Algebraic Operations on Differentials in Liebniz Notation: An Abuse?

[Hurkyl]

There is no triangle right! at least there shouldn't be. not quite!

Then why do you tell them there is? From your PDF:

Hence the right-angled triangle we were using to write the line gradient (between 2 points on the curve) has DISAPPEARED! or has it??
Consider this to have become reduced to the molecular level ... Now imagine magnifying this until it’s clearly visible again ... Once the limit as \Delta x \rightarrow 0 has been evaluated, \frac{dy}{dx} = \frac{\Delta y}{\Delta x} for the tangent.​

Things like this are the reason you are being criticized. You seem to know full well that there isn't a triangle. But you tell them it really is still there, and really small. Even worse, the way you talk about the limit of \frac{\Delta y}{\Delta x} as if we were simply plugging the value "\Delta x \rightarrow 0". Yes, I know that doesn't make sense, but the students don't know that.

What makes this sad it would take very little to turn this into a series of actual, true statements that don't require magnifying mythical hypotenuses of zero length or infinitessimals or anything like that. You want to talk about scaled triangles? Then do that. Draw a circle Z around t and mark the point L where the tangent line at t intersects Z. Then mark the points where the segments ta, tb, tc intersect Z and demonstrate how that intersection approaches L.

And emphasize that it's approaching -- don't phrase things as if the tangent line is really just another secant line.

(Maybe you'd prefer using a vertical line segment at x_t+1 instead of the circle, so you can work more nicely with right-triangles. The circle is nice because it doesn't depend on a choice of coordinates and handles vertical tangents easily, but the line is nice because it 'measures' slope rather than angle and is simpler algebraically)

You see in school, students are taught to work with averages, then move on to calculus and at best many can perform the techniques without grasping the intricacy of calculating with an interval of zero,

You make it sound like calculus is just a clever way to divide by zero without error.

Try to get the "story" of it. That's what mathematicians have to do.

You're missing the point. Yes, the story is important. But it's hard to convey the story if you leave stuff out and don't tell the rest right. At this point, I'm not even sure if you have a well-defined story to tell!

e.g. what do you think a "triangle reduced to the molecular level" is?

 

[OrderOfThings]

I'm curious what you think "d^2 f/dx^2" means (along with "d^2 x") -- when I learned calculus, that literally meant "the second derivative of f with respect to x then x"

Take a curve in the x,y-plane and pick a tangent vector field along the curve. Then at a point (x,y), the tangent vector has coordinates (dx,dy) and the derivative of the tangent vector has coordinates (d^2x,d^2y). The derivative of y as a function of x can be calculated as

\frac{dy}{dx}

and the second derivative is

\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d^2y\,dx-dy\,d^2x}{dx^3}.

If the curve is parametrized by a multiple of x, then dx=\textrm{const} and d^2x=0 and the second derivative formula reduces to

\frac{d^2y}{dx^2}.

For any affine reparametrization u=ax+b, then also d^2u=0 and

\frac{d^2y}{dx^2}\cdot\left(\frac{dx}{du}\right)^2=\frac{d^2y}{du^2}.

 

[jambaugh]

jambaugh, is it possible to rigorously develop calculus using differentials? I thought that they were more a calculational trick than anything else, but perhaps not?

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.

 

[jambaugh]

Followup note:
Once we define differentials we can then define the differential operator d.

We understand differentials of variables as new variables with the caveat that constraints on the original variables imply specific constraints on the differential variables. For example:
z\doteq f(x,y) \to dz \doteq \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial x}dx

Remembering that differentials are variables we may then apply differentials to differentials:
d^2 x,\quad d^3 y,\quad ...

