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1. ### I Several Questions About Smooth Infinitesimal Analysis

Hello. I read about smooth infinitesimal analysis and I have several questions: 1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6) 2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2)...
2. ### I Proving Continuous Functions in Smooth Infinitesimal Analysis

Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.
3. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

I read in this source: http://www.bndhep.net/Lab/Math/Calculus.htm The fact that x^2 becomes insignificant compared to x for very small values of x is a fundamental principle of infinitesimal calculus. We say x is infinitesimal when we allow its value to approach zero, but never actually reach...
4. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

What is relation between ##\Delta x## and ##dx##? If we want to get term with dx then we discard ##\Delta x^2## and other high-order terms in expression. Is it true?
5. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

I didn't fully understand what does it mean? What is it used for?
6. ### B Neglected terms in integral sum

I read "A treatise on infinitesimal calculus" Bartholomew Price. Integral sum, for ##2xdx##:
7. ### B Neglected terms in integral sum

Hello. As is known, we can neglect high-order term in expression ##f(x+dx)-f(x)##. For ##y=x^2##: ##dy=2xdx+dx^2##, ##dy=2xdx##. I read that infinitesimals have property: ##dx+dx^2=dx## I tried to neglect high-order terms in integral sum (##dx^2## and ##4dx^2## and so on) and I obtained wrong...

9. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Ok. We have 2 similar expressions: 1. ##2x \Delta x + \Delta x^2## where ##\Delta x## is variable and 2. ##x+x^2## where ##x## is variable. In 1. case we'll get ##2xdx## when ## \Delta x## tends to zero. In 2. case we'll get ##x## when ##x## tends to zero. But in the first case we'll get...
10. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Sorry, I forgot to say that ##x+x^2## is just expression where x is variable. Not differential.
11. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Hello. Let's assume that we have ##2x \Delta x + \Delta x^2##. When ##\Delta x## tends to zero we can neglect ##\Delta x^2## and we'll get ##2xdx##. Let's assume that we have ##x + x^2##. When ##x## tends to zero we can neglect ##x^2##. Will we get an infinitesimal ##x## as such as ##dx##? Thanks.
12. ### B What is the meaning of dx?

Hello, @fresh_42 You wrote that "In the first case ##dr \approx \Delta \, r## is an infinitesimal small change in ##r## and in the second it is an infinitesimal small piece of ##r##, so ##dr \approx h##." I have difficulty understanding this topic. How is it possible that ##dr \approx \Delta...

Yes
14. ### B Representation of infinitesimals in different ways

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...