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

learn more… | top users | synonyms

75
votes
11answers
15k 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 ...
-5
votes
1answer
575 views

Could we assign a numerical value to an infinitesimal?

This is a rewrite of the original question. I asked that one because I was trying to work out the meaning of the nilsquare infinitesimals used in traditional calculus (most like SIA) but my thoughts ...
11
votes
5answers
473 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 +dz^...
2
votes
4answers
607 views

Is this a correct/good way to think interpret differentials for the beginning calculus student?

I was reading the answers to this question, and I came across the following answer which seems intuitive, but too good to be true: Typically, the $\frac{dy}{dx}$ notation is used to denote the ...
25
votes
2answers
832 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 ...
7
votes
6answers
2k 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 ...
101
votes
7answers
8k 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 ...
38
votes
15answers
4k views

Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ \...
10
votes
4answers
1k 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 $4x$ ...
1
vote
10answers
1k views

What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
5
votes
4answers
212 views

First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
4
votes
2answers
383 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
votes
1answer
5k 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 ...
30
votes
7answers
2k views

Category-theoretic description of the real numbers

The familiar number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ all have "natural constructions", which indicate, why they are mathematically interesting. For example, equipping $\mathbb{N}$ with ...
19
votes
1answer
737 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
2k 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 ...
7
votes
6answers
749 views

How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
16
votes
8answers
888 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 ...
14
votes
3answers
1k 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 ...
7
votes
1answer
610 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 ...
10
votes
6answers
4k 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 ...
4
votes
1answer
209 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 ...
13
votes
2answers
174 views

What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes ...
1
vote
5answers
322 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 ...
4
votes
0answers
100 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 ...
2
votes
5answers
535 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 ...
0
votes
0answers
64 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 $C(\mathbb{R})$ ?
0
votes
1answer
230 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 ...