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

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3
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2answers
170 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?
0
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1answer
20 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\\ ...
0
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1answer
26 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 ...
0
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0answers
33 views

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 ...
13
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8answers
539 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 ...
8
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3answers
450 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 ...
15
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0answers
410 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 ...
3
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3answers
133 views

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 ...
1
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0answers
46 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 ...
0
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2answers
37 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
88 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 ...
1
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1answer
55 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.
0
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2answers
63 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 ...
1
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4answers
85 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 ...
1
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1answer
67 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 ...
1
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1answer
32 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 $$ds^2 = dx^2 + dy^2$$ ...
12
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2answers
280 views

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 ...
1
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1answer
53 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). ...
0
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3answers
72 views

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 ...
9
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2answers
190 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, ...
5
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3answers
157 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 ...
-3
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1answer
250 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 ...
0
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1answer
26 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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0answers
24 views

$\approx$ 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 ...
0
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1answer
43 views

If a square has a concrete area of $2 m²$, how long is its side?

If we make a square with an area of $2 m²$, its side is square root of $2$ then. Wouldn't that mean that the square root of 2 has a concrete length (and therefore point in the real numbers axis). We ...
2
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4answers
178 views

What is the answer to the paradox of the infinitesimal?

I just read this article on npr, which mentioned the following question: You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're ...
0
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5answers
194 views

Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
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6answers
233 views

Definition of tangent

What is the formal definition of a tangent to a curve? The only one I can find is that it is a straight line drawn between two infinitely close points on the curve.
2
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2answers
134 views

Surface Area of a Sphere

I'm having trouble with finding the surface area of a sphere, without using any calculus. What I thought, was that the surface area of a sphere is fundamentally an infinite number of rings, ...
0
votes
1answer
37 views

In-depth explanation of infinitely small differences?

As in this question's title, could anyone give me an explanation of an infinitely small difference and how one would calculate it (if one could even do that)? I've been trying to learn calculus for a ...
0
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1answer
65 views

If 0.99…=1 What about 0.89…=0.9?

I notice the general pattern is that ?.??999... equals to 0.??1 more than the repeating 9 part. Is it true?
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0answers
86 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
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1answer
52 views

Infinitesimals in gradients

Take the function $y(\vec v)$ such that $y:\mathbb R^n\to\mathbb R$. Given it's gradient $\nabla y = \left(\frac{\partial y}{\partial v_1},\cdots,\frac{\partial y}{\partial v_n}\right)$, it is ...
0
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2answers
35 views

Muddled Notions Regarding the Measurement of Quantities in Different Dimensions

In my calculus class, I've learned a lot about finding areas/volumes of various shapes by summing up infinitely small slices of 'something' and adding them all up. This is very interesting to me, but ...
2
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3answers
87 views

Functions and Infinitesimals?

My textbook says the following: $ads = v dv$ where $a$, $s$, $v$ are functions. I was introduced to integral calculus a semester ago, and Riemann Sums (and integrals) are the only notion of ...
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5answers
290 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 ...
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0answers
36 views

Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing ...
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2answers
1k views

Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
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5answers
256 views

What is the name of $0.\overline{0}1$

Short question: What is the name of the number closest but not equal to zero? Long question: Some programmers were discussing about the smallest number close to zero, which is ...
1
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1answer
67 views

Limits and common sense

I'm stuck in understanding of limits. It all makes sense, but at a certain point my answers which seem logical to me are not true. Please can somebody explain why as a huge number gets divided by a ...
4
votes
1answer
275 views

infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
1
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1answer
70 views

Small Lorentz Transformation

This is very simple and I can 50% understand it but would like to properly understand why it is. If we have an infinitesimal Lorentz transformation $\Lambda^\mu _\nu = \delta^\mu _\nu + \omega^\mu ...
4
votes
1answer
122 views

Non-standard analysis way of proving that derivative of $e^x$ is $e^x$

What is the non-standard (infinitesimal) analysis way of proving that the derivative of $e^x$ is $e^x$? I tried to prove it myself, but I am having a hard time proving this without recourse to ...
14
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7answers
304 views

Intuition for a real line vs. a “hyperreal line”

I am a student of pure mathematics but I have no formal background in nonstandard analysis. I came across the concept of a hyperreal field recently, read just a little about them, and followed the ...
4
votes
2answers
138 views

How do mathematics define a point?

I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified.
4
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1answer
82 views

Perturbation in characteristic p, or Why, really, does Lie's theorem fail?

While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ...
14
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3answers
602 views

Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ...
5
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2answers
147 views

How can this result in Thermodynamics be rigorously proved?

In Fermi's "Thermodynamics" there's a proof of the formula: $$W=\int _{V_1} ^{V_2} p\,\text dV,$$that is, the work done by the pressure of a gas that expands from a volume $V_1$ to a volume $V_2$ on ...
2
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1answer
400 views

Proof of Chain Rule using Nonstandard Analysis

I am trying to make an introduction and make myself comfortable with the Nonstandard Analysis in order to gain intuition for derivatives and integration. I am trying to prove myself the famous Chain ...
4
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1answer
177 views

Surface infinitesimals and its intuitive manipulation?

The excess pressure in the concave side of any liquid bubble or drop with surface tension of the liquid being $T$ is $\frac {4T}r$ and $\frac {2T}r$ respectively. I wanted to derive it using a ...