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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Definition of Continuity in Non Standard Analysis (Internal Set Theory)

In Alain M. Robert's book, he affirms that continuity is implicitly defined by S-continuity and give the following example: Let $I$ be a standard interval and $\mathcal{F}$ be the space of numerical ...
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How $\frac{1}{2}\Delta(\epsilon \vec E^2)=\epsilon (\Delta \vec E) \cdot \vec E$

How $\frac{1}{2}\Delta(\epsilon \vec E^2)=\epsilon (\Delta \vec E) \cdot \vec E$ When I saw the expression, I thought they were differentiating (taking small changes in electric field) $$\frac{1}{2}\...
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Going from $t^{1-\varepsilon}$ for all $\varepsilon>0$ to $t$

Say $\{S(t)\}_{t>0}$ is a family of subsets in $\mathbb{R}^n$ such that for all $\varepsilon>0$ we may find $t_\varepsilon>0$ and $M(\varepsilon)>0$ such that $$ |x|\leq M(\varepsilon) t^{...
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Is there a natural, operations-preserving bijection between Levi-Civita field and (a subset of) divergent integrals?

For instance, would the following definition of multiplication on divergent integrals correspond to the multiplication in Levi-Civita field? $\int_0^\infty f(x)dx \cdot \int_0^\infty g(x)dx =\int_0^{\...
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Differentiating by a function

I'm trying to learn some physics, and so it has come to pass that I came across an example in Arnol'd's book, which reads $$ x''(t) = \frac{dU}{dx}, $$ where $U(x) := gx$ is a function of $x$ (this ...
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106 views

Defining the Derivative using Internal Set Theory (Non Standard Analysis)

As an engineer, I have found NSA (Non Standard Analysis) to be much closer to our intuition than traditional calculus. Since there are basically two approaches to NSA, one using Hyperreals and another ...
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Intuition for derivatives of trigonometric functions

I’m trying to understand the geometric/infinitesimal ‘derivations’ of the formulae for derivatives of trigonometric functions here. What I don’t understand is how the similarity of the triangles gets ...
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Infinitely small quantities of various orders explained via a geometric illustration involving a circle ( Bouasse, Cours de mathématiques générales)

In his Cours de mathématiques générales Bouasse illustrates the idea of infinitely small quantities of various orders via a geometric example involving a circle. He resorts to a property of the figure ...
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Definite integral of a differential : $\int_{a}^{b} dy$ ( More generally: variables of integration versus differentials)

Suppose that $y$ is a function of a variable $x$ ( shortly : $y=f(x)$) ; in that case $ dy = f'(x) dx $ ( this is just the definition of a differential). With this definition in hand, I try to ...
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Infinitesimal Generator for the G/G/1 queue

I read the infinitesimal generator for the M/M/1 queue and thought to generalize to the G/G/1 queue. More specifically, though the queue length process is not Markovian anymore, we could consider an ...
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48 views

Calculating path integral "with infinitesimal quantity"(?) [closed]

This example is on "Fundamentals of Statistical and Thermal Physics" by Reif. I didn't know what exactly to write on the title, the subject is quite new to me. My question is how they got ...
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Chain rule in Partial derivatives Proof

I am trying to prove chain rule for Partial derivatives: Let $z=f(x,y)$ and $x=x(t),y(t)$. We need to prove that: $$\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x} \frac{\partial x}{\...
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38 views

Closure under reverse tetration

The natural numbers are closed under addition, but not subtraction The integers are closed under multiplication, but not division The rationals are closed under exponentiation, but not roots The real ...
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45 views

Integration which includes a diffferential(infinitesimal?) element of combined function , not a kind of $~ dx, d\theta ,ds,dt ~$

I think that this is the first wime when I handle an integration with infinitesimal(differetial?) element with combined function . $$ a \in \mathbb R_{> 0} $$ $$ \int_{0 }^{\frac{\pi}{2} } \frac{...
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How does the definition of the differential dy imply we are talking about infinitesimal change?

I know differentials are seen as infinitesimal changes but I am confused because the definition of the differential doesn't seem to imply anything infinitesimal: If $y = f(x)$ and you are in a point $...
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Can you (briefly) have a real, non-zero infinitesimal?

