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70
votes
6answers
4k 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 ...
49
votes
11answers
5k 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 ...
17
votes
1answer
320 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 ...
14
votes
3answers
489 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 ...
14
votes
2answers
890 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 ...
12
votes
7answers
263 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 ...
11
votes
2answers
265 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
274 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
1k 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
186 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, ...
7
votes
1answer
151 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
226 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
997 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
120 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
143 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
2answers
77 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
71 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
0answers
144 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
135 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
180 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
155 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
116 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 ...
4
votes
1answer
172 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
votes
0answers
84 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
81 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
5answers
538 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 ...
3
votes
3answers
714 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
1answer
246 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 ...
3
votes
1answer
179 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
301 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
3answers
169 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
184 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
4answers
356 views

Is there a scientific name for 0.infty?

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
1answer
80 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
160 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
361 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) = ...
2
votes
2answers
231 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 ...
2
votes
2answers
112 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, ...
2
votes
3answers
83 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 ...
2
votes
0answers
69 views

differentiable function in R but non continuous derivative for any point in R

Need your help with the next Question (I thought about her and could not find an answer) Is there differentiable function for all R , so that the derivative is non continuous for any point in R ? ...
1
vote
5answers
245 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
vote
1answer
1k views

little-o and its properties

I know that $f(x) = o(g(x))$ for $x \to \infty $ if (and only if) $\lim_{x \to \infty}\frac{f(x)}{g(x)}=0$ Which means than $f(x)$ has a order of growth less than that of $g(x)$. 1) I'm still ...
1
vote
1answer
51 views

What are the analogues of 'transfinitesimal' numbers?

There are such transfinite numbers as $\aleph$ and $\omega$. Do they have infinitesimal analogues? I'm especially interested in ...
1
vote
1answer
91 views

Am I talking right?

I'm trying to describe expected value. My paragraph goes: From probability theory we have $E[f(x)] = \int{f(x)p(x)dx}$. That is, the expected value of $f(x)$ is equal to the sum of infinitesimals ...
1
vote
1answer
38 views

What is $(\operatorname{monad}(0), \leq)$ isomorphic to?

Suppose, given $\epsilon \in \operatorname{monad}(0)$ and $\epsilon \neq 0$, is it true for each $x \in \operatorname{monad}(0)$, $$x = \sum_{r_i \in s \subset \Bbb R, s \text{ is finite}}a_i ...
1
vote
1answer
31 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.
1
vote
1answer
48 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 ...
1
vote
1answer
67 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 ...
1
vote
1answer
361 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 ...
1
vote
1answer
144 views

More than one dx under integral

Normally, integrals look like $\int{f(x)dx}$. But, I occasionally get something like $\int{dx\over 1+f(x)dx}$. Is it a sign of an error? Partcularly, I tried to relativistic constant force ...