Infinitesimal Distances - A Question

[Lajka]

Hello,

I've just recently stumbled upon this forum, in search for an answer for my little dilemma, so I hope someone can help me. This is the question:

Given that $$dx$$ is an infinitesimal interval on set $$R$$, does it mean that it has infinitly many points in itself as well?

If I understood correctly, its length does approaches zero (but never seems to quite get there), however, it's still an interval by its nature, and all intervals have the cardinal number c.

The reason I ask this is because a friend told me that there is a theorem, or a consequence of some theorem, that an infinitesimal region can contain at most one real number, and this pretty much rocked my world.

Here's some reasoning on my behalf:

1) Like I sad, it's an INTERVAL. If it merely contains a one real number, what else does it contain then? Seems not much of an "interval" to me...

2) This again is more of a question, really. If i take dx, and divide it by two, I should have gained two new (even) smaller intervals. I don't know if I have a right to do this. But if I do, then each one of them can contain at most one real number, so the original interval can now containt at most two real numbers. You can imagine what happens if I continue dividing these smaller intervals. So, it's a paradox. Something tells me this is logic is flawed, so I would appreciate if someone could point out where I made a mistake, thanks.But then again, even I sometimes, intuitively, believe that dx can't be "normal" interval.
Let me give an example of this:
If I look at the concept of integration, but not in the ordinary "area under the curve" sense, but in the sense that reflects the Fundamental Theorem:
$$\int^{b}_{a} f'(x)dx = f(b) - f(a)$$
i.e. I imagine that I'm summing up a bunch of $$dy's (dy=f'(x)dx)$$, e.g. $$dy_{1}+dy_{2}+...$$, and in that process the upper boundaries of the previous $$dy$$ and lower boundaries of the next one cancel each other. Following this process, and going from $$x=a$$ to $$x=b$$, we finally get some finite distance: $$f(b)-f(a)$$
What I'm trying to say, is that these intervals, these $$dy's$$, they don't overlap. But now I wonder. Why wouldn't they? If I imagine $$dx's$$ as "normal" intervals, I would actually expect that they do. I am not sure if I made myself clear enough, so I drew a little picture:

http://imagebin.ca/img/CFuUFxr.png

The lower part in the picture illustrates the way I reason the process of integration, and the upper part my new dilemma.
Again, if my logic is flawed, I would really appreciate if someone could point out why.

So, that's it. If I imagine a $$dx$$ as an infinitesimal distance, but still with infinitely many points, I have this problem. Then again, if I don't, I have dilemmas that I explained in the beginning. A bloody nightmare.

Any help is appreciated. Also, please bear in mind that I study to be an electrical engineer, so, although I like to think that I'm not a newbie in mathematics, I''m sure I can't compare with you guys. So be gentle, please. :D

Much obliged,
Lajka

 

[HallsofIvy]

Take a close look at your own picture. For a one-to-one function, the only way the "y" intervals can overlap is if the "x" intervals overlap- and they are defined not to overlap- they form a partition of the base interval.

 

[Lajka]

Ah, right, stupid me. Thanks.

Unfortunately, it would seem that that doesn't help me with my problem, only changes it a little bit. Because, you see, then I could ask the following:

Again, under the assumption that dx is a "normal" interval: I form the partition, then I form the product$$f(x') \Delta x$$, and then I take the limit and let it converge. So far, so nice. But exactly WHAT $$x$$, in $$f(x')$$, is $$x'$$ converging to? There are infinitely many $$x's$$ in $$dx$$.

I hope I am not confusing anyone by this, I'm not exactly known for my ability to be concise and good explainer at the same time.


So, here we are.

If I take "dot after dot" on my curve $$f(x)$$ (aka my 1st approach) and form $$dx's$$ accordingly, they should overlap.

If I form partition, and thus "create" $$dx's$$ first, the question which arises is what $$x$$ am I actualy "taking" for $$f(x)$$? There are infinitely many $$x's$$ in my $$dx$$, if $$dx$$ is truly a "normal" interval.

