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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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How can infinitesimals be invertible in SIA?

I read that infinitesimals in SIA can be invertible: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis In typical models of smooth infinitesimal analysis, the infinitesimals are not ...
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Doubt on the meaning of $dx$

I have only recently began to study calculus and I got this doubt. My teacher had quoted that $dx$ is an infinitely small change in the variable $x$ and $dy$ is an infinitely small change in the ...
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How to prove that $ \underset{\varDelta x\rightarrow 0}{\lim}\frac{\left| MN \right|}{\overset{\frown}{MN}} $

In the derivation of the arc differential formula, why is the limit of the ratio of the length of the line segment between two points to the length of the arc considered to be 1: $$ \underset{\...
Torsor-L's user avatar
3 votes
2 answers
980 views

Is it problematic to define the line integral in terms of infinitesimals

I am reading Griffiths’ Introduction to Electrodynamics wherein the author defines a line integral as $\int^b_a\mathbf{v}\cdot d \mathbf{l}$. However, this definition in terms of the dot product seems ...
Joa's user avatar
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Solve an integral, e.g, $I = \int_{p}^{p+\delta p} f(x) dx=\int_{p}^{p+\delta p}\frac{e^{ax}(1-e^{-ax})}{x(1-x)}dx$ [duplicate]

I am trying to solve an integral over an infinitesimal interval, such as: $\begin{align}I = \int_{p}^{p+\delta p} f(x) dx=\int_{p}^{p+\delta p}\frac{e^{ax}(1-e^{-ax})}{x(1-x)}dx&\tag{1}\end{align}...
CafféSospeso's user avatar
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How is it possible that infinitesimal $h$ in IDG and infinitesimal $\varepsilon$ in SDG are the same point on axis? [closed]

There are first order infinitesimals $h$ in Infinitesimal Differential Geometry and: $f(h)=f(0)+hf'(0)$ There are infinitesimals $\varepsilon$ that indistinguishable from zero in Synthethic ...
Mike_bb's user avatar
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What is the distance (metric) between $0$ and $\varepsilon$ on smooth real line in SDG/SIA?

As is known, infinitesimals don't have concrete value (size) in SIA/SDG. What is the distance (metric) between $0$ and $\varepsilon$ on smooth real line in SDG/SIA? Thanks.
Mike_bb's user avatar
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Is there distance (metric) between two points on the hyperreal line in Nonstandard analysis?

As is known, there is distance between two points on the real line. It's obvious. But if we imagine hyperreal line (see pic.) then we'll have infinitesimals and their corresponding points. For ...
Mike_bb's user avatar
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Is my explanation of why we can make $\frac{dy}{dx} {dx} = {dy}$ in the separable differential equations valid? [duplicate]

I'm trying to understand why we can separate $\frac{dy}{dx}$ in separable differential equations when rigorously it's not a fraction but an operator (AFAIK). I have an assumption, but I'm not sure ...
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Is this a valid basis for a transfinite number system?

I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
Aidan Simmons's user avatar
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How is it possible that graph of function $y=x^2$ doesn't coincide with its infinitesimally small segment of tangent line in Non-standard analysis?

As is known, graph of function $y=x^2$ touches the axis $X$ not only at point $(0,0)$. There is infinitesimally small segment of tangent line (at the point $(0,0)$) that coincides with this graph (it'...
Mike_bb's user avatar
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Help with using "infinitesimal Riemann sums" to arrive at the formula for arclength

I am trying to arrive at the formula for arclength using infinitesimals. So far, I have a definition which says: $\displaystyle \mathrm{Re}\sum_{k=0}^{\omega}f(x_k)\Delta x:=\int_{a}^{b}f(x)\mathrm{d}...
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Why don't infinitesimals in nonstandard analysis have concrete size but infinitesimals in surreal numbers have?

I read on Wiki (https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis#Overview) that infinitesimals in NSA don't have concrete size but infinitesimals in surreal numbers have. How is it possible?...
Mike_bb's user avatar
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How do I understand that there is no infinitesimal element in $\mathbb{R}$?

I read a problem in my book that if there is an element $o>0$ in $\mathbb{R}$, such that for any $x>0$, we will have $o<x$, and we call $o$ as an infinitesimal element. Prove that there is no ...
Young's user avatar
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How are we to interpret the differential in the integral?

I have been working a lot with infinitesimals lately and related concepts such as derivatives and integrals. The for a function $y=y(x)$, the differential $\mathrm{d}y$ can be defined to be the change ...
Alice's user avatar
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How is it possible that Cauchy sequence represents two different numbers?

In the book on Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, p.4) there is such term as "standard infinitesimals". This ...
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How to correctly choose number for Cauchy sequence for infinitesimal functions? Will it be $x=y=[sin(t)]$ or $x=y=[t]$?

I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions: ...
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Suggestions for defining the exponential function for infinitesimals

I am currently working on "my own" system of infinitesimals. They are pretty much analogous to dual numbers, except for the fact that mine are not nilpotent. My main question concerns ...
Alice's user avatar
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1 answer
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Let $x$, $y$ be positive hyperreal numbers. Can $\frac{x}{y}+\frac{y}{x}$ be infinite? finite? infinitesimal?

I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler. I'm doing an even numbered one and there are only answers for odd numbered problems. Q....
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$\epsilon$, $\delta$ are positive infinitesimal, is the following expression infinitesimal, finite, or infite? [duplicate]

I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler. I need some help with question 39 in Section 1.5 problems. $\epsilon$, $\delta$ are ...
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proving that a polynomial has at least two roots

I have the following question in real analysis course: let $p(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$. Prove that if $x_{0}$ is a root of P, meaning $p(x_{0})=0$ and $p'(x_{0})\neq0$, then p has ...
perplexed's user avatar
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Is it possible to extend the domain and range of a function that maps from R to R to other sets?

