For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).

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Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
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6answers
100 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
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Why is zero the only infinitesimal real number?

I am currently reading Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler and was wondering if someone could help me with an aspect treated in the book. On page 24 he says a ...
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30 views

Are Gaussians a basis for the vector space of continuous functions?

How can I prove (or disprove) that the Gaussian function family: $f_{\mu,\sigma}(x)=e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$ Are a basis for $C(\mathbb{R})$ ?
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2answers
64 views

Convergence of $\int\frac{\arctan x}{x} dx$

I can't find function to bound this integral in the intervals from $1$ to $+\infty$, to prove if it converges. $$ \int _1^{\infty }\frac{\arctan x}{x}dx $$ Any idea? How can I refute this if it is ...
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1answer
58 views

Convergent of integral of $1/x^x$

I need to prove if this integral converges (on interval $[1,+\infty))$: $$\int _1^{\infty }\frac{1}{x^x}\;dx$$ Has anybody any idea how to do it? thank you. :)
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37 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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1answer
120 views

What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \begin{equation} \forall a,b,c \in \Bbb ...
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Linear programming in Hilbert spaces

Let $H$ be a real Hilbert space. Let $b,c\in H$, $P\subset H$ be a convex cone and a continuous linear mapping $A:H\rightarrow H$. Consider the following sets: $$ B:=\{(Ax, \langle c,x\rangle:x\in ...
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3answers
115 views

$dx=\frac {dx}{dt}dt $. Why is this equality true and what does it mean? [duplicate]

$dx=\frac {dx}{dt}dt $. I know that this deduction is obvious from the chain rule, given that we treat our dx and dt as just numbers. But I find it quite unsatisfactory to think of it in that sense. ...
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40 views

Why generator in Lie Algebra is defined as the coefficient in taylor expansion of map

Booth defines the infinitesimal generator of a lie group (denote the manifold it defines by $M$) using flow $\theta_t(p)$ by calculatng the limit (mainly the derivation for $f$ in each point $p\in M$) ...
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4answers
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What's an intuition behind $\lambda=1$ for $dx\;dy = \lambda r\;dr\;d\theta$?

I've read a bunch of articles here on converting between rectangular and polar coordinates in integrals. I get the intuition about how the natural infinitesimal area segment in rectangular coordinates ...
4
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3answers
131 views

If $dx$ is an infinitesimal, can't you list all real numbers as the sequence each whole number times dx?

I'm taking calculus right now. If the difference between each real number and the next is an infintesimal, then wouldn't the following sequence $\{0\,dx, 1\,dx, -1\,dx, 2\,dx, -2\,dx, \ldots\}$ be a ...
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1answer
46 views

Derivative of an Infinitesimal?

I am currently studying calculus of variations (for my classical mechanics course). I have, on multiple occasions, seen the derivative of an infinitesimal quantity defined like below $$\frac{d}{dt} ...
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2answers
532 views

Has anybody ever considered “full derivative”?

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which ...
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3answers
106 views

infinitesimal intervals in physics

The density of states of a system in an interval $[E, E+dE]$ is given implicity by $dV = D(E)dE$ (Or I suppose explicitly, by $D(E) = \frac {dV}{dE}$, but we'll be integrating it anyway, so it doesn't ...
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0answers
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Size of a geometric point

It is well known that the geometric points do not have any length, area, volume, or any other dimensional attribute, also geometric object (for example "line") is made up of a infinite number of ...
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3answers
61 views

Does it make sense to ask if there are as many real numbers between 1 and 2 as between 0 and 1?

If yes, how does one go about finding the answer? Can the same question be asked for the hyperreals too?
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2answers
60 views

Using differentials to optimize a function

I've read in a paper by Tevian Dray an alternative way to solve optimization problems manipulating "differentials". Here is an example of how it works (next I quote the paper). Consider the ...
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1answer
40 views

hyperreals standard part inconsistency

$\def\st{\operatorname{st}}$ I'm studying non-standard calc from Keisler's book. Taking "standard part" rule doesn't make sense... its not commutative. e.g. $a$ is finite non infinitesimal $b,c$ ...
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0answers
64 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
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1answer
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When “magnifying infinitesimals” why dont they have curvature ? (non standard) Infinitesimal calculus

Im reading https://www.math.wisc.edu/~keisler/calc.html. If you open up the chapter $2$ pdf... The $2$ diagrams (1st on page $14$ of the pdf (not the text book), 2nd on page $15$) have me confused. ...
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1answer
92 views

What is the limit ${{\lim }_{x\to\infty}}x^\epsilon$ for an infinitesimal $\epsilon$?

What is the limit $${{\lim }_{x\to\infty}}x^\epsilon$$ for an infinitesimal $\epsilon$? Does it give zero or infinity? Note that I'm considering the infinitesimals described in ...
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Is this a valid thing to do to this differential equation?

Don't tell me how to solve it, but I've been trying to use the following equation to get r which is distance) as a function of t. $$\frac{GMm}{r^2}= -ma$$ I've been working on it for about a year ...
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1answer
51 views

Confused between infinitesimally small and $0$

Consider these two cases: 1) Let $I_n$ be the closed interval $[\frac{1}{n}, 1]$. Then $\bigcup_{n = 1}^{\infty} I_n = (0,1]$. 2) Take the two sets $y = e^x$ and $y = 0$ (the $x$-axis) in ...
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2answers
74 views

Proving: if $|a|<\epsilon \forall \epsilon>0$ then $a=0$ using a direct proof

I am asked to prove: if $|a|<\epsilon,\forall \epsilon>0$, then $a=0$ I can prove this as follows. Assume $a \not= 0$ I want to show then that $|a| \geq \epsilon$ for some $\epsilon$ We ...
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2answers
188 views

Is $\textrm{d}x \in \mathbb{R}$?