That would mean that there are a whole bunch of $$x's$$ that remain "unused", because I only "take" one $$x$$ per $$dx$$ to figure in my $$f(x)$$, and then just "shift" to the next $$dx$$ (god, I hope someone will understand me.). And that doesn't seem right to me.

So, my question basically stays the same, and it's revolving around the nature of $$dx$$.

Thank you for your help tho, I really appreciate it mate.

 

[Studiot]

Have another look at the limiting process.

It tells us what happens as we approach a limit ie as we make the difference between where we are and the limit itself smaller and smaller.

But it does not tell us what happens when we get there.

This is the trick in analysis, to prove that this shrinking process arrives at what we want.

 

[Lajka]

Yeah, I think I understand what you're trying to say.
And it was definitely helpful to reimagine the problem with having that in mind, thanks.

 

[Lajka]

One question, though: can anyone tell me why can't we just define $$dx$$ as:

$$lim_{\Delta x \rightarrow 0}\Delta x = dx$$?​

What's wrong with this? I can't figure it out.
Thanks.

 

[LCKurtz]

One question, though: can anyone tell me why can't we just define $$dx$$ as:

$$lim_{\Delta x \rightarrow 0}\Delta x = dx$$?​

What's wrong with this? I can't figure it out.
Thanks.

That's just a complicated way of saying dx = 0, which is pretty pointless. I don't find it helpful to think of dx as an infinitesimal in the first place. If you are doing integration where

$$lim_{\Delta x \rightarrow 0}\Sigma_{i=0}^n f(x_i^*)\Delta x_i = \int_a^b f(x)\ dx$$
the integral is just a number. You can think of the integral sign and dx as just symbols representing where the number came from.

 

[Lajka]

But isn't the point of a limit in the first place to express that some quantity tends to some number (and thus we foresee where it should "end up", so to speak), but never actually reaching it?
So, in that context, why can't I use $$lim_{\Delta x \rightarrow 0}\Delta x = dx$$ to say exactly that? I didn't say that $$\Delta x$$ becomes zero, I just said that it tends to zero ($$\Delta x \rightarrow 0$$). What am I missing here?

Also, I'll probably start to interpret $$dx$$ as you suggested, it seems that the notion of "infinitesimal interval", which I picked from my engineering classes as something that is totally legit, doesn't go all that well with mathematicians. I went to a professor of a near-by mathematics faculty, and she was looking at me like I'm a complete idiot every time I mentioned the "infinitesimal interval".
She was all like:
- Excuse me, what do you mean by that?
- Well, it's not that hard, just imagine it as an interval which length tends to zero.
- I'm sorry, I have no idea what are you talking about.
- Errrm... okay.

It was interesting. :D

 

[LCKurtz]

But isn't the point of a limit in the first place to express that some quantity tends to some number (and thus we foresee where it should "end up", so to speak), but never actually reaching it?
So, in that context, why can't I use $$lim_{\Delta x \rightarrow 0}\Delta x = dx$$ to say exactly that? I didn't say that $$\Delta x$$ becomes zero, I just said that it tends to zero ($$\Delta x \rightarrow 0$$). What am I missing here?

You are right, it is the $$\Delta x$$ tends to zero ($$\Delta x \rightarrow 0$$). It tends to 0, not "dx". Just think of the integral sign and the dx as reminders that the number that the integral represents came from the corresponding approximating sum.

It is possible to extend the real number system to rigorously include infinitesimals. It isn't part of mainstream analysis and many mathematicians never study it. It certainly isn't necessary in the development of calculus. And there is nothing to be ashamed of by being puzzled by the glib use of the word infinitesimal. Don't lose any sleep over it at this stage of your mathematical experience.

 

[Lajka]

I think I'm beginning to understand it a little better now.
That kind woman also gave me a book on Analysis 1, and I got to the part where the author, after few derivations, wrote something like this:
"blabla using this and doing that blabla, finally we come to the conclusion: $$dx = \Delta x$$."
I really became nervous there for a minute, because it was totally against everything I thought I understand. But then, if I look at $$dx$$ as some function, as the book instructs, instead of perceiving it as a distance, I guess it makes little more sense. It also makes things more complicated, but at least it starts to make little more sense.