Then given y = y(x) (using y both as the function name and the variable name.)
dy = y'(x)dx
d^2 y = d(y'(x)dx) = y''(x)dxdx + y'(x)d^2 x
If we later say x = x(t) we then have:
d^2 y = y''(x)x'(t)x'(t) + y'(x) x''(t)
or we can simply consider:
\frac{d^2 y}{dx^2} = y'' (x) + y'(x)\frac{d^2 x}{dx^2}
We go back to the geometric definition to see that:
\frac{d^2 x}{dx^2} = \frac{d}{dx}\frac{d}{dx} x + \frac{d}{dx}1 = 0
thence the notation is consistent with the Leibniz notation for the second derivative:
\frac{d^2 y}{dx^2} \doteq y''(x)

There is a very subtle bit of math here, what's going on is that the decision to allow x to be the independent variable [/i](and to allow the Leibniz notation for second derivative correspond to the literal interpretation of the fraction)[/i] imposes the constraint that its second differential be zero. This is a constraint identity not a definitional identity. One can get into serious trouble by changing around the independent variables and forgetting to "unconstrain" this condition.

Possibly a "better" notation for the second ordinary derivative would be:
y''(x) = \frac{\partial^2 (d^2 y)}{\partial (dx) \partial (dx)}
But this gets silly and just passes the issue on to the partial derivative notation.

I recommend not using higher order differential notation until one is very very well practiced. I've made embarrassing mistakes trying a cold "derivation" in class using the higher order notation and since stopped. But the Leibniz notation taken as notation is fine and has the virtue of showing the relationship of the units in physical applications.

 

[chrisr999]

I will answer your three questions, Hurkyl,
but please try to think it through.

I said "there is no triangle" at the point of tangency.
Can you see one?
The point of tangency is a "point" and you know what the definition of a point is,
it's a place of zero size.

If a student got to wondering about the apparently contradictory statement of there being a triangle at a point where logically there cannot be one, they will develop the skill to find calculus extremely easy.

It's not a very difficult riddle to resolve.

As I mentioned, the contradiction arises in attempting to write the gradient of the tangent to the graph.
We know the derivative is the gradient of the tangent and "unfortunately" the tangent intersects the curve only at a single point, hence there is no direct way from the function equation to express the gradient.

This is why we start by writing an approximation using two points on the curve itself, which initially is an inaccurate answer.

The overlapping triangles on the right of the tangent that I drew are vanishing as we move to the point of tangency. At the point of tangency we "appear" to have lost that triangle due to the mathematical definition of what a point is or a real number.

Had we not gone this route (though it's a necessary one to formulate the math), we could say it does not matter what size triangle we use against the tangent, since the ratio of the perpendicular sides is constant.

It's the same as looking at something close to you, say a bird for instance.
Now that bird flies away to a remote location. On it's path, there comes a point where you can no longer see it, even if we didn't have the horizon to contend with, but that doesn't mean it's not there.
At the point of tangency, according to the definition of real numbers, there can be no triangle there, but such definitions only mask the truth of the situation.

I chose to illustrate in that manner to encourage the student to exercise his imagination and resolve the apparent contradiction and not get tangled up in definitions where those definitions become a barrier.

The triangles on the left of the tangent help resolve the confusion as the student can easily imagine this shape reducing in size ad infinitum WITHOUT EVER DISAPPEARING COMPLETELY, even though the ones on the right do "appear to be disappearing", though approaching a limit which is still worded in a regimental way.

To answer the "divide by zero" question... Did you not see in the .pdf file that i said "We are not actually dividing by zero at all"?

That problem is resolved again by examining both types of triangle on both sides of the tangent.

Remember 2x/x is always 2 no matter how small x is but if you say 2x is zero as x goes to zero and x is zero as x goes to zero, so 2x/x is 0/0 as x goes to zero means we've gone a step too far and have forgotten to keep an eye on what the ratio is.

Use your creativity to try the 3rd question,
I guarantee you, a student can improve rapidly when you ask them to be imaginative.
I've gotten students to improve by 3 grades in a few lessons, where they had been floundering within the established educational system.

 

[Hurkyl]

Take a curve in the x,y-plane and pick a tangent vector field along the curve Then at a point (x,y), the tangent vector has coordinates (dx,dy) and the derivative of the tangent vector has coordinates (d^2x,d^2y).

So... you have in mind having some implicit parametrization of the curve (which I will call s) and when you say "dx", what you really mean is "the derivative of x with respect to s", and similarly "d²x" is shorthand for "the second derivative of x with respect to s".

If you actually write that out in Leibniz notation rather than using shorthand, what you mean by d²y / dx² is

\frac{\frac{d^2y}{ds^2}}{\left( \frac{dx}{ds} \right)^2}

is that correct?

