from $0$.
Let's consider function $g(x)$. From Kock-Lawvere axiom we have:
$g(\varepsilon)=g(0)...
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How is it possible that $\varepsilon$ in Dual numbers and $\varepsilon$ in SDG are different? [closed]
There is proof in SDG that if $\varepsilon^2=0$ then $\varepsilon$ is nilsquare infinitesimal indistinguishable from zero.
But in Dual numbers $\varepsilon^2=0$ and $\varepsilon \neq 0$. $\varepsilon$ ...
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How to prove that there are nilsquare infinitesimals that indistinguishable from zero?
As is known, non-zero infinitesimals exist. It can be proved.
In the book "A Primer of Infinitesimal Analysis" John Bell introduced infinitesimals that indistinguishable from zero. He did it ...
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What is difference in approaches between using standard part in NSA and limit?
What is difference in approaches between using standard part in NSA and limit? I don't mean technique of differentiation or integration.
Can somebody explain it on example?
Thanks.
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Why do we put $\mathrm{d}x$ in the integral?
Firstly, we studied that integration is always with $\mathrm{d}x$.
But in physics, I have seen this thing
$$V(t) = \frac{\mathrm{d}x}{\mathrm{d}t}$$
$$\mathrm{d}x = V(t)\ \mathrm{d}t$$
Then here he ...
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Using infinitesimals in multivariable calculus
In ordinary calculus,if we don't make any absurd notation,the use of infinitesimals is extremely useful and intuitive which ended up giving correct results. For example $\frac{dy}{dx}=\frac{\frac{dy}{...
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How to prove that we can't define discontinuous function on $R$?
I tried to do it like this:
Let for any map $g : \Delta \to R$ we have one element $b \in R$ that satisfies following:
for every $\varepsilon \in \Delta$, the equality $g(\varepsilon)=g(0)+b\...
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Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones?
Below, we interpret divergent integrals as germs of partial integrals at infinity:
$$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$
where $\operatorname{bigpart}$ means taking ...
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Discontinuous function on $\Delta$ in Smooth Infinitesimal Analysis
As is known, there isn't discontinuous functions in Smooth Infinitesimal Analysis.
I tried to define discontinuous function on $\Delta$:
$f(x) =
\begin{cases}
1, & \text{if $x$ = $0$} \\
0, &...
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Use of elemental length in volume of a truncated cone
The typical way to compute the volume of a truncated cone is to slice into discs and calculate the volume of a differential cylinder. While doing that we first take the area of the disc $\pi f(x)^2$ ...
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How to prove that hypotenuse of differential triangle coincides with the segment of the curve?
I read in one article that "for any curve given by an algebraic equation, the hypotenuse of the differential triangle generated by an infinitesimal abscissal increment $\varepsilon$ coincides ...
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Is it possible to say that $dx$ indistinguishable from zero in classical analysis as $\varepsilon$ in Smooth Infinitesimal Analysis? [closed]
$dx$ isn't nonzero infinitesimal in classical analysis.
Is it possible to say that $dx$ indistinguishable from zero in classical analysis as $\varepsilon$ in Smooth Infinitesimal Analysis?
Thanks.
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How is it possible that $dx = \varepsilon$?
Let's consider graph of function $y=x^2$ ("Calculus with infinitesimals" Efraín Soto Apolinar, see picture below).
We have the point $(dx, (dx)^2)$ which coincide with point $(dx, 0)$ and $...
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Does "$\varepsilon$ is indistinguishable from zero" mean that $\varepsilon$ is less than any positive real number?
I read book "A primer of infinitesimal analysis" John Bell. Author wrote about infinitesimals which indistinguishable from zero.
Does "$\varepsilon$ indistinguishable from zero" ...
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Nondegenerate triangle of zero area. How is it possible?
I read book "A primer of infinitesimal analysis" John Bell. I was confused when I saw example with area under curve. In that example author mentioned about nondegenerate triangle of zero ...
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Does not the "inexistence of infinitesimals in the real number system" conflict with the epsilon-delta definition of limit in the set of real numbers?
One of the propositions given in Advanced Calculus of a Single Variable is:
If |a - b| < ε for each ε > 0 then a = b.
Would not this conflict with the epsilon-delta definition of limit?
The ...
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Infinitesimally small area $(dx)^2$ as a point
I read in one book that infinitesimally small area $(dx)^2$ is a point. But point has no length in general sense. It's dimensionless.
How is it possible? And how to understand it?
Thanks.
