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9
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2answers
179 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, ...
4
votes
2answers
73 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 ...
0
votes
1answer
24 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: ...
1
vote
0answers
17 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
votes
1answer
38 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
votes
4answers
141 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 ...
1
vote
2answers
72 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 ...
7
votes
6answers
220 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
votes
2answers
91 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
31 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 ...
1
vote
0answers
52 views

find the the variable that maximizes a function

I have a function that I am trying to find for what input it maximizes. $$ f(n) = {\binom{S}{2}}^{n/S}$$ I need to find the $S$ for which this function maximizes (for infinite $n$). more generally, ...
0
votes
1answer
46 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?
4
votes
0answers
82 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 ...
1
vote
1answer
44 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
votes
2answers
31 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
votes
3answers
80 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 ...
10
votes
5answers
261 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 ...
1
vote
0answers
29 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 ...
14
votes
2answers
692 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 ...
0
votes
5answers
232 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
60 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 ...
3
votes
1answer
224 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
vote
1answer
62 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
107 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 ...
11
votes
7answers
236 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
133 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
1answer
78 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 ...
13
votes
3answers
445 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
votes
2answers
140 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 ...
1
vote
1answer
323 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
votes
1answer
170 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 ...
2
votes
0answers
68 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 ? ...
4
votes
3answers
142 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 ...
1
vote
1answer
97 views

sum of legs of inscribed right triangle

Consider a right and isoceles triangle ABC inscribed in a circle such as its hypotenuse forms the diameter AB of the circle (the right vertex is thus at the "apex" of the circle). If we procede to an ...
3
votes
2answers
276 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
486 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 ...
0
votes
3answers
244 views

What is the epsilon-delta definition of limits, exactly?

I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition ...
7
votes
6answers
864 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 ...
6
votes
6answers
781 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 ...
17
votes
1answer
311 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 ...
2
votes
1answer
79 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 ...
-4
votes
1answer
127 views

Proof that $0 = 1$ or that $\frac1\infty = 0$ [closed]

Proof that $0.9\ldots = 1$: Let $x = 0.999\ldots$ $$10x = 9.999\ldots\\ 9x = 9.999\ldots - 0.999\\ x = 1$$ Proof $0 = 1$: $$1 - \frac{1}\infty = 0.999\ldots\\ 1 - \frac{1}\infty = 1\\ ...
2
votes
2answers
175 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
3answers
275 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) = ...
1
vote
0answers
65 views

rigorous treatment of infinitesimal reparametrizations

my first post :) I am asking this directed to mathematicians or mathematical physicists since I don't like the usual physics approach. Reading some string theory books I always find that the ...
2
votes
2answers
208 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 ...
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 ...
7
votes
1answer
147 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 ...
1
vote
1answer
142 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 ...
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 ...