Suppose you have the set of the rationals between 0 and 1, let's call it $\mathbb{Q}_1$. Now I want the probability of choosing a number in $\mathbb{Q}_1$ and getting 1/2. If I add this to the ...
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Equation of infinitesimal elements of absolute component and rotation angle component with right triangle

I've drawn the below diagram. $$ dx= a \cdot \sec^{2}\left( \phi_{} \right) d \phi_{} ~~ \leftarrow~~ \text{?}$$ $$ x= a \tan \left( \phi_{} \right) ~~ \leftarrow~~ \text{ok} $$ $$ \phi_{} = ...
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Markov chain: probability of getting to one of the states before another

Let's say we have a 3-state CTMC represented by $$ \require{enclose} \begin{array}{ccc} & & \Large{\enclose{circle}{1}} \\\ & \lambda_1 \Large{\nearrow} & \\\ \Large{\enclose{...
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197 views

Is the Pythagorean Theorem true for infinitesimal leg?

In Newtonian Physics (Classical Mechanics) a body rotates around another (massive) body with constant circular speed ω (stationary trajectory) eternally. The linear velocity v is perpendicular to the ...
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54 views

What's the criterion for neglecting terms "much smaller than" other terms?

In a physics exercise relating some electric forces. I found the following equation for the restoring force: $ F = \frac{kQq}{(y_0 + \Delta y)^2} - \frac{kQq}{y_0^2}$ Where $ \Delta y << y_0 $ ...
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1answer
52 views

Do these statements about analysis with dual numbers make any sense?

I am reading Color for the Sciences by Jan Koenderink, and in Ch. 3 he introduces the dual number system to define the space of possible power spectra for a beam of light. However, his statements do ...
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What is the motivation behind the definition of infinitesimal probability $p(x)dx$'s definition?

I have had this question for a very long time now, which is:- "Why do we have $p(x)\,dx$ to be defined as the probability of $x$ lying in the interval $[x, x +dx]$ and not the interval $[x - \...
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1answer
107 views

On the existence of infinitesimals.

Creating the first infinitesimal by use of Compactness Theorem In this question someone tried to create the first infinitesimal using the compactness theorem, but I don't know much about compactness ...
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Can the integral $\int e^{dx}$ be solved?

Can the integral $\int e^{dx}$ be solved? I took Calculus I course, so I can compute easy integrals, but not observe and develop new things, like this particular integral. As far as I know, the $\frac{...
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2answers
49 views

How do you sum difference between two values of a function separated by an infinitesimal?

I have a function, $n_p(T)$ that gives me the number of protons at a given temperature. I would like to calculate the difference between the number of protons at a given temperature, $T_1$ and the ...
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191 views

Cognitive view of Infinitesimals in Keisler's Book through Cauchy's idea of infinitesimals?

(Quoting Keisler)$^{1}$ - ' $\mathbb{R}^{*}$ has a positive infinitesimal, that is, an element ε such that 0 < ε and ε < r for every positive r $\in \mathbb{R}$ '. (Please note - I happen to be ...
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1answer
32 views

Intuitive error in finding Volume of sphere using single integration.

To calculate the volume of a sphere of radius $\mathbf R$, I considered a thin disc of radius $ r= \mathbf Rsin\theta$ ,where $\theta$ is the angle radius vector on the circumference makes with the ...
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1answer
59 views

Can we have nonstandard/transfinite "decimal" expansions?

Can we take the decimal (or any arbitrary base) representation of a real number and just append some more digits beyond it? Is there a theory that covers this, maybe some kind of non-standard analysis?...
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Can numbers smaller than infinitesimals exist?

I have a good idea of infinitesimals to some extent.( A bit of non standard analysis) I am reading the book of keisler on non standard analysis and calculus. I am okay with them all but, if "a&...
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49 views

Order of infinitesimal $\int_0^{|x|^2}e^{-c_1 r-c_2\frac{|x|^2}{r}}dr$

How could I find the right order of infinitesimal of $$\int_0^{|x|^2}e^{-c_1 r-c_2\frac{|x|^2}{r}}dr,\qquad c_1,c_2>0,$$ as $|x|$ tends to $+\infty$? Should it be trivial? For an upper bound, I ...
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How to think of the location concept of a point?

It is said that points are to be thought of as 0-dimensional. It is also said that they represent a position, i.e., a location. In the physical world, location obviously encompasses some area, say, ...
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1answer
54 views

Are independent variable infinitesimals positive?

A calculation I'm doing involves the inequality $$\frac{dt}{dx}\leq \frac{-kt}{x}$$ and I want to shift the dx and t to the other sides, so I can integrate. Now t is known to be positive (it's $xf(x)$,...
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How to interpret $xdx$ intuitively if $x$ represents time?

From the book Calculus Made Easy Now in the calculus we write $dx$ for a little bit of $x$. These things such as $dx$ (...) are called “differentials”. If $dx$ is a small bit of $x$, and relatively ...
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2answers
109 views

On counting the areas covered by holes in a function in integration

As far as I know, holes in a function at the endpoints of an interval aren't usually given any importance while integrating over that interval. For example, while calculating the area under the ...
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226 views

Why 1 is not the limit of 0.999... but equal? [duplicate]

$\lim_{x\to0}x=0$ $\lim_{x\to1}x=1$ Limit is unreachable, isn't that mean for x→0 can never be 0? That's why infinitesimal isn't 0, because its limit is 0. (By Wikipedia: infinitesimals or ...
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87 views

What is difference between derivative in standard and non standard analysis?

I am reading the book on complex analysis by Tristan Needham. In that book he explains derivative in an intuitive way as a quantity by which dx is expanded to get dy in both complex and real number ...
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1answer
95 views

Does a higher dimensional space have more points than a low dimensional space? And in some way are some infinities bigger than others? [duplicate]

How many points can exist in one dimension? ANS: Infinite! How many lines can exist in a two dimension? ANS: Infinite! Now my question is how many points can exist two dimensions or even higher ...
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How to prove that hyper reals are not reals?

I was told by my professors that infinitesimals are numbers which are smaller than all the reals. And I was also told that it is the sequence, $\{1,1/2,1/3,1/4,\dots\}$. And I am comfortable with ...
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72 views

Any Equivalence between Limit computed via L'Hospital's rule and 0/0 i.e $(\frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)})$?

In Standard Analysis, Does $$\lim_{x \to c}\frac{f(x)}{g(x)}=\frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)}$$ Given that $$\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0$$ Yes, we can get the computed answer ...
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1answer
70 views

Relative scheme structure on a specific scheme related to the symmetric product of curves.

Given a smooth algebraic curve $C$ with a closed point $p$. The symmetric product $\text{Sym}^i(C)$ has a closed subvariety $\text{Sym}^{i-1}(C)$ which its embedding is given by adding the point $p$. ...
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183 views

Proof for how .9999999...=1 [duplicate]

I was just curious for the proof/theorem for a very close decimal (so when you keep adding a decimal) to equal the next integer such as .999999999999....=1. Thank you.
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1answer
66 views

Asking if an infinite series will converge or diverge

I have two questions regarding the convergence of infinite series. One, if $\sum_{n=1}^{\infty}a_n$ converges to L, where L is defined, does $a_n < L$ for each n? Two, if $a_n$ is defined for each ...
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70 views

On the infinitesimals. [duplicate]

I have completed my calculus course a few months back, I was introduced to the idea of infinitesimals as just a symbol in that course. Now I see that they are being used not only in math but in many ...
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1answer
45 views

Summation of function subtraction with a limit as $\epsilon \rightarrow 0$

I got stuck in a problem in the middle of my calculations of integrals and sums. $$\lim_{\epsilon \rightarrow 0} \sum_{n=1}^\infty f(n-\epsilon)-f(n+\epsilon)=0$$ where $f$ is continuous on all of the ...
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1answer
93 views

What is an isolated subgroup? [closed]

" As in the case of finite numbers, the infinitesimal numbers form an isolated subgroup  of $R_{inf}$ of $^*R$"page 152 What does this sentence mean? What is an isolated subgroup?
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1answer
55 views

A question about a limit

I have the next $\lim_{x \to \frac{\pi}{4}} \frac{f(x)-f(\frac{\pi}{4})}{x-\frac{\pi}{4}}$ when the function is $$f(x)=\frac{x+\sin(x)}{\tan(x)}.$$ I don't know how to even start. I am sorry that this ...
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1answer
39 views

Continuity of Functions with Vertical Tangents

I'm running into some confusion regarding properties of continuous functions. I'm comfortable with the epsilon-delta definition of limits and the basic definition of continuity at a point $a$ ($\lim_{...
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3answers
69 views

If $dx=dy$, is it true that $x=y$? [closed]

If $dx=dy$, is it true that $x=y$? I am in a dilemma whether this is true.
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1answer
129 views

Does dx in dA=dx*dy represent change, or is it a notation that denotes an infinitesimal arbitrary length? Change vs. an arbitrary physical length

I'm not sure how to explain this, but I have a gap in my understanding of infinitesimals/differentials. I've so far had calc 1 and 2, and have been taught that dy/dx represents a slope, which ...
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283 views

$\pi$ & $\phi$ (Golden ratio), Pentagon inscribed in unit circle

Everyone is aware that square inscribed in unit circle and infinite product giving rise to $\pi$. One of the simplest way to represent $\pi$ with the help of nested radical as follows $$\pi = \lim_{n\...

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