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

learn more… | top users | synonyms

1
vote
0answers
34 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 ...
1
vote
2answers
25 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 ...
4
votes
3answers
168 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 ...
6
votes
4answers
164 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 ...
0
votes
4answers
58 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. ...
0
votes
0answers
22 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 ...
0
votes
3answers
107 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 = ...
0
votes
3answers
63 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, ...
38
votes
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 ...
0
votes
2answers
68 views

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

$$\tan(\frac{\pi}{2}-\epsilon)$$ By epsilon I mean an infinitesimal.
0
votes
2answers
78 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 ...
3
votes
1answer
77 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 ...
0
votes
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 ...
1
vote
1answer
55 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. ...
4
votes
3answers
668 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
votes
2answers
83 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 ...
0
votes
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
votes
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
votes
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, ...
1
vote
10answers
970 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, ...
37
votes
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 \ \ \ ...
5
votes
1answer
76 views

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} ...
2
votes
4answers
579 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 ...
1
vote
1answer
32 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: ...
2
votes
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 ...
3
votes
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 ...
0
votes
0answers
31 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 ...
1
vote
2answers
93 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 ...
2
votes
1answer
106 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 ...
0
votes
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 ...
1
vote
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
votes
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 ...
1
vote
0answers
30 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. ...
1
vote
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 ...
1
vote
1answer
64 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 ...
3
votes
2answers
60 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 ...
5
votes
1answer
70 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 :) ...
1
vote
2answers
93 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
votes
2answers
66 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 - ...
3
votes
1answer
81 views

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) ...
2
votes
6answers
112 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
16
votes
3answers
2k views

Why is zero the only infinitesimal real number?

I am currently reading Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler and was wondering if someone could help me with an aspect treated in the book. On page 24 he says a ...
6
votes
3answers
195 views

Are there concepts in nonstandard analysis that are useful for an introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense. Can someone elaborate on this ...
0
votes
0answers
54 views

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 $C(\mathbb{R})$ ?
2
votes
2answers
78 views

Convergence of $\int\frac{\arctan x}{x} dx$

I can't find function to bound this integral in the intervals from $1$ to $+\infty$, to prove if it converges. $$ \int _1^{\infty }\frac{\arctan x}{x}dx $$ Any idea? How can I refute this if it is ...
0
votes
1answer
67 views

Convergent of integral of $1/x^x$ [closed]

I need to prove if this integral converges (on interval $[1,+\infty))$: $$\int _1^{\infty }\frac{1}{x^x}\;dx$$ Has anybody any idea how to do it? thank you. :)
4
votes
0answers
49 views

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 ...
6
votes
1answer
197 views

What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \begin{equation} \forall a,b,c \in \Bbb ...
1
vote
0answers
43 views

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 ...
5
votes
2answers
261 views

How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this ...