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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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what do infinitesimals look like?

i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions (1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a ...
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Adding changes or derivatives between two points

I've playing with the limit definition of derivative and I've to somewhat confusing conclusions. To clarify, I'm from an Engineering background so I don't think that an instantaneous rate of change ...
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Have I discovered a new significance to a previously discovered constant?

I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged ...
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1answer
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Calculate $\lim_{x\to 0}\left(\frac{3\sin(x)-3x \cos(x)}{x^3}\right)^\frac{1}{x}$

This is an exercise from today's exam and I think I did something wrong: Calculate the limit: $$\lim_{x\to 0}\left(\frac{3\sin(x)-3x\cos(x)}{x^3}\right)^{1/x}.$$ To tackle this problem I used ...
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1answer
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What are the prime ideals in the ring $\mathbb{Z}[i](\epsilon)/(\epsilon^2) $?

Let $\mathbb{Z}[i,\epsilon] \simeq \mathbb{Z}[i](\epsilon)/(\epsilon^2)\simeq \mathbb{Z}[x,\epsilon]/(x^2 +1, \epsilon^2)$ be the Gaussian integers with an infinitesimal number $\epsilon^2 = 0$. What ...
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Finding the infinitesimal order of a function as $n \to \infty$

I have to find the infinitesimal order of a function $f(n)$ as $n \to \infty$, this is what I did: $$ \begin{split} f(n) &= \sqrt{n+1} - \sqrt{n} + \frac{1}n \\ &= \frac{n\sqrt{n+1} - n\...
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Notation Question for “Generalization of L’hopital’s Rule” PDF by V. V. Ivlev and I. A. Shilin

Within this brief text starting on page 2, the author(s) have a habit of employing a sort of "mixed" notation of differentiation, mixing the "prime marks" of Lagrange's notation for differentiation of ...
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1answer
81 views

What is the product of infinitely many infinitesimals?

In general, taking the sequence for example, if $\lim\limits_{n \to \infty}a(n)=0$, we call the sequence $a(n)$ is an infinitesimal. It's well known that, the product of a finite number of ...
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50 views

When $x$ is given, what is $\mathrm{d}x$? [closed]

In the spherical coordinate, we know that $x = r\sin\theta\cos\phi$. Why does it imply that $$\mathrm{d}x = r\cos\theta\cos\phi \mathrm{d}\theta - r\sin \theta\sin \phi \mathrm{d} \phi?$$ It is first ...
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Two infinitesimals infinitely close to each other compared to a third one?

On keisler's book, Elementary Calculus, he presented the following definition: $\epsilon\ \approx\ \delta$ compared to $\Delta x$ if $\epsilon/ \Delta x \approx\ \delta\ / \Delta x$ and used it to ...
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Can we make it equal to x?

$\sqrt{6 +\sqrt{6 +\sqrt{6 + \ldots}}}$. This is the famous question. I have to calculate it's value. I found somewhere to the solution to be putting this number equal to a variable $x$. That is, $\...
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$x^2+y^2=0$ implies $x^2$ for every $x,y\in R$ is false

In Exercise 1.9 (iii) in A primer of infinitesimal analysis by John Bell we are asked to show that the following assertion is false: "$x^2+y^2=0$ implies $x^2=0$ for every $x,y\in R$", where $R$ is ...
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Proof that “$\uparrow$ is the unique solution of $tiny(G) = G$”

Tiny & miny games can be defined as: $$tiny(G) = \{0||0|-G\}$$ $$miny(G) = -tiny(G) = \{G|0||0\}$$ From the Wikipedia page for tiny and miny: Similarly curious, mathematician John Horton ...
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How to guess the output of $F(x)$ for $x$ that causes $0/0$

As an example: $$ F(x) = \frac{(1-x)^2-1}{(1+x)^2-1} $$ When $x=0$ this involves $0/0$ and calculators output 'undefined'. But when looking at a graph of the function it is intuitively clear that ...
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If $f'(x)=0$, is then $f(x+dx)=f(x)$?

I am always struggling with infinitesimals, and not sure I'm getting this right. The title basically states the simplest version of my question: If a function has zero slope at some point, is it ...
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The value of the infinitesimal in integral doesn't matter?