Is an infinitesimal a real number? Can "abuse of Leibniz's notation" be justified by claiming that an infinitesimal is a real number? If not, what is an infinitesimal?
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1answer
29 views

Method for computing polar coordinates surface element?

I have tried to compute the "classical" surface element in polar coordinates for volume integration (i.e. $dx\ dy=r \ dr\ d\theta$) through this method: $$ \left\{ \begin{array}{c} x=r \cos \theta\\ ...
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1answer
76 views

Multivariable functions limits and paths

In order to approach a point as (0,0) there many directions to do so. A whole 360 degrees actually. So between [0,360) degrees there are actually infinite directions. My question is why does it ...
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Order of infinitesimal of $\frac{\cos x}{x}$b for $x \to \infty$

I can prove that $\frac{\cos x}{x}$ is an infinitesimal for $x \to \infty$ with the squeeze theorem. But trying to find the order of infinitesimal, I'm not sure if my reasoning is valid. Here's what ...
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8answers
644 views

Is $0$ an Infinitesimal?

For the definition of Infinitesimal, wikipedia says In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or, so small ...
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3answers
488 views

Transcendental a infinitely close to rationals?

Apologies that this question is rather vague, but I do not know how to state it more precisely. Is, say pi, infinitely "close" to some rational number? More importantly, are all transcendental numbers ...
17
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1answer
601 views

Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
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3answers
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Why those division by zero are formalized?

Easy example first: $f(x) = nx$ $f'(x) = (f(x+0)-f(x))/0 = (nx+0n-nx)/0 = (0n)/0 = n$ Hard one: $f(x) = a^x$ $f'(x) = (f(x+0)-f(x))/0 = (a^{x+0}-a^x)/0 = (a^x(a^0-1))/0 = (a^x(e^{\ln(a^0)}-1))/0 ...
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0answers
56 views

How to Interpret Exterior Derivative as Infinitesimal

In Riemann Integral, one can intuitively interpret $dx$ as infinitesimal, and it makes sense, but in differential forms, it lost this interpretation, is there a way to make connection between these ...
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2answers
40 views

$\sqrt{1\pm10\varepsilon+\varepsilon^2}=1\pm P(\varepsilon)$. Is there a better way than mine to find $P(\varepsilon)$?

Some days ago we did a classwork, and there was this exercise: Using the limit definition, verify $$\displaystyle \lim_{x\to0} \frac{3x^2-1}{x+1}=-1.$$ From $\displaystyle ...
5
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1answer
98 views

Integrating over a power of the infinitesimal

I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go. Suppose I have a piecewise constant function $y=f(x)$ with $x,y\in\mathbb{R}^+$, described ...
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1answer
72 views

Non-Standard analysis and infinitesimal

Can someone please explain how Non Standard Analysis is used to justify infinitesimals? I am not very clear about this but apparently it has something to do with hyperreals.
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2answers
69 views

Can we operate on the real numbers in calculus?

For a set theory class, I was reading into the definition and properties of real numbers. Real numbers are Archimedean. That means there are no infinitely large real numbers or infinitesimally small ...
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4answers
98 views

Products of Infinitesimals

In my physics class my professor was abusing the derivative, as per so many physics classes I've been in. This time, he took the quantity $(x+dx)(y+dy)$ and argued that the $dxdy$ term should ...
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1answer
79 views

$dxdy=-dydx$ using Jacobian determinant. Why?

How do you reslove the contradiction due to the fact that $dxdy = dydx$ as per definiton of hyperreals ? Is this abuse of notation and by $dxdy$ its is actually meant $dx \wedge dy$ in both ...
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1answer
41 views

infinitisimal part and the directional integral

In the paper A circle detection approach based on Radon Transform by Erman Okman and Gozde B. Akar. I have a few questions on some basics. first of all what does $$ds^2 = dx^2 + dy^2$$ ...
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What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
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1answer
66 views

Generalization for Stirling numbers 2nd kind to negative column-indexes?

The exponential generating functions for the Stirling numbers 2nd kind are the n'th powers of $f(x)=\exp(x)-1$ (where this is understood as formal power series, Abramowitz&Stegun, 26.8.12). ...
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3answers
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Infinitesimal Unit of Measurement

This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even ...
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2answers
209 views

Newton's “Famous Blunder”?

On page $225$ of Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini (see here for a link), I read In the following demonstration... Newton made a famous blunder... He wrote, ...
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228 views

What problems arise when using infinitesimals in calculus?

In contemporary real analysis we use a limit definition in terms of deltas and epsilons. Before that, people used infinitesimals to calculate limits. Is there a specific non-philosophical reason why ...
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1answer
338 views

Could we assign a numerical value to an infinitesimal?

The common definition of 'infinitesimal' is that they are quantities that are too small to be measured or perceived. If we base the mathematically definition entirely on the common definition it ...
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1answer
27 views

Help with this simple limit question

We have this definition: $$f(h)=O(h^p) \quad\text{if} \quad \lim_{h\rightarrow0}\frac{f(h)}{h^p}=K\neq 0$$ show this: $$O(h^2)+O(h^3 )=O(h^2)$$ So what we need to do is follow the definition: ...
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$\approx$ and $\ll$ for different-order infinitesimals

This seems like a pretty basic question, but I've been searching around and haven't come across the answer. Consider two infinitesimal numbers, $\epsilon$ and $\epsilon^2$. On the one hand, it would ...