The problem is that this doesn't mesh with how people actually use Leibniz notation. The specific counterexample I had in mind when I wrote my post could be formulated as

x=s³
y=x³

and (AFAIK) most people would expect to have

\frac{d^2y}{dx^2} = 6x

whereas your interpretation would result in 8x.

One specific case where this might arise (and where I've seen trip up even knowledgeable people) is doing a change-of-variable for second derivatives. (edit: ah, I see jambaugh pointed that out already)

 

[OrderOfThings]

So... you have in mind having some implicit parametrization of the curve (which I will call s) and when you say "dx", what you really mean is "the derivative of x with respect to s", and similarly "d²x" is shorthand for "the second derivative of x with respect to s".

If you actually write that out in Leibniz notation rather than using shorthand, what you mean by d²y / dx² is

\frac{\frac{d^2y}{ds^2}}{\left( \frac{dx}{ds} \right)^2}

is that correct?

Well yes, since ds=1, the above fraction is identical to the fraction d^2y/dx^2. This is not a shorthand notation. But this fraction is only equal to the second derivative of y as a function of x when d^2x=0.

The problem is that this doesn't mesh with how people actually use Leibniz notation. The specific counterexample I had in mind when I wrote my post could be formulated as

x=s³
y=x³

and (AFAIK) most people would expect to have

\frac{d^2y}{dx^2} = 6x

whereas your interpretation would result in 8x.

The second derivative is computed by

y''(x) = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d^2y\,dx-dy\,d^2x}{dx^3} = 6x.

Nothing wrong here I think.

 

[Hurkyl]

I said "there is no triangle" at the point of tangency.

Yes, you did say "the triangle ... has DISAPPEARED" -- which you immediately deny it by saying "or has it?", followed shortly by "it's really still there".

What exactly is the "story" do you want the students to see? Use a definition if you have to -- precisely conveying ideas is one of the things those are really good at.

 

[chrisr999]

you see, the way you are wording your question when you say..."I say the triangle has disappeared" and then I say "or has it",

this is just a way to ask the student to hold on for a second and wonder about what's really happening,
encouraging them to think it through,
or rather, to be more precise, to ask them if they can begin to imagine what is happening to both the triangle that is "changing shape" and the one that can be drawn at any size you like.

True, if the student has trouble imagining it, I can develop an animation for them, but only if their visual modality is not very active.

the mathematics of calculus is easy to understand non-verbally, without reference to number systems or notation of any kind. Bear in mind, to write a book or speak we must use some kind of symbols.
Also, when you are driving along in your car at various instantaneous speeds, you do not need any diagrams or notation, you only need your senses.

Now, the triangle that is "changing shape", the one from which we write the gradient of the line using 2 points on the curve itself (the crude gradient) is the one that is "disappearing", "appearing to disappear", "imploding", "approaching a galactic black hole" or however we want to illustrate it's vanishing act, will of course cause trouble for students that have become "notationally dependent".
They try math by working "in the dark" so to speak and end up in trouble with the description.
it is not the description that is important to understand but THE DESCRIBED". There are numerous ways to describe the described.

Bringing up the topic of the vanishing triangle, which is at the heart of the conflicting views regarding "dividing by zero", "limit of fractions", "limits" etc which are all incomplete ways to approach the problem, introduces the problem that is a side-effect of the "notation".

So you examine all of that, you don't give up the analysis but continue on because someone points out that "the problem of the vanishing triangle is resolved by the gradient of the tangent itself".
This is a visual comprehension, whereby the student is asked to imagine reducing it to as small a size as possible and compare it to the the smallest size they can imagine for the vanishing triangle.
It's not really all that Lazarus-like.

If you never got into notation at all, you could simply say, the function derivative is always the gradient of the tangent that you can move all around the curve. Of course that would not be mathematically efficient!

However it does offer instantaneous comprehension from which you can write the understanding in numerous ways, hopefully while being aware of the confusion introduced by choosing only one of the notational modalities available.

 

[chrisr999]

I will write up the story as an attachment, am a little busy at the moment

 

[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!

