Han, Taking the limit as h goes to zero is equivalent to examining the behaviour of the function as h becomes infinitessimally small.
It may be "equivalent" but it is not logically the same thing. Unfortunately, physicists use the term "infinitesmal" in a way that would make mathematicians cringe.- And engineers are worse! I've seen a engineering text that referred to 0.001 as an "infinitesmal"!
Why not divide numbers with null HallsofIvy: Please give me a definition of the first derivative of the function f(x) which does not involve infinitessimals. Since you asked me (another person has already given this) "The first derivative of a function, f, at x= a, is the slope of the tangent line to the graph of y= f(x) at the point (a, f(a))" If you really meant "formula" rather than "definition" lim(h->0) {(f(x+h)- f(x))/h} is the one that's normally given nowadays and does not involve "infinitesmals". There are a variety of ways of finding the tangent to to a curve. Fermat (before Newton) showed how to find the center of a circle that will be tangent to a curve at a given point. Although Fermat was actually interested in finding the normal to the curve (a diameter of the circle) you could also find the tangent to the curve that way. Newton DID use what he called "infinitesmals". In order to find the derivative of y= x^2 at a specific, he would argue that increasing x by the infinitesmal dx, y increases from x^2 to (x+dx)^2= x^2+ 2xdx+ (dx)^2 so that the increase is dy= (x+dx)^2- x^2= 2xdx+ (dx)^2. That is a rate of increas of dy/dx= 2x+ dx. Since dx is an infinitesmal and can be neglected when adding to "regular" numbers, dy/dx= 2x. There are serious logical difficulties with this process. During the nineteenth century, the whole process of "infintesmals" was replaced with the limit. Any calculus book would do the derivative above as: f(x+h)= x^2+ 2xh+ h^2, so f(x+h)- f(x)= 2xh+ h^2 and (as long as h is not 0), (f(x+h)- f(x))/h= 2x+ h. Now, take the limit as h-> 0 (It having already been showed that "if f(x)= g(x) for x not equal to a, then lim f(x)= lim g(x) where the limit is x->a") we arrive at f'(x)= 2x. The two methods are algebraically the same- the crucial difference is that h is never said to be "infintesmal" or anything other than an ordinary real number! During the twentieth century (300 years after Newton) mathematicians working in symbolic logic showed that it is possible to extend the real number system to include infinitesmals and that all calculus can be done in that. But as I said, only a relatively small number of mathematicians seriously work with "infinitesmals". Certainly one wouldn't want to inflict non-standard analysis on beginners in calculus!
now I din't work the math on your equation, but it looks funny to me. maybe just because I didnt learn it this way. I have this gut feeling telling me that unless we let h->0 then we will end up with a secant line having length "h" in dimension "x". could you elaborate?
HallsofIvy, i think that both analyses of the derivative that you have presented amount to exactly the same thing. As I said before, taking a limit as h goes to zero is equivalent to examining the behaviour as h becomes infinitessimally small. If you can show me any case in which a limiting process cannot be replaced with an equivalent infinitessimal argument, I might change my mind on this.
I have some difficulties with the understanding the concept of the infinitesimals. There is an example: I make a line l size l = infinitesimal l is a legitime object because l has a value l>0 Now a will draw a square ABCD with l as diagonal between A en C Wat is the size AB?
REPlease Register or Log in to view the hidden image!n Radioactive Waves AB would be smaller than l. Smaller than l can be only 0. Right?
Thats why it would be infinitesimal. Its like the concept of infinity being large, only it is infinitly small. Have you ever learned about fractals? There are shape that can encompass a finite area but having an infinite peremiter. If you can grasp this concept you should be able to understand infinitesimals
Merlin: Its kind of the same sort of deal. In a sierenpinski triangle, after iterating n->infinty the end result is infinitly many intersecting lines. It is not quit a plane, yet more than a line. It has a fractal dimension between 1 and two. Cantor's Dust has a fractal dimension between 0 and 1.
O.K./K.O. I was supposing that maths should be "right" in its own nature. By looking at (supper)small numbers and infinity it became pretty obvious that the fault lies in my modeling of the problem (of the square ABCD). To use infinitesimal you must first specify some basic elements of the model, in case of the square ABCD it would be that only AB (as the smallest element) could be specified as infinitesimal (I guess). Maths is then just an adjustable tool to get to some info about the (theoretical) world. You can adjust the tool to your needs but while doing that you yourself must take care that you do not step out into the absurd. Not an easy job I can say...because the absurd is soo iiiinnnnnvvviiiittttttiiiiiiiinnnnnnnngggggggggggg
James R said "i think that both analyses of the derivative that you have presented amount to exactly the same thing. As I said before, taking a limit as h goes to zero is equivalent to examining the behaviour as h becomes infinitessimally small. If you can show me any case in which a limiting process cannot be replaced with an equivalent infinitessimal argument, I might change my mind on this." Of course they AMOUNT to the same thing. The point is that in order to use an "equivalent infinitesmal argument" you first have to DEFINE "infinitesmal". That was not properly done until until the twentieth century and requires more "machinery" than most mathematicians want to deal with. "Non-standard" analysis is a subject taught in some graduate schools, primarily to students who are interested in symbolic logic and "foundations of mathematics". Since limits and infinitesmals give the same result, it is much simpler to use limits instead of infinitesmals. I'm not particularly concerned about changing your mind (and I just finished saying that any limiting process CAN be replaced with an equivalent infinitesmal argument) but I get little annoyed at people who throw "infinitesmals" around without realizing how very, very difficult it is just to DEFINE the things! Please Register or Log in to view the hidden image!
HallsofIvy, What's hard about defining them? Can't I just say "There exists <font face="symbol">e</font> > 0..." and go from there?
some mathematicians have devised an extension of the reals to include infinitesimal quantities? and they can do calculus in this space? i d like to hear more about this. can you give some references?
yes please enlighten us. the more i hang around here the more i realise the less i know (did that statement even make sense? math makes me go crazy after a while)
if i get what halls is saying, james, i think the problem with the epsilon delta method is that as soon as you choose an epsilon, its not really an infinitesimal. epsilon is a small but finite real number. now whenever i think of infinitesimal, i think of it just like that: i.e. a small finite number, that i can choose to be as small as i like. if i think of it that way, then of course there is no difference between the two approaches. but it is perhaps an abuse of language. something that is an infinitesimal should not be finite, right? i have never heard of another way of thinking about it, or of constructing the calculus, but if there is one, i am intrigued.
i thought epsilon was an infinitesimal, by means of taking the limit. could someone please define it? a search at wolfram left me empty handed. dictionary.com produced this: in·fin·i·tes·i·mal: adj 1. Immeasurably or incalculably minute. 2.Mathematics. Capable of having values approaching zero as a limit. infinitesimal \In`fin*i*tes"i*mal\, a. [Cf. F. infinit['e]simal, fr. infinit['e]sime infinitely small, fr. L. infinitus. See Infinite, a.] Infinitely or indefinitely small; less than any assignable quantity or value; very small. Infinitesimal calculus, the different and the integral calculus, when developed according to the method used by Leibnitz, who regarded the increments given to variables as infinitesimal. infinitesimal calculus n : the branch of mathematics that is concerned with limits and with the differentiation and integration of functions [syn: calculus, the calculus] infinitesimal calculus n. Differential and integral calculus. now I ask, how is it any different? please provide another refference.
epsilon is finite. if you take the limit as epsilon goes to zero, it becomes zero. neither the finite number nor the zero that it became was evern an "infinitesimal" according to the definition that you gave.
i think we are getting off topic, and i also think this is getting sort of interesting. maybe a new thread is in order?
