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

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Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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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 ...
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Do we have $\int f dxdy = \int fdydx$ or $\int f dxdy = -\int f dydx$?

If $f : \mathbb R^2 \to \mathbb R$ is an integrable function, then do we have $$ \int f dxdy = \int f dydx $$ or $$ \int f dxdy = -\int f dydx? $$ (I am leaving the domain of integration as it does ...
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Infinitesimal generator as a derivation in $SO_2$

Suppose we are looking at $SO_2(\mathbb{R})$. The infitismal generator can eb found using Taylor approximation in form of a matrix, $$X=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$ However I saw ...
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$dxdy=-dydx$ 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 statements ...
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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 $$ds^2 = dx^2 + dy^2$$ ...
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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 ...
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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\\ ...
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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 ...
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49 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 ...
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Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
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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 ...
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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 ...
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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 ? I'...
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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 ...
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Infinitesimal Transformation. Importance of Infinitesimal Transformation

My teacher ask me to start to study about Hamilton systems, Noether theorem. I am not advanced in that kind of study, so the teacher want from me to see some easy research, some proofs, step by step. ...
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Linear programming in Hilbert spaces

Let $H$ be a real Hilbert space. Let $b,c\in H$, $P\subset H$ be a convex cone and a continuous linear mapping $A:H\rightarrow H$. Consider the following sets: $$ B:=\{(Ax, \langle c,x\rangle:x\in P\},...
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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 $p\in M$) ...
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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 ...
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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 ...
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$\approx$ 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 ...
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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 theory....
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Infinitesimal area element in polar coordinate

We know, that the infinitesimal area element in Cartesian coordinate system is $dy~dx$ and in Polar coordinate system, it is $r~dr~d\theta$. This inifinitesimal area element is calculated by measuring ...
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Combine probability and infinitesimals

If we pick a real number from $[0, 1]$ (with uniform probability measure), this number had probability zero to be picked. Can we canonically assign an infinitesimal $\epsilon$ to the event $A_x$ that ...
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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 $C(\mathbb{R})$ ?
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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 ...
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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 ...