Questions tagged [infinitesimals]

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

419 questions
Filter by
Sorted by
Tagged with
60 views

Definition of Continuity in Non Standard Analysis (Internal Set Theory)

In Alain M. Robert's book, he affirms that continuity is implicitly defined by S-continuity and give the following example: Let $I$ be a standard interval and $\mathcal{F}$ be the space of numerical ...
16 views

13 views

48 views

Can you (briefly) have a real, non-zero infinitesimal?

Suppose you have the set of the rationals between 0 and 1, let's call it $\mathbb{Q}_1$. Now I want the probability of choosing a number in $\mathbb{Q}_1$ and getting 1/2. If I add this to the ...
29 views

197 views

Is the Pythagorean Theorem true for infinitesimal leg?

In Newtonian Physics (Classical Mechanics) a body rotates around another (massive) body with constant circular speed ω (stationary trajectory) eternally. The linear velocity v is perpendicular to the ...
54 views

What's the criterion for neglecting terms "much smaller than" other terms?

In a physics exercise relating some electric forces. I found the following equation for the restoring force: $F = \frac{kQq}{(y_0 + \Delta y)^2} - \frac{kQq}{y_0^2}$ Where $\Delta y << y_0$ ...
52 views

Do these statements about analysis with dual numbers make any sense?

I am reading Color for the Sciences by Jan Koenderink, and in Ch. 3 he introduces the dual number system to define the space of possible power spectra for a beam of light. However, his statements do ...
54 views

49 views

How do you sum difference between two values of a function separated by an infinitesimal?

I have a function, $n_p(T)$ that gives me the number of protons at a given temperature. I would like to calculate the difference between the number of protons at a given temperature, $T_1$ and the ...
191 views

Cognitive view of Infinitesimals in Keisler's Book through Cauchy's idea of infinitesimals?

(Quoting Keisler)$^{1}$ - ' $\mathbb{R}^{*}$ has a positive infinitesimal, that is, an element ε such that 0 < ε and ε < r for every positive r $\in \mathbb{R}$ '. (Please note - I happen to be ...
32 views

Intuitive error in finding Volume of sphere using single integration.

To calculate the volume of a sphere of radius $\mathbf R$, I considered a thin disc of radius $r= \mathbf Rsin\theta$ ,where $\theta$ is the angle radius vector on the circumference makes with the ...
59 views

Can we have nonstandard/transfinite "decimal" expansions?

Can we take the decimal (or any arbitrary base) representation of a real number and just append some more digits beyond it? Is there a theory that covers this, maybe some kind of non-standard analysis?...
92 views

Can numbers smaller than infinitesimals exist?

I have a good idea of infinitesimals to some extent.( A bit of non standard analysis) I am reading the book of keisler on non standard analysis and calculus. I am okay with them all but, if "a&...
49 views

Order of infinitesimal $\int_0^{|x|^2}e^{-c_1 r-c_2\frac{|x|^2}{r}}dr$

How could I find the right order of infinitesimal of $$\int_0^{|x|^2}e^{-c_1 r-c_2\frac{|x|^2}{r}}dr,\qquad c_1,c_2>0,$$ as $|x|$ tends to $+\infty$? Should it be trivial? For an upper bound, I ...
43 views

How to think of the location concept of a point?

It is said that points are to be thought of as 0-dimensional. It is also said that they represent a position, i.e., a location. In the physical world, location obviously encompasses some area, say, ...
54 views

Are independent variable infinitesimals positive?

A calculation I'm doing involves the inequality $$\frac{dt}{dx}\leq \frac{-kt}{x}$$ and I want to shift the dx and t to the other sides, so I can integrate. Now t is known to be positive (it's $xf(x)$,...
88 views

How to interpret $xdx$ intuitively if $x$ represents time?

From the book Calculus Made Easy Now in the calculus we write $dx$ for a little bit of $x$. These things such as $dx$ (...) are called “differentials”. If $dx$ is a small bit of $x$, and relatively ...
109 views

On counting the areas covered by holes in a function in integration

As far as I know, holes in a function at the endpoints of an interval aren't usually given any importance while integrating over that interval. For example, while calculating the area under the ...
226 views

Why 1 is not the limit of 0.999... but equal? [duplicate]

$\lim_{x\to0}x=0$ $\lim_{x\to1}x=1$ Limit is unreachable, isn't that mean for x→0 can never be 0? That's why infinitesimal isn't 0, because its limit is 0. (By Wikipedia: infinitesimals or ...
87 views

What is difference between derivative in standard and non standard analysis?

I am reading the book on complex analysis by Tristan Needham. In that book he explains derivative in an intuitive way as a quantity by which dx is expanded to get dy in both complex and real number ...
95 views

Does a higher dimensional space have more points than a low dimensional space? And in some way are some infinities bigger than others? [duplicate]

How many points can exist in one dimension? ANS: Infinite! How many lines can exist in a two dimension? ANS: Infinite! Now my question is how many points can exist two dimensions or even higher ...
59 views

How to prove that hyper reals are not reals?

I was told by my professors that infinitesimals are numbers which are smaller than all the reals. And I was also told that it is the sequence, $\{1,1/2,1/3,1/4,\dots\}$. And I am comfortable with ...
72 views

Any Equivalence between Limit computed via L'Hospital's rule and 0/0 i.e $(\frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)})$?

In Standard Analysis, Does $$\lim_{x \to c}\frac{f(x)}{g(x)}=\frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)}$$ Given that $$\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0$$ Yes, we can get the computed answer ...
70 views

Relative scheme structure on a specific scheme related to the symmetric product of curves.

Given a smooth algebraic curve $C$ with a closed point $p$. The symmetric product $\text{Sym}^i(C)$ has a closed subvariety $\text{Sym}^{i-1}(C)$ which its embedding is given by adding the point $p$. ...
183 views

Proof for how .9999999...=1 [duplicate]

I was just curious for the proof/theorem for a very close decimal (so when you keep adding a decimal) to equal the next integer such as .999999999999....=1. Thank you.
66 views

Asking if an infinite series will converge or diverge

I have two questions regarding the convergence of infinite series. One, if $\sum_{n=1}^{\infty}a_n$ converges to L, where L is defined, does $a_n < L$ for each n? Two, if $a_n$ is defined for each ...
70 views

On the infinitesimals. [duplicate]

I have completed my calculus course a few months back, I was introduced to the idea of infinitesimals as just a symbol in that course. Now I see that they are being used not only in math but in many ...
45 views

Summation of function subtraction with a limit as $\epsilon \rightarrow 0$

I got stuck in a problem in the middle of my calculations of integrals and sums. $$\lim_{\epsilon \rightarrow 0} \sum_{n=1}^\infty f(n-\epsilon)-f(n+\epsilon)=0$$ where $f$ is continuous on all of the ...
93 views

What is an isolated subgroup? [closed]

" As in the case of finite numbers, the infinitesimal numbers form an isolated subgroup  of $R_{inf}$ of $^*R$"page 152 What does this sentence mean? What is an isolated subgroup?
I have the next $\lim_{x \to \frac{\pi}{4}} \frac{f(x)-f(\frac{\pi}{4})}{x-\frac{\pi}{4}}$ when the function is $$f(x)=\frac{x+\sin(x)}{\tan(x)}.$$ I don't know how to even start. I am sorry that this ...