 

[chrisr999]

my discussion was on continuous functions with tangents at all points.
i discussed a basic discontinuous one also.
Try not to say it doesn't make sense in a different area.
To avoid confusing yourselves, you should closely examine the graph of the function you mentioned.
If you decide to say "derivatives arent always tangent gradients" then SHOW CLEARLY WHY THAT IS SO. If you add further complexity to a student's analysis who is clearling up something he wants to understand, you are only going on to another level before he's ready.
Stick to the fact that the derivative of simple continuous functions is given "exactly" by the gradient of a tangent at the point the derivative is calculated.

If you don't understand that and go onto compound shapes for which you don't show what you are now classifying a derivative as, you end up wasting the person's time and it is a hopeless discussion to have.

But you don't listen anyway, so what's the point continuing this?

fractions are fractions.
is speed a fraction?
are growth rates fractions?
are gradients fractions?
is a derivative a gradient?
don't let words be your masters.

 

[chrisr999]

You also misquoted me JA4Coltrane,
not only do you not present your graphical analysis,
which any of us can do for you,
but you proceed to work around zero without showing the limit or why you would want the limit or discuss what you are looking for in this case etc etc etc.
I will not waste my time discussing a half answer unless you are prepared to "take it to the limit".

 

[chrisr999]

Have a look at your curve. Increasing amplitude and reducing distance between the localised peaks.
At what "point" can you not have a tangent?
The closer you get to zero, the harder it becomes to view in this case, even with the real number system, and which system is modeled unambiguously with that mathematics function?
The fact that it may take immense computational power to zoom in is not the point.
The point is "do you want clarity or confusion"?
and at what proximity to zero?

 

[Hurkyl]

If you decide to say "derivatives arent always tangent gradients" then SHOW CLEARLY WHY THAT IS SO.

Nobody has said that. (At least, I don't think anyone has)

The main thing that people are saying is that derivatives are not difference quotients. The equivalent geometric statement is that derivatives are not slopes of secant lines. Furthermore, if you want to interpret "dy/dx" as the quotient of two things -- "dy" which somehow relates to changes in y and "dx" which somehow relates to changes in x -- then you have to introduce some new mathematics, be it differential forms, infinitessimals, or something else novel.

(Note that if you switch to the tangent line to talk about "dy/dx", "dy" no longer has any bearing on the function/curve that we were studying)



Also, do keep in mind that many people have strong algebraic intuition, often much stronger than their geometric intuition. There's an old joke that half of the people who study algebraic geometry do it so that they can apply their geometric intuition to study algebra. The other half do it so they can apply their algebraic intuition to study geometry.

(I assert that the ideal is to be adept in both pictures)

If you look back over the history of mathematics, you can see lots of examples of cases where people were trying to study geometry, but could only make progress by turning geometry problems into algebra problems. e.g.
* Descartes invention of coordinate geometry
* The algebraists figured out projective geometry first
* Algebraic topology

 

[chrisr999]

that was directed to the person that made the statement, Hurkyl,
if I'm going to be quoted out of context, I ask the person discuss with me, not discuss what they erroneously thought i was talking about and misquote me to others.
If you want, read his statements again and if you want to respond to him, please do.
Professionalsm if possible.

 

[chrisr999]

I'm sorry for being impatient sometimes, Hurkyl,
I appreciate people taking part in these discussions,
I appreciate your input,
I'm very busy with a lot of projects,
I don't mind someone saying "look i don't see it like that, this is how i see it" or whatever, but to say, "thats all wrong" etc and sticking with that, correct or not, gets tiresome.
What should matter is the subject itself and not the characters,
the revelation of the subject, which clearly will be situationally dependent.
Definitions may be worded slightly differently for specific cases.
this is a colourful world, not all black, white and shades of grey.

 

[chrisr999]

the thread was originated exploring the nature of the "derivative" of the function y=f(x) which in Calculus is written dy/dx.

the question basically was.. do the normal mathematics of fractions still apply?

the answer depended on the type of derivative.

Derivative means "derived from".
gasoline is a derivative of oil, orange juice is a derivative of orange etc etc.
The derivative under analysis originally was dy/dx.

Definition means "definite", "clearly defined", "unambiguous".

the original derivative in question is "the rate of change of a function derived from the formulation of the function itself".

It was correctly pointed out, that if you combine partial derivatives, you will get exceptions. We went into that and the reason for it.

The question that remains is...
Is the fractional nature of dy/dx still under scrutiny or is it resolved?
If not, is there any point discussing other types of derivatives until it is?
If it's not resolved, why not? what's unclear?

 

[Hurkyl]

The original point of contention, "Is dy/dx a fraction?", I assert is the algebraic analog of the question "Is the tangent line a secant line?" And unless we reinterpret the question in a framework other than plain Euclidean geometry / real arithmetic, the answer is a definite "no". (Although, I don't think you yet agree with that)

I also assert that there are (at least) three general ways of thinking of geometry "in the small". Applied to the question of computing a tangent line via secant lines:

* The standard analysis picture: there is no infinitessimal geometry. Our method is to approximate the tangent line via secant lines, and then take a limit as the error in approximation goes to zero to get the tangent line.

* The nonstandard analysis picture: infinitessimal geometry looks exactly like ordinary geometry. We take a secant line whose points are infinitessimally separated, and then round that to the nearest standard line to get the tangent line.

* The differential geometry picture: infintiessimal geometry is affine. The tangent line is a secant line through two points whose separation is a nonzero infinitessimal.

There may be other pictures, but these seem by far the most prevalent and well-developed.

I assert that the fact we have such well-developed foundations means that we should use them when teaching calculus! 150 years ago, it would have been appropriate to present calculus using vague and poorly-defined notions. Today, I assert it is not.

I assert that whichever foundation is used, it should actually be taught to the student -- no fair invoking thinks like hyperreal infinitessimals without explicitly teaching the student enough to be able to manipulate and reason about them on his own.


Do you consider any of these assertions fair?

 

[D H]

Just curious, why is this thread still here? After all, we don't allow personal theories at this site.

 

[chrisr999]

That's more like it, Hurkyl.

I consider your assertions very fair and it is from this position that we can examine the relative merits of learning calculus.
As this is highly involved, especially when dealing with a range of students of varying levels and ability.
When dealing with advanced calculus, the framework of reference needs to be seriously accurately defined and definitions, terms, number systems, degrees of freedom must be mapped out.

In learning the subject initially, there is flexibility of expression, but the frame of reference still has to be fully "defined" so that 2 or more people are referring to the exact same thing.

first point: "Is the tangent line a secant line?"
my answer: tangent has a single point of contact, secant has two. there is a difference.
are there any alternative definitions?

"Is dy/dx a fraction?" is answered by "Is dy/dx the tangent gradient?"
is a gradient a fraction? if it isn't and it's neither vertical nor horizontal, what are we
referring to?

Perhaps your analysis is pointing to the secant line gradient as being a fraction in terms of delta(y) divided by delta(x) and somehow differing in quality to the tangent gradient. No matter which units we use, the secant line, which is used to initiate the written mathematics of the "inaccurate instantaneous gradient of the curve" has a gradient expressed as a fraction. The gradient of the tangent gives the ratio of the exact value being sought.
That's the exact ratio.
the secant ratio is the inexact ratio. It's used to formulate the maths, the mathematical computation then proceeds to eliminate the error that was introduced by the secant, to arrive at the gradient of the tangent. But I've already presented all this.

second point: standard and non-standard analysis.
whatever a guy likes to order at the bar!

Once a student doesn't get confused with the terminology and understands what's going on, and once the descriptions are very clear, there shouldn't be a problem.

A kid rolling a ball around on a table has enough visual representation to silently lead into the geometry.

What i wanted to show at the infinitesimal level, though it's very basic, is the "merging" of the ratios the secant and tangent gradients, at a small enough level to show the student that the goal is the tangent ratio.

At the non-verbal level, the tangent is simply the secant pivoted on the point of tangency itself and rotated until there is one point of contact between the line and curve.
The mathematics expresses this rotation.
The limit is revealed by the tangent.
As the student explores what's happening at the "infinitesimal" level around the point of tangency is akin to a kid learning to ride a bicycle with training wheels.
If he understands the geometry clearly, he can work easily with necessarily agreed terminology later, particularly in working through the mathematics without referring to geometry.