I am studying calculus by the infinitesimal approach using "Elementary Calculus: An Infinitesimal Approach" textbook. in page 187, the author proved that the value of the infinitesimal we integrate ...
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Validity of allowing $\epsilon$ to vanish in Baby Rudin Theorem 3.10a

In Theorem 3.10a of Rudin's PMA, we prove that $$\text{diam } \bar E = \text{diam } E$$ by [fixing] $\epsilon>0$ and [choosing] $p \in \bar E, q \in \bar E$. By the definition of $\bar E$ there ...
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Find infinitesimal transformation and generator

We have asked to find the infinitesimal transformation and generator for the following group of transformation. $ T_a: \bar{x}=ax$ and $\bar{u}=\frac{1}{a} u$ Please tell me how to find it. Thanking ...
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Paradox about the volume of a cylinder

Trying to apply Cavalieri's method of indivisibles to calculate the volume of a cylinder with radius $R$ and height $h$, I get the following paradoxical argument. A cylinder with radius $R$ and ...
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Partial derivative of Algebra on infinitestimal

I'm seeing an infinitestimal equation in thermaldynmaic: $dE=TdS-PdV$. However, whats so interesting is that it's partial deritative was $\displaystyle(\frac{\partial E}{\partial V})_T=T(\frac{\...
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Is there a proper term for the rational and irrational infinitesimals [closed]

There are a different infinities, such as the countable infinity of the rationals, and the uncountable infinity of the reals, Aleph null and Aleph one. The conceptual infinitesimals of 1/infinity ...
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How Euler arrived at power series for $a^x$ and $\ln x$

I'm reading a book that reproduces Euler's arguments, and I have a few questions about a few things he does. Below are parts of the argument: Let $a > 1$. Consider an "infinitely small quantity" $...
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Order and principal part of $f(x)=\, e^{x^2} \int_{x}^{+\infty}e^{-t^2}\,dt$, infinitesimal as $x\to+\infty$

Determine if $f(x)$ is an infinite or an infinitesimal as $x\to+\infty$, its order and the principal part $$f(x)=\, e^{x^2} \int_{x}^{+\infty}e^{-t^2}\,dt$$ I can write, and then, using L'Hôpital's ...
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Can $\mathrm{d}x$ be thought of as a derivative and differentiation or it's just a small change in $x$ and nothing more? [duplicate]

The $\mathrm{d}x$ appears on integrals. I saw conflicting views regarding it. People sometimes write it does have a connection to differentiation and derivatives. Does it or does it not?
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1answer
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Maximizing the value of a two variable function along any curve

I read that, of all the points on an origin-centered circle in the x-y plane, the function $z=ax+by$ is maximum (or minimum) at the point where $\frac{x}{y}=\frac{a}{b}$ I think this is too specific. ...
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1answer
71 views

Limit/Infinitesimal Expression for Probability of a Continuous Random Variable at Any Single Value in Its Support

The following is a lecture slide from a machine learning class: I already have basic understanding of probability, including continuous random variables. And I'm familiar with the typical ...
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2answers
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Doubling dx and dy

Suppose we have a function $f(x,y)$. Thus the total differential is $df=f_x dx + f_y dy$. Does doubling $dx$ and $dy$ double $df$. I want to multiply everything by $2$ but since we are dealing with ...
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How to explain a continuous curve to a layman

In trying to explain irrational numbers, I am now wondering how to explain continuous curves. (Related to limits and such). In discrete systems (topology, geometry, etc.), a curve makes sense because ...
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Equivalent infinitesimal for $\log(\cos(x))$

I recently came across a limit problem which replaces $\log(\cos(x))$ with an equivalent infinitesimal $\cos(x) - 1$. How do we prove that $\cos(x)-1$ is the equivalent infinitesimal for $\log(\cos(...
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Given $f(x,y)=5+2x+4y+x^2+y^2+(x^2y^4)^\frac15$, show that it is differentiable at $(0,0)$.

I was given the function: $f(x,y)=5+2x+4y+x^2+y^2+(x^2y^4)^\frac15$ I need to show it is differentiable at $(0,0)$. I started using the method of differentials and infinitesimal functions: $\...
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A question about nonparametric (kernel function) estimation

Suppose $X_1,\cdots,X_n\overset{iid}\sim F$, the $q$th quantile $\theta=F^{-1}(q)$. Let $K$ denote an $r$th order kernel. That is for some integer $r\ge2$ and constant $C\neq0$, $K$ satisfies $$$$ \...
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Are infinitesimals, i.e. $dx = …$, rigorous and correct notation? [closed]

In many fields of physics and engineering, when we want to describe an infinitesimal, for example, the electric field, we could say $dE(x_i,y_i) = e^{jkr}...dx_0dy_0$ Since derivatives are not ...
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Is the center of mass of infinitesimal pie slice $R/\sqrt{2}$?