So, to clarify all this, according to mathematicians, $$dx$$ is NOT an interval, not even the one whose length tends to zero?
So, it's actually some kind of function defined on a set of $$\Delta x's$$, or something like that? I got to say, I never thought about it in that way before. From day one, even before I had any calculus whatsoever, we were instructed to interpret these $$dx's$$ as infinitesimal intervals, infinitesimal volumes etc.

Can anyone tell me then what gives us the right (engineers, physicians) to walk around and freely interpret these infinitesimals as distances, volumes, whatever, without any fear that we may screw something up?

And now I have to ask this too: is $$dx$$ from the definition of differential the same $$dx$$ which appears in the definition of integral? I'm not sure of anything anymore :D
Thanks.

 

[Studiot]

I am not sure from your posts where you are coming from.

If you are an engineer/scientist wanting to enquire more deeply that's to be applauded. You also seem to suggest you have some experience of maths beyond elementary calculus ie are you familiar with the process of establishing a control area or volume around a desired parameter such as flow or charge and shrinking it to zero. Some idea would be helpful to providing an answer to your questions and exploring why we have the different symbols

$$\Delta x$$ , $$\delta x$$ , $$dx$$ , $$\partial x$$

What the differences are (they do represent different ideas), what they are used (correctly) for and so on.

In particular it is a bad idea to take

$$\Delta x$$ as the same as $$dx$$

 

[Lajka]

Well, I'm from Serbia, Europe, and I study to be an engineer. I'm currently in my second year at the department of physical electronics (that would be the literal translation, maybe the better one is engineering physics).

I have gone through some "advanced" courses, but nothing too much, really: functions of several variables, vector analysis, ordinal differental equations systems, equations of mathematical physics, some complex analysis, Fourier, Laplace and Z transformations, and things like that. So, nothing too advanced.

Like I said, from day one here at school, before we had any course in calculus, we were teached, in courses like Fundamentals of Elecrotechnics and likewise, to interpret these infinitesimals as distances, volumes etc. in order to buiid various integrals, use charge densities, construct Maxwell equations etc.

These pure mathematical descriptions of infinitesimals and differentials are pretty new to me, so lately I find myself going back to the basics, trying to comprehend it a little better, go more deeply into this and "upgrade" my knowledge, but at the same time trying to combine all this new information with what I thought I already knew, and that seems to be a pain the ***, if you pardon my expression.

 

[Studiot]

Let us start by looking at the difference operator $$\Delta$$. This idea was introduced by Newton when he was working on his version of the calculus. It’s principle features are that it is

Finite
Fixed
Has one specific value which can be calculated or measured
Is the difference between two values of some quantity

It may be large or small and appears in many guises. Newton himself did much work on finite differences.
$$\Delta ({x_n})\quad = \quad {x_{n + 1}} - {x_n}$$
Another use is in the Heisenberg Uncertainty Principle. This demonstrates the definite specific finite nature of the beast.
$$\Delta p\Delta q\quad \ge \quad h/4\pi$$

The are examples of small but finite. On the other hand the pressure head from the reservoir above my house is the difference in altitude between them. This is a larger figure of 150 metres.
$$\Delta H\quad = \quad ({h_2} - {h_1})$$

There may, of course be lots of delta somethings, each with its own fixed value, but I think you will agree this is not the ideal gadget to be doing limits and calculus with.

To answer another of your questions – This delta can be regarded as an interval and often is.

So moving on we would like a variable rather than a constant and one that stands for small quantities only.

This is where $$\delta$$ comes in and is defined to be differences in some quantity like its big brother, but is now variable and small.

So $$\delta x$$ means a small change or variation in x and is a variable.
Splendid, splendid, that is enough for engineers and scientists.

So what of dx? And how do we work with this quantities, what are their algebras like?

Well when Leibnitz was working on his version of the calculus he introduced the idea of the slope of a curve ( the tangent ) and the notation
$$\frac{{dy}}{{dx}}$$ for this slope.

Clearly this has one specific value at the point of tangency, which is the one point a curve and its tangent have in common. Of course a tangent is a straight line whose slope is defined by change in y over change in x

Leibnitz approach was to make these changes ever smaller (what we now call taking the limit) so the curve and its tangent became ever closer until they coincide.

So if $${\delta y}$$ and $${\delta x}$$ are the values from small sections of the curve near the point of tangency then

$$\frac{{\delta y}}{{\delta x}}\quad = \quad \frac{{dy}}{{dx}}\; + \;\varepsilon$$

As $$\varepsilon \to 0$$ we have

$$\delta y\quad = \quad \frac{{dy}}{{dx}}\delta x\; + \;\varepsilon \delta x$$

Since $$\varepsilon \delta x$$ is the product of two vanishingly small quantities it is negligible and we ignore it.

So

$$\delta y\quad = \quad \frac{{dy}}{{dx}}\delta x$$

Is the often seen equation.
It is this equation which gives rise to the proper name for

$$\frac{{dy}}{{dx}}$$ as the differential coefficient and also the name for the differential of y – $${dy}$$.

Now this level of analysis is fine for a first course in calculus and in particular for single variable calculus.

But it is unsatisfactory to mathematicians for several reasons.

Firstly because we are not working on a function and mathematicians like to work on functions.
Having said that we are working on the tangent, not the function itself.
Working on the tangent or the curve (function) is generally no big deal in single variable calculus as there is only one tangent at any point. But as soon as we go to functions of several variables the curve can be twisted in space and there are infinitely many tangents available to work on – we have too much choice.

At this point mathematicians go back to Newton and talk about a point x on a curve f(x) and another point $$\xi$$ and f($$\xi$$) and let the second approach the first thus

A function is differentiable if and only if the limit

$${\lim }\limits_{x \to \xi } \frac{{f(x) - f(\xi )}}{{x - \xi }}$$

exists.

If it exists the limit is called the derivative at $$\xi$$ and is denoted by
$$f'(\xi )$$
This is called the derived function.

All this has established functions, derivatives, differentials and their algebra so that when we move on to multivariable calculus and trade

$$\frac{{\delta y}}{{\delta x}}$$ for $$\frac{{\partial y}}{{\partial x}}$$

we have something to work with.

 

[Lajka]

Wow, this is great, I just came back from school and read it all in one breath. Thank you very much for taking the time to write this.

So, if I understood correctly, when we were using these little distances and volumes in those various engineering classes, we were actually reffering to $${\delta x's}$$ and $${\delta x}{\delta y}{\delta z's}$$?

In fact, if I may be so bold to say, it seems to me now that I am actually operating with these $${\delta x's}$$ all the time, and only with them I can operate, because, in fact, that's the only thing I can imagine in this little head of mine; some very small, but finite in length, variable distance, and $$dx's$$ now seem like a mathematical idealization of that term, as they progressively become infinitely smaller, and now it seems to me that that's something I actually can't imagine. Please correct me if I'm wrong.

However, one questions still remains: why do people hate to refer to $$dx's$$ as intervals? Okay, if all the above is true, then fine, we have $${\delta x's}$$ and we can live happily ever after, or maybe even the notion of them $$dx's$$ as intervals perhaps isn't useful. Okay, but $$dx's$$ STILL seem to be legitimate intervals, only infinitesimaly small ones; maybe, if my reasoning is sound, maybe they're something that's not even possible to imagine, but STILL, intervals.
I guess my question is: why did that professor looked at me like I'm an idiot when I talked about them? I now recall her saying that length of these $$dx's$$ isn't even defined, so there is no point in saying that their length reaches zero (I remembered this little detail later, I should've asked her to clarify this to me, but oh well...)

Also, one more thing: I just remembered that I saw these notions of "small deltas" before, in fact, here:
http://en.wikipedia.org/wiki/First_law_of_thermodynamics
But I wasn't really sure what their exact meaning was back then when I saw them for the first time. Would this also be a good example of usage of this notions? (or if I reformulate the question, do quantities in this article have the same meaning which you described? I think they do, but I just want to check, do forgive me)

 

[Studiot]

I am not sure why you are so worried about intervals. Differentials like dx etc are not intervals. In one dimensional analysis there is no distinction to be made, it is only when you come to multidimensional analysis that it becomes important.
In modern topology parlance they are called 1-forms, but I doubt this helps. The

When you first meet a subject your teachers insulate you from the akward parts of the theory by only giving examples that 'work out'. There are even difficulties with the simple version of the one dimensional approach.

I said earlier to check your understanding of limits.
Limits are not about intervals they are about sequences. Sequences of points (or values).

I think the confusion arises because in some presentations there is ambiguity about what is meant by x.
x is a specific point, but it is also and axis (and therefore and interval).
When we refer to 'delta x' we are in fact referring to a small chunk of the x axis. This concept is useful in Physics for example the to decribe the control volume in fluid mechanics.

But it is unsound for mathematical rigour. Here when we talk about 'x and 'delta x' we are referring to some point x (not the whole axis) and to a sequence of points getting successively closer and closer to x so that we can show convergence.

So here are some things to think about:

In elementary texts it is common to define the differential coefficient as

$$\frac{{dy}}{{dx}}\quad = \quad {\lim }\limits_{\delta x \to 0} \quad \frac{{f(x + \delta x) - f(x)}}{{\delta x}}\quad$$

and then derive formulae for simples cases as for instance with y = ^x

$${\lim }\limits_{\delta x \to 0} \quad \frac{{f(x + \delta x) - f(x)}}{{\delta x}}\quad = \quad \frac{{{x^2} + 2x\delta x + {{(\delta x)}^2} - {x^2}}}{{\delta x}}\quad = \quad \frac{{2x\delta x}}{{\delta x}} + \frac{{{{(\delta x)}^2}}}{{\delta x}}$$

$$= \quad{\lim }\limits_{\delta x \to 0} \quad 2x + \delta x{\kern 1pt} \quad = \quad 2x$$

However we get a problem if we try this approach with the y = modulus of x. This function is defined and continuous for all x, but it's derivative is not.

$${\lim }\limits_{\delta x \to 0} \quad \frac{{f(x + \delta x) - f(x)}}{{\delta x}}\quad = \quad \frac{{\left| x \right| + \delta x - \left| x \right|}}{{\delta x}}\quad = \quad \frac{{\delta x}}{{\delta x}}{\kern 1pt} \quad = \quad 1$$

 

[alQpr]

...

The reason I ask this is because a friend told me that there is a theorem, or a consequence of some theorem, that an infinitesimal region can contain at most one real number, and this pretty much rocked my world.

Here's some reasoning on my behalf:

1) Like I sad, it's an INTERVAL. If it merely contains a one real number, what else does it contain then? Seems not much of an "interval" to me...

2) This again is more of a question, really. If i take dx, and divide it by two, I should have gained two new (even) smaller intervals. I don't know if I have a right to do this. But if I do, then each one of them can contain at most one real number, so the original interval can now containt at most two real numbers. You can imagine what happens if I continue dividing these smaller intervals. So, it's a paradox. Something tells me this is logic is flawed, so I would appreciate if someone could point out where I made a mistake, thanks.

Previous responders (and the Analysis book you referred to) have helped clarify the use of dx to denote a differential, and although I tend to agree with Studiot who says it is a "bad idea" (to identify dx with $$\Delta x$$) it is in fact standard notation used as you describe when you say we should "look at dx as some function, as the book instructs, instead of perceiving it as a distance".

But I think the theorem quoted by your friend was referring to the concept of an infinitesimal interval in non-standard analysis (See the Wikipedia entry and books by Robinson and Nelson for more detail on this if you want). This introduces a number system much larger than the reals including infinitesimal and infinite quantities and in that system it is true that an infinitesimal interval includes at most one real number. This is because if it included more than one it would have a non infinitesimal width. The problem with your problem with this is that you have missed the key words "at most". If you split an infinitesimal interval into two parts at least one of them will include no pure standard real numbers (but it may include many non-standard numbers that are infinitesimally different from a real number).

 

[Studiot]

Previous responders (and the Analysis book you referred to) have helped clarify the use of dx to denote a differential, and although I tend to agree with Studiot who says it is a "bad idea" it is in fact standard notation used as you describe when you say we should "look at dx as some function, as the book instructs, instead of perceiving it as a distance".

Perhaps you would like to clarify this paragraph?
I don't think that was what I said at all.

 

[alQpr]

. . .

In particular it is a bad idea to take

$$\Delta x$$ as the same as $$dx$$

Defining df(a,$$\Delta x$$)=f'(a)$$\Delta x$$,
and identifying x with the identity function x(x)=x,
leaves us with dx(a,$$\Delta x$$)=$$\Delta x$$;

but (I thought I was agreeing with you that) it would be a bad idea to suggest that this justifies taking $$dx$$ as the same as $$\Delta x$$.

PS I have edited my original to, I hope, better identify the "bad idea".

 

[ShinyRedCar]

...I'll probably start to interpret $$dx$$ as you suggested, it seems that the notion of "infinitesimal interval", which I picked from my engineering classes as something that is totally legit, doesn't go all that well with mathematicians...

I think this is mostly true which is why we have the definition of "differentials" that appear in current calculus textbooks. Be that as it may, engineers and physicists generally do not use them in that very limited fashion. They use them the way that you've been taught. It is very unfortunate that when students get beyond calculus to statics, dynamics, etc., they are confused by the use of differentials and how to manipulate them. All they know about differentials is how to use them to find tangent-line approximations.

 

[Lajka]

Unfortunately, I couldn't read all new posts till today, but I've been giving some thoughts to this subject in the past few days.
I guees I'll have to "erase" some concepts I've been using in the past, and start all over again. I'm not sure if that's going to be easy though, but I guess I have no choice. For example, I was always wondering why do people say that the derivative of the function, $$dy/dx$$ is not a fraction. It always seemed to me that, if you perceive $$dy$$ and $$dx$$ the way I did, imagining the derivative as the ratio of these two infinitesimal distances is perfectly logical. Maybe now I'll get it right this time.
So, I'll continue to read that book on Analysis, and God help me.

 

[Lajka]

Unfortunately, I couldn't read all the new posts till today, but I've been giving some thoughts to this subject in the past few days.
I guees I'll have to "erase" some concepts I've been using in the past, and start all over again. I'm not sure if that's going to be easy though, but I guess I have no choice. For example, I was always wondering why do people say that the derivative of the function, $$dy/dx$$, is not a fraction. It always seemed to me that, if you perceive $$dy$$ and $$dx$$ the way I did, imagining the derivative as the ratio of these two infinitesimal distances is perfectly logical. Maybe now I'll get it right this time.
So, I'll continue to read that book on Analysis, and God help me.

 

[Studiot]

Strictly dy/dx is not the derivative it is the differential coefficient. I believe I mentioned this.

However the distinction is only important for the calculus of several varaibles.

Good luck with your reading.

 

[Lajka]

Yeah, you did. I didn't quite get it, though. :D
In one variable calculus, I always called that a derivative. In calculus of several variables, we of course have directional derivatives, and their special case, partial derivatives (e.g. $$\partial f/\partial x$$), that are multiplied with their infinitesimals respectfully (e.g. $$dx$$) and summed up to make "the complete change" (e.g. $$df$$). At least, that's how I imagine it at the moment, perhaps that'll change too.
Maybe you were referring to something like this.
Meh, I guess I'll find out when I become better versed in all this.

Good luck

Thanks, I'll need it. :D