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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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$x^2+y^2=0$ 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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finding the formula and the bound of a series according to 2 given conditions

give an examble for a sequence of numbers that so that : 1. for ever n : -10 < a(n) < -10 + 1/n 2. what is the bound of this sequence ( to what value it is getting closer and closer ? )
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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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1answer
59 views

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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28 views

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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1answer
47 views

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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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
58 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
48 views

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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1answer
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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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1answer
55 views

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
49 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
104 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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33 views

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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$dx$ and Planck length.

It seems that our Universe is "pixellated". The smallest possible distance is the famous Planck-length. Instead in the realm of maths, the smallest possible distance is $dx$. So $dx$ should belong to ...
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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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1answer
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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
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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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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 ...
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Prove that a function have no maximum

I want to prove that $\frac{\arctan(x)}{x} $ have no maximum in the interval $(0,\infty)$ ? I did proved that $\lim \limits_{ x \to 0+} \frac{\arctan(x)}{x} =1$ and $\lim \limits_{x \to \infty}\frac{\...
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How does exponentiation work with infinitesimal hyperreal numbers

Given a hyperreal infinitesimal number $\epsilon$ , is it meaningful to take its square root, $\sqrt{\epsilon}$ or any other root? What about using it as an exponent, as in $2^{\epsilon}$ ? And what ...
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Why can we replace an infinitesimal in a limit with an equivalent infinitesimal?

I read the following in a website. I want to know why we can replace one infinitesimal with an equivalent one. The idea seems intuitive but is there a formal proof?
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Manipulating infinitely small $dx$

From basic infinitesimal calculus, we know that the $$\frac{dy}{dx}=1$$ implies $$dy=dx.$$ However, what is the exact argument behind this result? Now, consider $$\frac{d^2y}{dx^2}=1.$$ Why can't we ...
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About the Left Hand Derivative = Right Hand Derivative condition of differentiability

Somehow I'm getting the conclusion that the only curve which can satisfy this condition is a straight line. My reasoning is: The right hand derivative of a function $f(x)$ at point $x=a$ is $$\lim_{...
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Differential Equations and Infinitesimals

Say we have a differential equation such as $ \frac{dy}{dx} =xy$ I was taught you would solve by rewriting the equation as $\frac{dy}{y} = xdx$ Then you integrate both sides $ \int\frac{dy}{y}=\int ...
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Dealing with double infinitesimal quantities?

I have been working on some physics problems, and have realised that I sometimes write a single infinitesimal (or delta) quantity that actually is a product of two independent delta quantities, e.g. $...
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Total Second Derivative in Leibniz Notation - Issues and Questions

I have been thinking about the second derivative from the Liebniz perspective. Take the equation $y = x^3$. The first derivative is $\frac{dy}{dx} = 3x^2$. Now, the second derivative is commonly ...
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Prove: If $x$ has the property that $0\leq x<h$ for every$ h>0$, then$ x=0$.

I'm going through Apostol's Calculus I introduction, and I'm trying to prove this, but I'm having a little trouble doing it. It's proposed as an exercise in section I 3.5: order axioms. So, what I ...