I was recently working on a physics problem from edx.org. I think I can show below that the center of mass of an infinitesimal pie slice of a disk is $\frac{R}{\sqrt2}$ from the center. ($R$ is the ...
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1answer
52 views

Infinitesimal Approaches To Differential Geometry As Conservative Extension

When studying differential geometry, I often feel that infinitesimal approaches would do a deal for the intuition. There also seems to exist various examples like synthetic differential geometry or ...
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1answer
115 views

Do surreals prove reals are countable?

If the surreal number $\epsilon = 1/\omega$ is the lower bound of the difference between any two real numbers (since it is smaller than any real number), and there are a countable number of these ...
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Is there an analog with multiplication for the way the limit of a series (additive) becomes an integral?

Roughly speaking - as in the introductory definition of the Riemann integral - we have a whole notation and apparatus for dealing with an infinite sum of ever smaller 'widths' becoming the integral - ...
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If the sum of any finite terms in a sequnce is less than or equal to epsilon, then prove that the infinite sum is also less than or equal to epsilon.

We have a sequence of sequences which converges to $a_{i}$ i.e. $$ \{ x_{i}^m \}_{m=1}^{\infty} \rightarrow a_{i} \quad \forall \; i \in \mathbb{N} $$ for any $\epsilon >0$ and for any $M$, if ...
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1answer
99 views

Strict infinitesimals and concept of scales.

When differential and integral calculus were first discovered, in the 1600-1700s, they were proven to be immensely useful in so many applications it is almost mind boggling. But as far as I know, it ...
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Which branch of mathematics rigorously defines infinitesimals?

I have some trouble doing standard computations in calculus because of the notion of a differential, otherwise known as an infinitesimal, being rather ill defined, in my experience. Are there any ...
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How to prove set of hyperreals {1, 2, 3, …} are > every real number

I'm working through Infinitesimal Calculus by Henle and Kleinberg on my own and I'm having some trouble with one of the exercises. Here's what I have currently: Exercise: Let $j$ be the hyperreal $\{...
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Identity function on a smooth world

For anyone who is familiar with the concept of smooth world (from Bell), is the identity function $f(x)=x$ continuous in that world? Cheers!
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1answer
117 views

Didn't understand a step in Einstein's paper on special relativity

I didn't understand a step in Einstein's paper(special relativity). Suppose we have a function $f(x', t)$ such that: $$\frac{1}{2}[f(0, t) + f(0, t + \frac{x'}{c - v} + \frac{x'}{c + v})] = f(x', t +...
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Is $1-\cos^2(x)$ a greater infinitesimal than $\sin^3(x)$ as $\to 0+$?

What I did was to see that both are $\sin$ functions and the first one acts as $x^2$ while the second one goes to $x^3$. So the order of infinitesimal of the first one is $2$ and for the second it is $...
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Attempting to express infinitesimals using Arabic numerals

I've been researching infinitesimals in my spare time, and have come to an hypothesis. Is it fair to say that $\varepsilon = 0.\bar01$? In English: the infinitesimal $\varepsilon$ is equal to an ...
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How can skew-symmetric matrices be thought of as infinitesimal rotations?

I've recently stumbled upon the fact that skew-symmetric matrices represent somehow infinitesimal rotations. Having never encountered them, I looked them up and learnt they have to do with Lie ...
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Calculate the limit $L=\lim_{x\to 0^+}\left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x$.

Question: Calculate the limit $$L=\lim_{x\to 0^+}\left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x.$$ I'm thinking of using infinitesimal, but I'm not used to these kind ...
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If $dx$ is just syntax and not an infinitesimal then why do we apply operations to it?

So apparently my understanding of this concept is either old, outdated, or nonstandard, etc, but I was under the assumption that in an integral, $dx$ represented the "infinitesimal change in $x$", ...
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1answer
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Using annother integral to find da for use in an iterated integrand.

When doing a surface integral like "$\phi=k\cdot\int \int _ {\text{lateral surface}} \frac{\sigma}{r} da$" along the lateral surface of a cone it is necessary to find $da$ in terms of some iterable ...
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What does the p-harmonic series converge to when p = 1 + ε?

In infinitesimal calculus, $\epsilon$ is an infinitesimal number, that is, it is defined to be a number smaller than any real number but greater than $0$. The p-harmonic series is: $\displaystyle \...
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1answer
36 views

Distance of closest neighbor points in a vectorspace ${\mathbb R}^n$ (infinitesimal or zero)?

The title should be almost entirely self explaining, but let me give the specific context and use case I'm thinking about while asking too, it's related to the HadwigerNelson problem and chromatic ...