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

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81
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6answers
5k views

Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
53
votes
11answers
7k views

What is $dx$ in integration?

When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board. $$\int f(x)\, dx$$ When he came to explain the meaning of the $dx$, he told us ...
21
votes
2answers
487 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 ...
18
votes
1answer
353 views

Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ...
16
votes
0answers
492 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 ...
15
votes
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 ...
14
votes
7answers
325 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 ...
14
votes
3answers
642 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 ...
13
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8answers
582 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 ...
12
votes
2answers
294 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 ...
10
votes
5answers
822 views

Justification of algebraic manipulation of infinitesimals

As an engineering student, I regularly see people making arguments like this: Consider a rectangle of dimensions $x\times 4x$. If we make $x$ bigger by a small quantity $dx$ then this will make $4x$ ...
10
votes
5answers
313 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 ...
9
votes
6answers
2k views

What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
9
votes
2answers
206 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, ...
8
votes
3answers
479 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 ...
7
votes
1answer
162 views

What is this limit called? Is it a different kind of derivative?

(first I should notice you this is not something I can look up in a textbook, because I'm learning partial derivatives, alike I do with most Maths, as a hobby. If something below is wrong, blame the ...
7
votes
6answers
243 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.
6
votes
6answers
1k views

What does limit actually mean?

I have been in a deep confusion for about a month over the topic of limits! According to our book the limit at $a$ is the value being approached by a function $f(x)$ as $x$ approaches $a$ I have a ...
5
votes
3answers
105 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 ...
5
votes
3answers
186 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 ...
5
votes
2answers
152 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 ...
5
votes
3answers
95 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 ...
5
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1answer
140 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 ...
5
votes
2answers
80 views

How to show $\chi_{{}^{*}P} ={}^{*}\chi_{P}$ by transfer principle?

Let $\mathfrak{R}$ be the real number system, $(\mathbb{R},+,\cdot,<)$ and ${}^{*}\mathfrak{R}$ be the hyperreal number system $({}^{*}\mathbb{R},{}^{*}+,{}^{*}\cdot,{}^{*}<)$. Transfer ...
5
votes
1answer
77 views

Determining hyperreal class for $\frac{\epsilon + \delta}{\sqrt{\epsilon^2 + \delta^2}}$

I'm solidifying my calculus by going through Keisler's book that uses a hyperreal/infinitesimal approach. I'm stuck on this problem. Given infinitesimals $\epsilon,\delta > 0$, deterimine whether ...
5
votes
1answer
93 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 ...
5
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0answers
157 views

Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals 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
votes
3answers
201 views

How $\frac{dx}{dy}=f(x)g(y) \Leftrightarrow \int \frac{dx}{f(x)} = \int g(y)dy$?

In my intro differential equations class we have often used the "equivalence" stated in title. It seems to me that somehow, the intermediate step $$ \frac{dx}{f(x)} = g(y)dy$$ is being used, in which ...
4
votes
3answers
182 views

non-archimedean in lay terms

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way ...
4
votes
1answer
319 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 ...
4
votes
1answer
188 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 ...
4
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0answers
88 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 ...
4
votes
1answer
88 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 ...
3
votes
3answers
145 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 ...
3
votes
3answers
857 views

Calculating limits using the $\epsilon$-$\delta$ definition.

Suppose you have a function $f(x)=( x^2-4)/(x-2)$. How then do we find the limit as $x\to2$ in accordance with the epsilon delta definition? I mean suppose we don't know how to calculate limit and we ...
3
votes
2answers
275 views

Nilpotent infinitesimals comparison

I'd like to understand better the advantages and disadvantages of various approaches to nilpotent infinitesimal numbers and their application to differential geometry in the context of physics and ...
3
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1answer
59 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 ...
3
votes
1answer
186 views

Why every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$

Let $\mathbb{Q}$ be the set of rational numbers. Show that every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$. This is an exercise on page 180, A ...
3
votes
2answers
181 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?
3
votes
2answers
386 views

Definition of the integral in non-standard calculus

In the sources I've seen, the integral is defined in non-standard calculus as the hyperreal extension of a function related to Riemann sums. E.g., Let $$ S(\Delta x) = \sum_{a}^{b}f(x)\Delta x$$ be a ...
3
votes
2answers
156 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, ...
3
votes
3answers
198 views

How can I show that $f(x) = (x^2)/(1-e^x)$ has global minimum at $(0, +\infty)$?

I showed that $\lim f(x) = 0$ at both the $0$ end and $+\infty$ end. What is the proper way to finish the proof?
2
votes
2answers
203 views

Defining infinitesimals

Can such definition of infinitesimals hold? $$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$ And, if the above definiton works, then obviously ...
2
votes
5answers
619 views

What's the importance of “infinitesimally small” whenever calculus is explained

First of all, i just like reading and understanding things related to math, but NOT at all any expert in math. So, I apologize if the question seems dumb. It always puzzles me whenever a basic ...
2
votes
5answers
433 views

Is there a scientific name for 0.infinity?

First of all I want to say that when coming to math - I know absolutely nothing - so please forgive me if my question is not "scientifically" correct, if it is not "syntax-correct" - or even too ...
2
votes
4answers
68 views

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 ...
2
votes
1answer
87 views

Co-transitivity of the constructive order relation

One of the first exercises in J.L. Bell's A Primer of Infinitesimal Analysis asks the reader to show that, for arbitrary real numbers $a$, $b$, and $x$, if $a < b$, then either $x > a$ or $x ...
2
votes
4answers
198 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 ...
2
votes
3answers
611 views

Formal definition of the Differential of a function

The formal definition of the differential of a differentiable function $f: x \mapsto y=f(x)$ is that it's a two-variable function, its name is $df$ and its value is $df(x,\Delta_X) = ...