1
vote
1answer
22 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 $$ds^2 = dx^2 + dy^2$$ ...
9
votes
2answers
180 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, ...
5
votes
3answers
103 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 ...
-1
votes
1answer
194 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 ...
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: ...
2
votes
4answers
152 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 ...
7
votes
6answers
222 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.
0
votes
1answer
33 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 ...
0
votes
2answers
32 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
81 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
262 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 ...
14
votes
2answers
746 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 ...
3
votes
1answer
229 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
110 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
244 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 ...
13
votes
3answers
457 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 ...
4
votes
3answers
143 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
98 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
282 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 ...
7
votes
6answers
918 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
819 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
316 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 ...
-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
177 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
291 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) = ...
4
votes
3answers
178 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 ...
1
vote
0answers
115 views

Proving $f$ is constant.

Let $f$ and $g$ be continuous function where $f,g:[a,b] \rightarrow \Bbb R $ and $\int_a^b g(x)=0$ and $\int_a^b f(x)g(x)=0$ , show that $f$ is a constant function. I tried a bunch of things ...
5
votes
1answer
69 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 ...
3
votes
5answers
521 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 ...
1
vote
1answer
949 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 ...
44
votes
10answers
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 ...