I am currently working on a project where I would like to define infintesimals that can be used in conjunction with the real numbers (similar to the hyperreals). Right now, I am working on an ...
Alice's user avatar
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2 votes
2 answers
98 views

Difference between integrand's variable and variable in integration's limits

If a function $G(x)$ is defined by $$G(x)=\int_a^xf(t)dt$$ Then by the Fundamental Theorem I can see that $G(x)$ is an antiderivative of $f(x)$ and thus $$G'(x)=f(x)=\frac{dG(x)}{dx}$$ My confusion ...
Phy's user avatar
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Timing of the Taylor expansion

This is probably going to be a silly question, but here it goes: assume you have a complicated derivation where some function $f(x,\lambda)$ is manipulated in various ways (for example, into something ...
Noobgrammer's user avatar
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Expressing trigonometric functions of infinitesimal arguments as algebraic quantities/elementary functions

I have recently been working with, and reading a bit about, infinitesimals and hyperreals and am currently trying to figure out how the trigonometric functions for infinitesimal inputs should behave ...
Alice's user avatar
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Thoughts about allowing arithmetic with infinitesimals to (mostly) solve limits, can it be done without contradictions?

I am currently working on a project for school where I would like to create an arithmetic framework that would make it easier to solve limits. Suppose we have a function $f:\mathbb{R}\smallsetminus \{...
Alice's user avatar
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0 votes
0 answers
37 views

Does cross product depend on the orthogonality of the basis vector [duplicate]

When I learn cross product, I find myself always using orthogonal basis vectors (e.g. $\hat{i}$, $\hat{j}$ and $\hat{k}$). But I am wondering does cross product depend on the orthogonality of the ...
Bruce M's user avatar
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8 answers
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if motion in infinitely small time interval is linear then why a tangent to a circle has only one common point with the circle

I cant understand the infinitely small linear relation in calculus. I don't yet know calculus. But my physics book use the conclusions from it without explanation. Edit as asked by someone: Instant is ...
sachchidaananda's user avatar
1 vote
1 answer
276 views

How to explain that area under the curve is real number?

As is known, one can consider area under the curve as area that consists of infinitesimally small areas bounded by $dx$ & $dy$ , in general terms. Why doesn't there exist an area that equal to $r+\...
Mike_bb's user avatar
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3 votes
1 answer
116 views

Infinitesimal Interpretation Of Exterior Derivative

I have a question regarding the conceptual understanding of the exterior derivative. I've read that one can view a $k$-form on an $n$-dimensional manifold as a collection of infinitesimal (oriented) ...
fweth's user avatar
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Is it correct to define nilpotent functions as equivalence classes?

I read in Giordano's paper on Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf , p. 3-5 ) that infinitesimal numbers can be considered as ...
Mike_bb's user avatar
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3 votes
3 answers
108 views

Infinitesimal step in different coordinates

I was wondering why an infinitesimal step in different coordinates are not the same size unless specifically stated. For example, dx and dy are both infinitesimal steps in spatial coordinates, but dx/...
Alexander Savadelis's user avatar
5 votes
4 answers
307 views

Cauchy sequence for $0$ in non-standard analysis

In real analysis Cauchy sequence for $0$ is $(1/2,1/4,1/8,...)$. But in non-standard analysis (hyperreal numbers) this sequence is infinitesimal $\varepsilon$. Since hyperreal numbers are extension of ...
Mike_bb's user avatar
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Why is $x(t):=|t|^a$ not an element of $D$? (Synthetic Differential Geometry)

If $a \in (1/2,1)$ then $x(t):=|t|^a$ is not element of $D$ but $x^2=0$ in *$R$. Why is it so? Thanks.
Mike_bb's user avatar
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0 votes
0 answers
103 views

Definition of first order infinitesimal number using equivalence classes

I read about model construction in infinitesimal differential geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf , page 3-6) I can't understand how equivalence class ...
Mike_bb's user avatar
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3 votes
2 answers
151 views

Are Distributions just functions with infinitesimal coefficients?

It is relatively well known that the Dirac delta, the classical example of a distribution, can be written as $$ \frac{a}{π(a^2+x^2)} $$ where $a$ is an infinitesimal such as a hyperreal. This can be ...
Daniel Schwartz's user avatar
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1 answer
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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 ...
Mike_bb's user avatar
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2 votes
1 answer
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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.
Mike_bb's user avatar
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4 votes
1 answer
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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 ...
Aymen's user avatar
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5 votes
1 answer
146 views

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}{...
madness's user avatar
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1 vote
1 answer
145 views

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\...
Mike_bb's user avatar
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1 vote
1 answer
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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 ...
Anixx's user avatar
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1 vote
1 answer
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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, &...
Mike_bb's user avatar
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2 votes
1 answer
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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 $\pi f(x)^2$ ...
a_i_r's user avatar
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1 vote
1 answer
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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 ...
Mike_bb's user avatar
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-1 votes
1 answer
122 views

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.
Mike_bb's user avatar
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1 vote
1 answer
201 views

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 $...
Mike_bb's user avatar
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2 votes
1 answer
174 views

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" ...
Mike_bb's user avatar
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2 votes
1 answer
258 views

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 ...
Mike_bb's user avatar
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2 votes
2 answers
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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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