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Intuitive explanation of why the area under the curve of a hyperbola (1/x) is infinite but not the area of a decreasing exponential?
In some of the videos I've watched on the Laplace transform, the authors say that if the exponential is decreasing, the area calculated by the transform is finite, and in control theory we can say ...
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Is it a good idea to learn about hyperreals and how they are related to limits before beginning to learn about calculus?
I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were ...
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Probability of dart hitting a point given that it hits a point in a finite set of points
I understand the probability of the dart hitting the center point is 0, and I think this is used as an example of how a probability of 0 doesn't mean something is impossible.
Now image we define 3 ...
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Sum of Two Infinitesimals being Infinitesimal
Any hyperreal number greater than 0 and smaller than all positive real numbers is infinitesimal.
We know the sum of two infinitesimals is infinitesimal.
Let $A$ be the smallest positive real number.
...
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Is a non-fractal, continuous curve made of tiny line segments?
EDIT: This question was in-part made unclear by my misconception of the hyperreals not being dense, since I thought there were no numbers between zero and consecutive infinitesimals
($0, 1/\infty, 2/\...
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What is the reciprocal of an infinitesimal?
"An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity." (https://mathworld.wolfram.com/Infinitesimal.html)
What is the ...
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Proving a Definite-Integral Lies Between Two Values
I'm trying to solve the following inequation but not sure what I'm supposed to do,
I know it's an integral of an even function over a symmetric interval, so I can double the area over the positive x-...
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Clarification on Nonstandard Analysis: Is $0.\overline{9}=1$, is it not, or is there some subtlety that allows both interpretations?
This, I hope, is not a duplicate; I am exercising my critical thinking here and I want to understand what going on, and the available content I have found online on this so far has not helped.
I'm ...
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Intuition behind Lie algebra being an "infinitesimal transformation"
I have been having some trouble intuitively understanding the "infinitesimal" behavior of a Lie algebra. Currently, I think of Lie groups intuitively being a group of continuous ...
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Pointwise and uniform convergence for particularly well-behaved functions
I'm asking myself...
if $f_n(x):\mathbb{R}\to\mathbb{R}$ are infinitely differentiable functions, and each $f_n$ is such that $f_n(x)\underbrace{=}_{|x|\to\infty}\mathcal{O}(|x|^{-N})$ for any $N\in\...
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The Function $1-(1-1/x)^{x}$
It is well known that the limit of the function $1-(1-\frac{1}{x})^x$ is $(1-1/e)$ when $x$ goes to infinity. In addition, I know that the function is always above its limit, i.e., for every $x \geq 1$...
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Is there a nice way to consistently define division by infinitesimals in the dual numbers?
I have been trying to extend faithfully extend the dual numbers, recreationally, of course.
Background: As a quick summary, our domain of discourse is $\mathbb{R}_\varepsilon=\{a+b\varepsilon:(a,b)\in\...
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Justification for a step in the derivative transform of infinitesimal transformation
When I was looking through p23-24 of Symmetry Analysis of Differential Equations by Daniel J. Arrigo, I wasn’t sure why we can do this:
$$\frac{\frac{dy}{dx} + [Y_x+Y_yy’]\epsilon+O(\epsilon^2) }{1+[...
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Nonstandard Complex Analysis?
I recently discovered Nonstandard Analysis and am slowly working my way through Kelsier's textbook and Foundations companion. However while I have found plenty of stuff about real nonstandard ...
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How to find infinitesimally small segments for graph $y = x^2$?
I read on Wiki:
Intuitively, smooth infinitesimal analysis can be interpreted as
describing a world in which lines are made out of infinitesimally
small segments, not out of points.
I tried to find ...
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How to prove that 2 points $\varepsilon$ and 2$\varepsilon$ are indistinguishable not only in sole case? (Smooth infinitesimal analysis)
There are 2 points on the Real line: $A$ and $2A$. They are indistinguishable in sole case - if $A$ is $0$.
But how to prove that $\varepsilon$ and 2$\varepsilon$ are indistinguishable not only in ...
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Discontinuous function in Smooth infinitesimal analysis
I read that there isn't discontinuous functions in Smooth infinitesimal analysis. But I tried to define discontinuous function ($\varepsilon$ is infinitesimal):
$f(x) =
\begin{cases}
1, & \text{...
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How do I take the infinite limit of this kind of update?
Given a non-negative normalized weight vector $v$ of dimensionality $d$ and a $(d-1)$ dimensional non-negative normalized weight vector $w$, I transform $v$ to the $(d-1)$ dimensional $v \leftarrow \...
