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

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17
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1answer
519 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 ...
21
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2answers
493 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 ...
0
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0answers
26 views

Does Thompson's Calculus hit all the essentials needed?

I found out in my preliminary research for my undergraduate degree that I need a bit of undergraduate-level mathematics for my topic at the very minimum to start with, up to ODEs. So I found a ...
2
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4answers
98 views

$dx=\frac {dx}{dt}dt $. Why is this equality true and what does it mean? [duplicate]

$dx=\frac {dx}{dt}dt $. I know that this deduction is obvious from the chain rule, given that we treat our dx and dt as just numbers. But I find it quite unsatisfactory to think of it in that sense. ...
1
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0answers
36 views

Why generator in Lie Algebra is defined as the coefficient in taylor expansion of map

Booth defines the infinitesimal generator of a lie group (denote the manifold it defines by $M$) using flow $\theta_t(p)$ by calculatng the limit (mainly the derivation for $f$ in each point $p\in M$) ...
2
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4answers
69 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 ...
4
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3answers
106 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 ...
0
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1answer
40 views

Derivative of an Infinitesimal?

I am currently studying calculus of variations (for my classical mechanics course). I have, on multiple occasions, seen the derivative of an infinitesimal quantity defined like below $$\frac{d}{dt} ...
-4
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1answer
314 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 ...
5
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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 ...
2
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0answers
54 views

Size of a geometric point

It is well known that the geometric points do not have any length, area, volume, or any other dimensional attribute, also geometric object (for example "line") is made up of a infinite number of ...
1
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3answers
51 views
1
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2answers
51 views

Using differentials to optimize a function

I've read in a paper by Tevian Dray an alternative way to solve optimization problems manipulating "differentials". Here is an example of how it works (next I quote the paper). Consider the ...
1
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1answer
34 views

hyperreals standard part inconsistency

$\def\st{\operatorname{st}}$ I'm studying non-standard calc from Keisler's book. Taking "standard part" rule doesn't make sense... its not commutative. e.g. $a$ is finite non infinitesimal $b,c$ ...
0
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0answers
61 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
2
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1answer
35 views

When “magnifying infinitesimals” why dont they have curvature ? (non standard) Infinitesimal calculus

Im reading https://www.math.wisc.edu/~keisler/calc.html. If you open up the chapter $2$ pdf... The $2$ diagrams (1st on page $14$ of the pdf (not the text book), 2nd on page $15$) have me confused. ...
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 ...
1
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0answers
40 views

Is this a valid thing to do to this differential equation?

Don't tell me how to solve it, but I've been trying to use the following equation to get r which is distance) as a function of t. $$\frac{GMm}{r^2}= -ma$$ I've been working on it for about a year ...
0
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1answer
42 views

Confused between infinitesimally small and $0$

Consider these two cases: 1) Let $I_n$ be the closed interval $[\frac{1}{n}, 1]$. Then $\bigcup_{n = 1}^{\infty} I_n = (0,1]$. 2) Take the two sets $y = e^x$ and $y = 0$ (the $x$-axis) in ...
0
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2answers
68 views

Proving: if $|a|<\epsilon \forall \epsilon>0$ then $a=0$ using a direct proof

I am asked to prove: if $|a|<\epsilon,\forall \epsilon>0$, then $a=0$ I can prove this as follows. Assume $a \not= 0$ I want to show then that $|a| \geq \epsilon$ for some $\epsilon$ We ...
0
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1answer
24 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\\ ...
3
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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?
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 ...
0
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1answer
56 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
38 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 ...
2
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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 ...
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 ...
8
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3answers
481 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 ...
10
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5answers
825 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 $4x$ ...
0
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2answers
64 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 ...
3
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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 ...
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0answers
49 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 ...
5
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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 ...
0
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2answers
40 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 ...
1
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1answer
64 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.
54
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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 ...
1
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4answers
90 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 ...
0
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5answers
209 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 ...
1
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1answer
74 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
36 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
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 ...
1
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1answer
63 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). ...
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3answers
83 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 ...
5
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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 ...
15
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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 ...
9
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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, ...
9
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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 ...
0
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1answer
27 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
28 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 ...
4
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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 ...