12/24/2023 0 Comments Infinitesimals to derive chain ruleThe other fundamental fact about infinitesimals is known as the Kock-Lawvere axiom: for any % function f, at any x∈ℝ, there is a unique "slope" s(x) ∈ℝ such that f(x+ε)= f(x) + εs(x). This is the first fundamental fact about infinitesimals: you can't ever tell them apart from zero. Whenever you have an infinitesimal ε, you can't assume ε≠0. In SIA, an infinitesimal is any number ε such that ε 2 = 0. The following will be about SIA infinitesimals: other systems work differently. One I'm particularly familiar with is Smooth Infinitesimal Analysis (SIA). There are other ways of doing analysis (which are, of course, non-classical ). In the usual formulation (styled classical analysis ), the only infinitesimal is zero. There are many different versions of infinitesimal quantities in mathematics. But maybe there is a non-standard measure theory that includes the concept of sets of infinitesimal measure. Standard measure theory does not involve infinitesimals, because a measure is a real-valued function, and the system of real numbers does not include infinitesimals. You should look into measure theory, which studies concepts like "length" in a rigorous way. If points of an infinitesimal size exist, could the length of a line be defined as sum of the length of an infinite number of infinitesimal points? I imagine this would cause some difficulty, as the number of points between 0 and 1 are mappable to the number of points between 0 and 2, though clearly the lines are not the same length. Yes, my understanding is that the definition of infinitesimals in non-standard analysis gives rise to different orders of infinitesimals.Ĥ. Are there also multiple orders of infinitesimals? Can one infinitesimal value be smaller than another? It is generally agreed that there are multiple orders of infinity. However, non-standard analysis is an alternative formulation of calculus that gives a rigorous definition of infinitesimals and uses this as the foundation.ģ. Infinitesimals do not exist in the system of real numbers. Not in the standard formulation of calculus. Does this mean, though, that the tangent line to a curve at a particular point is actually a line defined by two points separated by an infinitesimal distance? Is there a generally agreed upon, well-defined concept of infinitesimal values in mathematics? As I understand it, in differential calculus, we use the concept of the limit 1/n as n goes to infinity to define the concept of a derivative. ![]() I hope some of these questions made sense, and maybe you could point me to some relevant information on this topic because the Wikipedia entry on infinitesimals is rather poor.ġ. If infinitesimal quantities exist, would it be reasonable to posit a point which has an infinitesimal length in every dimension? Similarly, could a one-dimensional horizontal line be seen to have an infinitesimal height (as well as width and metrics in every other dimension)? If not, is the line's height in other dimensions 0? Is there a difference between having a height of 0 and an infinitesimal height? ![]() Most mathematical definitions that I've seen for a "point" assume that a point has no size in any dimension. ![]() I've recently been wondering about infinitesimal quantities, and there are a few things that I'm having difficulty wrapping my head around.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |