Algebraic Operations on Differentials in Liebniz Notation: An Abuse?

[Hurkyl]

Now, the triangle that is ... "disappearing" ... will of course cause trouble for students that have become "notationally dependent".

Notation has nothing to do with it -- it simply doesn't make sense to ask for the line through two points if the two points are equal. If the algebraically-minded student says "you can't do that becuase rise-over-run is 0/0" or "that doesn't work because 0m=0 doesn't have a unique solution", then good for him. If the geometrically-minded student says "you can't do that, because every line through one point is also a line through the other point", then good for him. If the student objects on some other (valid) grounds, then good for him.

And the correct response to the student? "Yes, you're right, I cannot do that. We're going to have to find some other method of computing the tangent line. But does this failed attempt give us any ideas?"

And ideally, the student springs forth with something involving limits, having just learned about them. I don't care if they come up with the limit of rise-over-run, or the limit of the angle the line makes with horizontal, or the limit of the position of where the secant lines meet some other auxiliary line, or something else. Even coming up with the idea of the limiting line is a good one, although that requires us to do some extra work to figure out what we mean by that.

If this was a course where they were actually being taught about infinitessimals, it would be enough for them to recognize that choosing the two points infintiessimally close should give us a secant line infinitessimally close to the tangent line.

But what I don't want them to get stuck in their heads is "oh, maybe everything I know about Euclidean geometry is wrong and there really is a tiny triangle of zero size there" or "if we just choose the second point really close to the first one, then that secant line is the tangent line". But those are exactly the ideas you are reinforcing.



Are you trying to get the student to honestly-and-truly think in terms of a triangle-like thing of zero size? Your PDF says both yes and no, but a clear answer would nice.

* If the answer is yes, then you have put the idea of infinitessimal geometry into their heads, and as the saying goes, "a little learning is a dangerous thing". It is a Bad Idea to do that unless you commit to the idea of fleshing out and teaching some form of infintiessimal geometry in parallel with the ideas from calculus. Are you doing that?

* If the answer is no, then the problem is that you never make clear that your zero-size triangle-like thing is a completely fictitious idea that you simply used to guide you towards some other method that works -- you never remove the triangle from the argument! The steps of the derivation is left in the form "first produce the mythical triangle, then change the triangle into something that really exists", and you never demonstrate how that gets turned into a new argument that doesn't involve any mythical objects at all.

And furthermore in the no case, I question the value of teaching the student to think in terms of mystical objects -- this is not frontier research in mathematics, this is something we've been working out for centuries! If you want them to think in terms of zero-size triangles, then you should teach them infinitessimal geometry. Otherwise, the fact we arrive at a zero-size triangle should be viewed as an obstruction to our calculation, and now the game is to find a way around/eliminate the obstruction.

 

[chrisr999]

You have a good sense of perseverence, Hurkyl,
I admire that about you,
I've attached a little piece here and no, I wasn't on peyote when i wrote it,
it's just a piece I put together today and I apologise in advance for it being far removed from text!
I hope it's enjoyable,
I want to promote the learning of calculus at as young an age as possible,
I find that this style can uncover some young kids who have ability that can be harnessed.
It won't be for everyone though, I guarantee that,
chris.

I will update it later, as unfortunately I didn't draw all the diagrams I should have.

 

[jgens]

Although I haven't thoroughly read through your newest installment, the geometric interpretation that you seem to be stressing doesn't seem to differ much from the treatment I've seen in other calculus books - though personally I find your format more difficult to follow. I'm still caught up on your introduction of "infinitesimal measurements." You don't develop what they actually are for the student and they don't exist as a subset of the real numbers.

What I'm most curious about is what a student should make of infinitesimal elements from your discussion, especially since what some students might define as an infinitesimal you define otherwise (and without discussion). Clearly, Lim_{x -> infinity} (1/x) = 0 from your example, however, why shouldn't a student take that as the definition of an infinitesimal? After all, if ε is a positive infinitesimal then ε < 1/2 and ε < 1/4 and ε < 1/100, hence it would seem that Lim_{x -> infinity} (1/x) = ε. If you plan on introducing infinitesimals, especially since you insist on self-discovery, you need to develop them more and remain consistent. Is Lim_{x -> infinity} (1/x) = 0 or is Lim_{x -> infinity} (1/x) = ε, or does ε = 0 (in which case your entire discussion around the ratio of dy to dx doesn't make much sense)?

Edit: Fixed < and > signs.

 

[jgens]

Let the coordinates (x,y) represent a point on a smooth curve.
Draw a line tangent to the curve at this point.
Now define a point on this tangent line with coordinates (x+dx,y+dy)

The differentials dx and dy are new variables (not necessarily infinitesimal) which express the coordinates of a point on this tangent line in a coordinate system parallel to the original but with origin (x,y).

Since in this construction the tangent line goes through this translated origin point (x,y),
its equation is dy = m dx + 0, i.e. by definition dy/dx = m = the slope of the tangent line.

This extends to arbitrary dimensions via tangent hyper-planes to hyper-surfaces.

Ultimately we define differentials as coordinates in the tangent space at some point on a manifold. Equivalently they are a basis for the co-tangent space.

Thanks for the insightful response. My calculus books have never given a rigorous treatment of differentials so this is very interesting.

 

[Hurkyl]

What I'm most curious about is what a student should make of infinitesimal elements from your discussion, ...

It's very interesting you would make those statements! They closely parallel one of the ways to go about defining the hyperreals (i.e. nonstandard analysis), and you've highlighted one of the major differences between that treatment of infinitessimals and the more naïve ideas I often see.

One of the methods of defining hyperreals really does start by positing the existence of a number (which I will call ε) that satisfies all of the axioms

0 < ε
ε < 1
ε < 1/2
ε < 1/3
ε < 1/4
...​

and *poof* the result is the hyperreals.

disclaimer: *poof* may not be as obvious as it appears. I assert that while it's straightforward, it's incredibly tricky if you haven't learned it[/size]

And the hyperreals have infinite numbers, such as H which I will define to be 1/ε. And it's easy to show
\lim_{x \rightarrow H} 1/x = \epsilon
but the bit that seems to diverge from more naïve versions of infinitessimals is that H is not actually +\infty. In fact, even in the hyperreals, we have
\lim_{x \rightarrow +\infty} 1/x = 0.

 

[chrisr999]

Epilogue added to the attached file to complete that piece that was a bit rushed yesterday, sorry, chris

 

[chrisr999]

Keisler's work is good, using very appropriate terms such as the hyperreals.
Even though we've only been discussing a small branch of calculus, it's worth the effort to know we have a solid foundation.

 

[chrisr999]

I've added a few notes to give credit to "infinitesimals" as being a far superior analysis than the notion of approaching zero alone.
Infinitesimals do not introduce ambiguity, they clarify it by virtue of the fact that derivatives deal with tangents, requiring only an analysis that falls "well short of true zero".
thanks for the thread,
sincerely,
chris

 

[HallsofIvy]

It's very interesting you would make those statements! They closely parallel one of the ways to go about defining the hyperreals (i.e. nonstandard analysis), and you've highlighted one of the major differences between that treatment of infinitessimals and the more naïve ideas I often see.

One of the methods of defining hyperreals really does start by positing the existence of a number (which I will call ε) that satisfies all of the axioms

0 < ε
ε < 1
ε < 1/2
ε < 1/3
ε < 1/4
...​

and *poof* the result is the hyperreals.

This, by the way, uses the "compactness" property of axiom systems: "If every finite subset of a set of axioms has a model, then the entire set has a model". A "model", here, is an actual logical system that satisifies those axioms. All of the axioms given here are of the form "there exist \epsilon&lt; 1/n" with n going over all positive integer. For any finite subset, there is a largest such n, say N, and there certainly exist a real number \epsilon&lt; 1/N. Thus, the set of real numbers is a model for any finite subset of these axioms and so there exist a model, the hyperreals, for the entire set of axioms.

disclaimer: *poof* may not be as obvious as it appears. I assert that while it's straightforward, it's incredibly tricky if you haven't learned it[/size]

And the hyperreals have infinite numbers, such as H which I will define to be 1/ε. And it's easy to show
\lim_{x \rightarrow H} 1/x = \epsilon
but the bit that seems to diverge from more naïve versions of infinitessimals is that H is not actually +\infty. In fact, even in the hyperreals, we have
\lim_{x \rightarrow +\infty} 1/x = 0.

 

[chrisr999]

hi jgens,

I've added a few pages to the end of the file to bring in more clarity to the "infinitesimals" and the exact ratio of the derivative.

Let me know how it feels to you.
There are other ways, of course, let's just see if we can clear up everything.

 

[jgens]

Well, having read through your newest installment relatively thoroughly, I have a few suggestions which you (and others) may or may not agree with:

1) Omit the discussion of infinitesimals. While this new version does give a slightly more adequate treatment of infinitesimals than previous versions, I still think that it leaves too much open for misconception and misunderstanding. Though a completely rigorous treatment of infinitesimals could probably be forestalled until the student has more mathematical maturity, I still think that the teacher/professor/instructor needs to work out several of the properties of infinitesimals (or carefully guide them there) to avoid misunderstanding the concepts. Drawing from an earlier example, why shouldn't a student take, Lim_{x -> +∞} (1/x) as the definition of an infinitesimal? In which case the student would find that Lim_{x -> +∞} (1/x) = ε. As Hurkyl pointed out earlier, this isn't the case, but the inquisitive student doesn't know that!

Since a lot of what you're introducing seems to be along the lines of differentials, your discussion of the derivative could probably stand without infinitesimals.


2) Assuming that the student is not familiar with derivatives, when you're introducing the geometric interpretation of the derivative, place more emphasis on the derivative as the limiting secant line. Your approach to do this with triangles works pretty well, but depending on the background of the student, may seem superfluous. Reorganize the discussion so that you develop the limit definition of the derivative and then define dy/dx = ∆y_tan/∆x_tan in terms of differentials as jambaugh posted earlier. This way, you remain consistent with the notation of calculus (using dy and dx instead of ∆y_tan and ∆x_tan) and you develop the derivative as a quotient of differentials rather than a ratio of infinitesimals. You may also want to mention that, Lim_{x -> a} [(f(x) - f(a))/(x-a)] is a perfectly acceptable definition of the derivative.

As an aside, I take issue with the statement that the derivative is not the limit of a quotient, especially since the derivative is defined in terms of limits. While you do argue that we could simply define the derivative in terms of tangent gradients, this provides no way to actually calculate the derivative. Additionally, by placing an inordinate focus on derivatives as the tangent gradient the student is led away from important concepts like the derivative as a function. Even though the derivative can be defined in terms of differentials, they don't provide a method for calculating derivatives. For these reasons, I still think that it's best to define - at least initially - the derivative as the limit of a quotient.

Hopefully you'll find these comments helpful!

 

[chrisr999]

hi jgens,
have a look at the last few pages of this updated discourse.
It will show how infinitesimals relate to the real number system.
Again, they vary in dimension and it is their ratios that ultimately matter.
Their exact ratio is obtained from the linear function.
Their varying ratio is what calculus eliminates.
chris

 

[chrisr999]

There's enough information in version 5 to answer all but one of those questions, jgens,
it can be worded differently for different students of various levels but it really is at an elementary enough level for young students.

I haven't discussed any of the mathematical techniques at all! hardly anyway!
but that becomes quite easy to do from here,
though you should know by now, there is no division by zero involved even if many get that impression before fully examining the geometry.

You're going to have to apply yourself though! to break through it.
I will be busy for a week.
sincerely,
chris

 

[chrisr999]

tell you what,
if i have time, i will work through an example for you, jgens,
if you have one that's really perplexing,
i will do a tutorial on it,
using mathematics only without the geometry!

 

[chrisr999]

I understand your position on the infinitesimals, jgens,
and, ok, it would be only appropriate to do that for you, I'm sorry i have quite a few things to do at the moment, however as you've seen, there are very proficient guys on this thread capable of lighting up the darkness in their unique ways and also from the perspective of accurate mathematical terms.

What you need is to find a way to handle these "units" that is very clear for you, where the words are expressed in your preferred learning modalities.

Let's say someone wanted to know what a papaya tastes like and they'd never encountered one. They know what it tastes like through experiencing it and could then describe it. But if i didn't have one to give them and started describing it to them, that would deny their own experience of it and would always be an inaccurate description, it would approach the true sensation without ever being completely accurate.

This is why I gave the experience of them before any description.
To me, they are "a pair of orthogonal nano-axes that do not cross" and their function is to zoom in on the point where you want to find the rate of change of a function where the gradient is measurable. The tangent is the other geometric tool.
You analyse a one-point situation with 2 points initially and zoom in on your point of interest, until your infinitesimals do not distinguish between the right and wrong value of the derivative. They have whatever length they have in that scenario.
You then zoom out, allowing these infinitesimals attach to the straight line.
Their lengths are real values, though not relevant. Their ratio during the zoom-out is relevant.
They are "tools" of geometric analysis.
You've got to have a sense of them, not a definition.
You can define them as you please after experiencing them.

No, the limit of 1/x as x approaches infinity is not an infinitesimal.
You are not paying attention to the graph of the curve!
The infinitesimals in that case are doing something I haven't discussed in the little piece i wrote but Hurkyl was showing you just how interesting it all is and these are non-complex examples.
For 1/x, there is one infinitesimal, because the gradient has no measure at the limit, it's zero.
That infinitesimal is the vertical one, the horizontal one is increasing out of bounds as the vertical one reduces to zero, but it really does not stop reducing!
i couldn't call the horizontal one "infinitesimal" as it's increasing to infinity.
Can you visualise it? If not, draw it.
The tangent is the x-axis which has a gradient of zero as the triangle I used has "melted" completely when we can't visually tell the difference between the axis and curve.
We don't have a final "measureable ratio" for situations in which the x and y axes are the tangents and also the point of intersection lies at unreachable infinity.
This is the case for "discontinuous" functions.
They are perfectly analyseable but require additional definition, as you say, a "rigorous" one, for completeness, but students can easily extrapolate them when they've got the spirit of the analysis.
We either have a continuous or discontinuous function.
If you like, you can define rigorous definitions for both cases, it shouldn't really be necessary though.
chris

 

[0xDEADBEEF]

Leibnitz

When do Americans learn it's Leibnitz and Wiener. English "i" is German "ei" or "ai" and English "e" is German "i" or "ie".

 

[jgens]

I do hope you realize my criticisms are not because I do not understand standard calculus or specific properties of numbers! However, based on most, if not all, of your comments directed towards me I don't get that impression.

In which case the student would find that Limx -> +∞ (1/x) = ε. As Hurkyl pointed out earlier, this isn't the case, but the inquisitive student doesn't know that!

No, the limit of 1/x as x approaches infinity is not an infinitesimal.
You are not paying attention to the graph of the curve!

I recognize that and I pointed it out earlier! My point was that from your discussion of infinitesimals, this is not clear and that nothing prevents the student from reaching a fallacious conclusion.

Aside: I still don't think that the newest treatment does justice to infinitesimals, nor does it adequetly introduce the derivative, especially since the definition of the derivative in terms of the tangent gradient at a point does not lend itself (at least as easily) to the interpretation of the derivative as a function!

 

[Hurkyl]

I'm not sure if it's clear, but we're not being hypothetical about our objections (at least, I'm not). People really do have the misconceptions that we've been talking about. There are people who fail to recognize the difference between "the limit of a sequence" and "a sequence". There are people who think there is a smallest, positive, real number. There are people who think it is impossible to obtain exact answers using limits. There are people who think infinity is just a large real number, and the limits of something as x->0 is nothing more than plugging in a small value of x.

Most (all?) of my criticisms are from these kinds of explicit examples: my impression of your exposition and specific way of phrasing things is that rather than dispelling some of these misunderstandings, it could actually reinforce them!

It's hard to give constructive criticism, because you seem to find the things I disagree with to be a key feature of your exposition! e.g. when I see phrases like

dx is the length of the horizontal side of the blue triangle when it’s shape cannot be distinguished from the red triangle​

or

When the two triangles become “indistinguishable for all practical purposes”, the ratio of the perpendicular sides is dy/dx​

or even

\frac{dy}{dx} = \lim_{\Delta x \rightarrow \color{red}{dx}} \frac{f(x + \Delta x) - f(x)}{\Delta x}​

(color added for emphasis) I am vehemently opposed, because it reads as an explicit endorsement of the idea that a limit is nothing more than an approximation, formed by plugging in some unspecified value really close to the target.

But I get the impression that you really do consider things like these to be the key features of your method of presentation! There really isn't anything for me to do than to argue that your approach is fundamentally flawed.

 

[chrisr999]

Though the "value" of 1/x as x goes to infinity is "infinitesimal" if zero is included.
But not from the geometric simplicity of the illustration.
I'm sorry lads, I thought this was an exploration, rather than a matter of trenching in and defending the territory.
No, I wasn't dictating to you,
but some of you are very inflexible in your thinking!

It's inappropriate to say such a thing as "leave out the infinitesimals" and so on,
small measurements and using them is so basic,
a kid would get it without trouble!

and similar impatient comments.

The forum is here for your exploration,
you should try to respect it and other people.
Use it foolishly if you want to, who cares.

I won't be contentious or argumentative which happens when someone is only prepared to go so far.
Your objections are basically,
"Don't want to think independently about this, sorry, just want to repeat, repeat".
"Don't ask me to think, I haven't given you permission to ask".
you've gone as far as you will go, i reckon,
this is my final thread,
sorry for wanting to contribute and discuss the truths of the subject!
If someone was really interested in this they wouldn't take to silly attacks against someone else's analysis and call it flawed without even trying to see the view. I tried to put it as simply as possible.
But you can only lead a horse to water as they say,
There is sincere analysis but you always have the lazy ones that couldn't be bothered to even look.
I mean there are complaints about the spelling of people's names!
complaints about the definitions!
crikey!

my sincere apologies lads,
take care,
chris

 

[chrisr999]

Hurkyl,
I can hardly believe it was possible to get the impression I was saying a limit is an approximation!
along with a few other things,
but that demonstrates you didnt even understand what i put forward.
You ought to be very clear that the approximation is away from the limit,
but to get that impression?
To be honest, that comment is really strange,
I really can't imagine this is a mature forum,
over and out,
happy wood-pushing lads

goodness gracious.

 

[chrisr999]

By the way, Hurkyl,
if there is anyone who thinks there is a smallest possible real number,
it only means they are young, not introduced to the concept of a continuum,
such as "distance", never heard of "pi" yet, they are only learning or totally disinterested in math!
they shouldn't be criticised for just getting used to it in the beginning but if they've been at it for a long time, maybe scientific disciplines are not their scene,
bur for God's sake, lads, would you come on!

Your objections are very very inflexible and narrow and unimaginative,
sorry!

good luck

 

[chrisr999]

dx is not a definate length!
the student doesn't have to hit zero to understand the limit!
the student needs only enough imagination to realize the limit is revealed by the tangent,
which I've tried in numerous ways to explain!
that is EXACT not approximate, the limit is THERE, not just in continually reducing all the way to zero until your imagination runs out.
therefore the student only needs to zoom in within distances defined by the real number system
TO THE POINT THEY REALISE WHAT THEY ARE LOOKING FOR IS HANDED TO THEM ON A PLATE BY THE TANGENT.
They don't have to reach zero. THAT'S THE WHOLE POINT!
The mathematics then weaves it's way around those observations.
I haven't objected to your posts, I've just been surprised how quickly you object to mine without thinking through what i wanted to show!
so, I didnt hold my patience but what the heck! i wouldn't be doing anyone any favours that's not copping themselves on!

This is why the mathematics is written using "as the limit of dx APPROACHES zero",
not "until dx reaches zero".

So we use a little imagination to jump from the curve to the tangent since the answer is there!
and the student should stay there until it's clear, as we've seen in this discussion,
there's no point moving on and confusing yourself even more.
Once the confusion is cleared up that's it, it's clear.
the objections have been considered and ultimately dismissed, lads!
sorry to disappoint you,
but that's life

surely, the english is clear?
cripes lads, i thought we were exploring together.

 

[Hurkyl]

Hurkyl,
I can hardly believe it was possible to get the impression I was saying a limit is an approximation!

When you're writing material for people to learn from, it doesn't matter what you really mean -- what matters is what will be learned from it.

 

[Hurkyl]

When looking at your own work, you have to put yourself in the mindest not of yourself who has spent years or decades studying mathematics, but instead in the mindset of a student who doesn't understand things -- one who might even have specific misunderstandings -- and evaluate what they might learn from your writings. Yes, I'm sure that some people will get the right idea -- but I'm equally sure that some people will get wrong ideas.

I'm somewhat baffled that you can't even begin to understand why I think that some students could get the wrong idea, especially since I've highlighted some key passages that lead to my perception. (Note that understanding why but disagreeing with me is something different than not understanding at all)

 

[Hurkyl]

Aside -- I would like to point out that when interpreting derivatives via differential forms/dual numbers, df(x)/dx really is computed by plugging in an infintiessimal nonzero value and computing the difference quotient. In particular, there is the strict equality

f(x + \epsilon) = f(x) + \epsilon f&#039;(x)

This works out because when using differential forms/dual numbers infinitessimal geometry is affine geometry: nonlinear effects are nonexistant.

 

[Hurkyl]

By the way, Hurkyl,
if there is anyone who thinks there is a smallest possible real number,
it only means they are young, not introduced to the concept of a continuum,
such as "distance", never heard of "pi" yet, they are only learning or totally disinterested in math!

Er, right. Hasn't this entire discussion been specifically about how to teach this stuff to people who are "only learning"?


(Or am I misunderstanding -- are you saying that it isn't worth trying to teach anyone who doesn't fully understand things like "continuum" by the time they reach their first calculus class?)

 

[okkvlt]

dy=y[x+dx]-y[x]
dx=dx

dy/dx=(y[x+dx]-y[x])/dx


looks like a fraction to me

 

[chrisr999]

thank you, Okkvlt,
it's amazing how it got to the point it wasn't obvious

 

[jgens]

dy=y[x+dx]-y[x]
dx=dx

dy/dx=(y[x+dx]-y[x])/dx


looks like a fraction to me

Still doesn't look like a fraction to me. If you want dy and dx to be infinitesimal you need the standard part function to actually define the derivative so your definition wouldn't be complete anyway. Or, assuming dy and dx aren't infinitesimal, you need to use a limit to actually compute and define the derivative.

 

[Ja4Coltrane]

For functions from R to R, a derivative is a limit of a fraction. Now I understand that dy/dx seems perfectly reasonably viewed as a fraction because of that, but then we run into trouble.

For instance, if f(x) = 3x for all x, then it is awesome to write df = 3 dx because it makes sense heuristically. In fact, even if f is non linear but well behaved, it still seems nice because of local linearity. However, derivatives are NOT always slopes of tangent lines.

Take f(x) = x + 2x^2*sin(1/x) for x non-zero, f(0) = 0. Then, f' is positive at 0 so by chris's interpretation, dy/dx>0 so dy is positive if dx is positive.

However, this is absurd because f is not monotone on any neighborhood of the origin!

I think this is a wonderful example of why the fracional heuristic is inferior to the precise definition.

Heuristics can be used to get ideas for theorems and their proofs, but they are not substitutions for definitions!