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

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What is the point of making dx an infinitesimal hyperreal?

It seems fairly common to describe $\mathrm{d}x$ in nonstandard analysis as an infinitesimal. But after thinking it through (and then skimming Keisler's text), I can't see the point and actually think ...
5
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1answer
187 views

How do we interpret and perform integrals using infinitesimals?

If $\mathrm{d}x$ is treated as a hyperreal infinitesimal we can easily do derivations. How do we interpret and perform integrals using infinitesimals? What is the $\mathrm{d}x$ in $\int x^3\,...
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2answers
28 views

Can you recommend me any books about infinitesimal probability?

Can you recommend me any books about infinitesimal probability? There are several good books about infinitesimal calculus but it is hard to find any books about infinitesimal probability...
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1answer
40 views

What are the grounds for treating 'dx(differential, infinitesimal)' as if they were numbers?

I'm studying calculus and sometimes I find it strange to treat dx(differential) like numbers! Substitution rule would be a good example. ( I will use the first example in this website http://tutorial....
2
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2answers
63 views

Different kind of infinitesimals or zeros

If there are different kind of infinities (aleph0 aleph1 and so on) then are there different kind of infinitesimals? Or should I consider zero the "opposite" of infinity if there is such a thing and ...
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2answers
172 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 ...
2
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1answer
71 views

Is a limit a formalized infinitesimal?

From what I understand after thinking about this, delta epsilon really seems to formalize the notion of an infinitesimal. The constraint $0<|\delta-c|$ combined with the fact that there is no real ...
2
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1answer
103 views

What's so different about limits compared to infinitesimals?

If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous ...
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1answer
31 views

How do I make a change of variables in an integral?

I was solving a PDE using a change of variables. The result is $$V(\xi,\eta) = e^{-\frac{1}{2}\xi}\int A(\eta)\;d\eta$$ where $A(\eta)$ is an arbitrary function of $\eta$. I had originally used the ...
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4answers
157 views

Using infinitesimals to find the volume of a sphere/surface area of a sphere

I've always thought of $dx$ at the end of an integral as a "full stop" or something to tell me what variable I'm integrating with respect to. I looked up the derivation of the formula for volume of a ...
1
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0answers
37 views

Clarification of meaning of dx in an integral [duplicate]

I would like to have some clarification on the physical meaning of $dx$. I already know the following in the context of the area under the curve: $\lim_{\Delta x \rightarrow 0} \sum f(x) \Delta x \...
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2answers
28 views

Density of real numbers and density function

In Quantum mechanics, given a certain material, it is possible to write the density of energy states $\rho (E)$ as a function of $E$. That is: let's consider all the real values contained in the ...
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3answers
185 views

Learning differential calculus through infinitesimals

In class, we've studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and ...
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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 ...
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4answers
61 views

Dividing derivatives by derivatives

We are often taught that $$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$$ Why are we allowed to say this? What about the case of higher derivaitves, i.e. $$\frac{\frac{d^ny}{dt^n}}{\frac{d^nx}{...
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0answers
24 views

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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3answers
108 views

the relationship between $f(x)$ and $dx$ in $\int_a^b f(x)\,\mathrm{d}x$

In this example, If we use the relation $F'(x) = f(x)$, this consequence of the fundamental theorem may be written in the form $$ f(b) - f(a) = \int_a^b F'(x) dx = \color{#c66}{\boxed{\color{...
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3answers
64 views

How does differentiation work?

I am a physics student and my teacher told me, to find the instantaneous velocity of an object, reduce the time interval to a very small extent. May the time interval be very very very close to 0, ...
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20answers
2k views

Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines ...
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2answers
69 views

What is the value of $\tan (\frac{\pi}{2} - \epsilon)$? [closed]

$$\tan(\frac{\pi}{2}-\epsilon)$$ By epsilon I mean an infinitesimal.
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2answers
86 views

Cauchy's contribution

Sometime, I believe perhaps 2 years, ago I asked a question about breakthroughs, such as those within mathematics and physics which may lead a whole discipline forwards in many ways. One example from ...
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1answer
80 views

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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2answers
72 views

proof that $ \lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = \infty $

I started learning infinitesimally math and I have the following question: Is the following sentence true $ \lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = 1 $ I can see that it tends to $\infty$...
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1answer
64 views

Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
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3answers
670 views

Why don't infinite sums make any sense?

Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4. However, how is this ...
3
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2answers
89 views

Definite Integral of a infinitesimal

I did not study math, but have some foundations in it. I have been looking through some books on nonstandard analysis, and have (what I consider to be) a pretty simple question which I haven't been ...
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0answers
9 views

Link to question regarding treating differential operator as a ratio [duplicate]

I have attempted to find the post which provides an explanation as to the circumstances in which we can treat $\frac{dy}{dx}$ as a ratio which appears to be used in solving separable DE's, but I have ...
2
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2answers
48 views

Ceiling function of an infinitesimal

I was working with infinitesimals and I came across the problem: what is the ceiling function of an infinitesimal? Wolfram Alpha says an infinitesimal equals zero, so therefore the ceiling function of ...
0
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1answer
16 views

Make $f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$ be the infinitesimal of the highest order

Here is the question: Find $a$ and $b$, letting $$f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$$ be the infinitesimal of the highest order when $x \to 0$, and find that order. According to the key, $a=-...
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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, ...
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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 \ \ \ \...
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1answer
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Interpretation of $d\phi(z)$ in differential geometry

In "Exercises and Solutions in Mathematics", Ta-Tsien, 2nd Edition, exercise 3343. Statement of the exercise Let $(\mathbb{H}, g)$ be the two-dimensional hyperbolic space, where \begin{equation} \...
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4answers
604 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 ...
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1answer
36 views

Dual number division

I was reading the Wikipedia article on dual numbers and I read the first part. Then on my own I tried to find how to divide two dual numbers. I came up with: $$\frac{a+b\epsilon}{c+d\epsilon}=\frac{a+...
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1answer
33 views

Does this idea make sense? (infinitesmals and positive change)

I didn't know where else to ask this, so here goes... (caveat: I've been away from math for awhile) I'm trying to "say" symbolically that continued positive infinitesimal changes eventually (not at ...
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4answers
84 views

Find $\lim_{x\to0}\frac1{x^3}\left(\left(\frac{2+\cos x}{3}\right)^x-1\right)$ without L'Hopital's rule

Find the following limit $$\lim_{x\to0}\frac1{x^3}\left(\left(\frac{2+\cos x}{3}\right)^x-1\right)$$ without using L'Hopital's rule. I tried to solve this using fundamental limits such as $\lim_{...
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32 views

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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2answers
98 views

Is there such a thing as “hypertopology” (analogous to the hyperreals)?

The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with rigor)....
2
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1answer
112 views

Mapping $\mathbb{Z}$ onto $\mathbb{Q}$

There is a problem that calls for the mapping the set of all the positive integers to the set of all positive rational numbers in order to prove they have the same cardinality. I know more the common ...
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2answers
25 views

Find Maclaurin expansion of $y=2^x$ to $x^4$

Find Maclaurin expansion of $$y=2^x\text{ to } x^4$$ This is my try. We have $\displaystyle 2^x=e^{x\ln 2} =\left[1+\frac{x^2}2+\frac{x^3}6+\frac{x^4}{24}+o(x^4)\right]^{\ln 2}$ with $o(x^4)$ is ...
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1answer
51 views

Does probability theory require infinitesimals to work?

I'm taking an intro to probability class, and my book lists this as an axiom: $$P(\bigcup_{k=1}^{\infty}A_k) = \sum_{k=1}^{\infty}P(A_k)$$ where $A_k$ is an event and $P(A_k)$ is the probability of $...
0
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1answer
28 views

Is there any relationship between Gödel numbers associated with proofs of undecidable theorems and infinitesimals?

I recall from my reading of the popular book, Gödel, Escher, Bach some years ago that Hofstadter speculated on the value of Gödel numbers of undecidable theorems. If I recall correctly, he suggested ...
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0answers
33 views

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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1answer
40 views

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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1answer
73 views

Non-standard analysis - infinitesimals and archimedean property

I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero. However, at the same time, they satisfy all the ...
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2answers
61 views

What happens if to introduce infinite and infinitesimel quantities this way?

What if to introduce $\varepsilon$ and $\omega=1/\varepsilon$ from the following equation as a definition? $$\left(1+\varepsilon\right)^{1/\varepsilon}=e$$ or such $\varepsilon$ that $e^\varepsilon=...
5
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1answer
75 views

Can this sequence be a hyperreal number? What would be its real part?

Consider the sequence $\{a_n\} = \{\sin(n) \mid n\in \mathbb N \}$. Can this sequence be viewed as a hyperreal number? What could be its real part? Any intuition would be highly appreciated :) ...
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97 views

Historical Approach to $\lim_{x \to 0} \frac{e^{\alpha x} - e^{\beta x}}{x}$, without L'Hospital's Rule

I encountered this problem, amongst others, in the slightly older Calculus textbook Piskunov's Differential and Integral Calculus when I was working with a student: Calculate the limit $$ \lim_{x \to ...
4
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2answers
70 views

Basic question about nonstandard derivative

I'm trying to understand how the nonstandard derivative works. For instance, consider the function $f(x) = \frac{1}{2} x^2$ The derivative is $f'(x) = st \left( \frac{\frac{1}{2}(x + \epsilon)^2 - ...
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1answer
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Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) \right]-V(x)}{...