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Infinitesimal: does $f(x + dx)$ being defined imples $f(x + dx) - f(x)$ to be infinitesimal too?
I am refreshing my calculus knowledge. One of the textbooks I use states what you can see down below (see the screenshot attached).
The question I have: must not the definition explicitly require $f$ ...
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What does a function of its own arc length look like?
Introduction
What does a function of its own arc length look like?
A strange question for sure, but first let me elaborate:
Imagine a function that starts at the point $\left( 0, 0 \right)$. If we now ...
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Strange integral sum. Where did I go wrong?
Let $f(x)=x^2$. $df(x)=2xdx$
For $x=0$: $f(0+dx)-f(0)=0$. Thus $f(dx)=0$.
I want to find area under the curve $f(x)=x^2$ from $0$ to $b$: $\int x^2dx = f(0)dx + f(0+dx)dx + f(0+2dx)dx + ... + f(0+(n-1)...
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$\epsilon$-method for directional limits.
Good evening. I'm going to ask a question which might appear dumb or worse, but I consider myself both a student, a researcher and a continuous learner and there are times in which even the tiniest ...
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Smooth infinitesimal analysis and law of excluded middle
I read about smooth infinitesimal analysis and I have several questions:
How to prove that in SIA every function on $R$ is continuous? (Every function whose domain is $R$, the real numbers, is ...
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Why is the diagonal of the infinitesimal square contained in $D(2)$?
The object $D(2)$ is the first order infinitesimal neighbourhood of the plane in synthetic differential geometry. For example in the algebraic geometric model over the base ring $\mathbb C$, the space ...
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What will infinitesimals of derivative do with the mean value theorem?
I think these three proofs of increasing/decreasing are already very convincing. What I don't understand is the last sentence of the text
"It's not a sure thing that these infinitesimals have ...
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Algebraic contradiction involving infinitesimals
Let $\Delta$ be infinitesimal, $\hat{e}=(1+\Delta)^{\frac{1}{\Delta}}$, and $log_{\hat{e}}(a)=\hat{ln}(a)$.
$$a^{\Delta}=({\hat{e}^{\Delta}})^{\hat{ln}(a)}=(1+\Delta)^{\hat{ln}(a)}$$
Consider $\frac{a^...
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How to determine the probability distribution of a variable obtained as function of another variable having its own probability distribution function?
I have a variable, $y$, having the following probability distribution function:
$P_y(y)=\frac{|y-a+2ab|}{a^2b^2}\exp(-\frac{y-a+2ab}{ab})$ defined for $x\in\mathbb{R};x>=a(1-2b)$
where $a$ and $b$ ...
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Convergence of derivatives near the boundary of an open interval
Suppose there are two continuously differentiable functions $a(x_1)$ and $b(x_2)$, where $x_1, x_2\in \mathbb{R}_+$.
I assume that $a'(x_1) >0$ if $x_1<x^o$ and $a'(x_1)<0$ if $x_1>x^o$. ...
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Neglected terms in integral sum
As is known, we can neglect high-order term in expression $f(x+dx)-f(x)$. For example, $y=x^2$: $dy=2xdx+dx^2$, $dy=2xdx$.
I read that infinitesimals have property: $dx+dx^2=dx$
I tried to neglect ...
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Will we obtain an infinitesimal when we neglect $x^2$ in $x+x^2$, as ${x \to 0}$?
Let's assume that we have $2xdx + dx^2$. We can neglect $dx^2$ and we'll obtain $2xdx$.
Let's assume that we have $x+x^2$ and ${x \to 0}$. We can also neglect $x^2$, but will we obtain an ...
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Has There Been Any Study of the Quotient-root Derivative Definition $\lim_{\Delta x \to 0} \sqrt[\Delta x]{\frac{f(x + \Delta x)}{f(x)}}$?
Has there been a study of turning the difference-quotient seen in the common derivative
$$
\frac d {dx} f(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
$$
into a quotient-root
$$
...
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Infinitesimals of a higher order and limit
I read in Bartholomew Price book ("A treatise of infinitesimal calculus") that "the last term of which equality must be neglected, because it contains infinitesimals of a higher order ...
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Algebraic property of infinitesimals $(m+dx=m)$
If $m$ is real number and $dx$ is infinitesimal, then $m+dx=m$ (if $dx$ is infinitesimal and $dx^2$ is infinitesimal, then $dx+dx^2=dx$).
I suppose that $m+dx=m$, because if $\dfrac{m}{dx}=N$ ($N$-...
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