infinitesimals Definition and 36 Threads

I was watching an explanation of why the spacetime interval is invariant in all inertial frames (even when it's not lightlike) and the author made the assertion that if we have the relationship ds'=f(ds), we can expand the function as A+B*ds+C*ds^2+... (where C is not the speed of light). That's...

Some sources state a similar format of the following $$\int_a^{a+da}f(x)dx=f(a)da$$ Which had me thinking whether the following integration can exist $$\int_a^{a+dx}f(x)dx=f(a)dx$$ I have difficulty grasping some aspects about these integrations 1. Regarding the 1st integration, shouldn't ##a##...

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See my insights article for those interested in an unconventional approach to doing Precalculus at an accelerated pace and beginning Calculus. It is different from the usual way that a precalculus is done text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and...

Hi Everyone I have been doing further investigation into infinitesimals since I wrote my insight article. I had an issue with the original article; the link to the foundations of natural numbers, integers, and rational numbers was somewhat advanced. I did need to write an insights article at...

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When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...

Hello. There are 4 types of infinitesimals: 1) dx=1/N, N is the number of elemets of the set of the natural numbers (letter N is used to indicate the cardinality of the set of natural numbers) 2) Hyperreal numbers: ε=1/ω, ω is number greater than any real number. 3) Surreal numbers: { 0, 1...

When comparing 2 infinitesimals, does the higher order one approach 0 faster or slower?

dl = Infinitesimal length of the segment. dlπ/2 = the semicircle length lim dl-> zero dl/(dlπ/2) = 2/π, no zero, so the answer would be yes. But second the book, the answer is no, where am i wrong?

I'm trying to derive the lever law by myself, however, I'm stuck. Please follow the logic of my calculations. Every object in the picture has the same mass. I want to prove that, under the effect of the gravitational force, I can replace the objects in A and C with the two objects in B, and...

I've understood the formal definition of limits and its various applications. However, I'm trying to dive more into the history of how the concept of limits were conceived (more than what Wikipedia tends to cover), and how to formally understand and visualise infinitesimals. For example, I know...

I've always thought of dxat 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 sphere, and here, dx is taken as an infinitesimally small change which is multiplied by the area of...

This doesn't seem to me like an accurate characterization of NSA. In a synthetic treatment of the reals, we posit some axioms about the reals, and we simply assume that there exists a systems that obeys those axioms. In a constructive description of the reals, we build them up using Dedekind...

All throughout calculus texts, the authors have always put conditions on the manipulation of differentials. They say that for the chain rule, the cancellation of the differentials is simply a way to remember the formula. When doing separation of variable for ODEs, texts always say something...

Euler was the master in analysisng anything. This can be seen in his words in the preface of his book "Mmathematica" (translated by Ian Bruce), where he speaks on the text of Hermann "Phoronomiam": Euler has given many insightful words on analysisng things in his preface of many other books...

Say ##A##, ##B##, ##C##,... are finite numbers; real, complex, quarternians, tensors, or what have you. "First Order" infinitesimals are finite variables prepended with the letter ##d##. Infinitesimal of any order, are prepended with ##d^n## where ##n## is the infinitesimal "order". Finite...

Ok so what I want to know is, is this valid? If so what does it mean?

in a certain problem it was written Δθ=∂θ/∂x*Δx + ∂θ/∂y*Δy + ∂θ/∂z*Δz + infinitesimals of order higher than Δx,Δy and Δz.can anyone tell me what is "infinitesimals of order higher than Δx,Δy and Δz?"

Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...

I'm trying to find a way to use calculus without infinitesimals and I'm stuck on this physics problem. It's a uniform charge distribution question. Basically a half circle with radius r and you have to find the electric field at a point that is along its x-axis. The E_y component will be 0...

Author: Jerome H. Keisler Title: Elementary Calculus: An Approach Using Infinitesimals Download Link: http://www.math.wisc.edu/~keisler/calc.html Prerequisities: High School Mathematics Table of Contents: Introduction Real and Hyperreal Numbers The Real Line Functions of Real...

Hi, I first had a question regarding infinitesimals. What does it mean when the infinitesimal is at the beginning of the integral? For example: ∫dxf(x) is this the same as ∫f(x)dx ? My second question was how to convert a summation to an integral and a summation into an integral...

Please see below link for the two different styles of solving a separable equation. http://en.wikipedia.org/wiki/Separation_of_variables#Ordinary_differential_equations_.28ODE.29 Which one is more proper? Why? My DE teacher told me that strictly speaking it's wrong to use the first method...

I've been playing around with a free PDF Calculus book lately. But, I have no way to check the logic used to get to a particular answer. I've been trying to find the standard part for: (1/ɛ)((1/sqrt(4+ɛ))-(1/2)) I've tried every way I could think of to algebraically manipulate this in...

Infinitesimals and "Infini-tesa-tesimals" Positive infinitesimals are defined as greater than zero, and less than 1/n, where n is any number 1,2,3... The set of negative infinitesimals is the same, but where negative infinitesimals are less than zero and greater than 1/-n. Infinities are the...

I am not sure into which rubric to put this, but since there is some Model Theory here, I am putting it in this one. First, I define the Cantor set informally: A(0) = [0,1] A(n+1) = the set of closed intervals obtained by taking out the open middle third of each interval contained in A(n)...

In Lectures on the hyperreals: an introduction to nonstandard analysis, pp. 50-51, Goldblatt includes among his hyperreal axioms that the sum of two infinitesimals is infinitesimal, that the product of an infinitesimal and an appreciable (i.e. nonzero real) number is infinitesimal, and that the...

"A real number is a Dedekind cut in the set Q of rational numbers: a partition of Q into a pair of nonempty disjoint subsets <L,U> with every element of L less than every element of U and L having no largest member. Thus 21/2 can be identified with the cut: L = {q in Q: q2 < 2}, U = {q in Q: q2...

The problem is to find the E field a distance z above the centre of a square sheet of charge of side length a and charge density \sigma. The solutions use the result for a square loop and integrate to get the result for the square sheet, however it's the change \lambda \to \sigma \frac{da}{2}...

Homework Statement Prove that infinitesimals are not a subset of R. Homework Equations N/A The Attempt at a Solution Well, I had two ideas about how to prove this but I'm really not sure about either. Proof 1 was the first idea I had but I think it's probably wrong since it has...

Have a look! Hi all, Well, i am posting the work that i have done in proving that theorem. Like i said it is nothing important, but rather it is important only for me, so if you could have a look at what i have done i would really appreciate it. The theorem that i have tried to prove...

Homework Statement I hate infinitesimals and differentials. When I learned calculus, we used Liebniz notation df/dx only as a convenience for using the chain rule. In physics, apparently, people just play around with differentials and infinitesimals and expect to get the right answer...

Is there a formulation of calculus that uses infinitesimals rigorously without introducing an additional number system (non-standard analysis) and without deviating from classical logic?

Compute the standard part of this, please: \frac{ \sqrt{H+1}}{ \sqrt{2H} + \sqrt{H-1}}, where H is positive infinite. It probably should be some algebra trick I'm not familar with.

I know what all of these are, but I’ve never seen or heard a formal definition for them, could someone please provide one? 1) a number 2) a real number 3) a integer 4) a rational number 5) a irrational number 6) a transcendental number 7) a infinitesimals number 8) a hyper real number...