Algebraic Operations on Differentials in Liebniz Notation: An Abuse?

[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.

 

[Ja4Coltrane]

Here is how I like to look at this. (df/dx) at a point x is a sort of generalization of a tangent line. That means that it isn't always one (consider f(x) = x^2 if x is rational, 0 otherwise which has f'(0)=0), but can be.

For that reason, since the *notation* df/dx looks like a difference quotient, it is extremely appealing and thought provoking. I like the notation, but I don't take it literally.

By the way: Chris, you got annoyed at me for using a not-very-nice function. I personally think that my function was pretty nice! It is, despite what you said, differentiable at every point (I'm referring to the function in post #90).

 

[chrisr999]

you didnt annoy me, JA,
thats your impression,
did you join to add clarity?
you saw DH object to "personal theories".
When you say things like "sort of", "generalisation of a tangent line",
you are veering too far away from mathematics without referring to what exactly
df(x)/dx is. It is something very simple and exact.
Your function is no bother to me and i can discuss it to your heart's content, it was simply that it doesn't help to be unspecific and divert the discussion. It's a waste of time.
Yes, your little function is very sexy, but...
If you are willing to say the derivative is a generalisation, then it's not, it's a formulation of exact mathematics and has been written as a fraction for a very specific reason.
The f(x)=x(squared) is very simple to examine around x=0.
The tangent is the x axis.

 

[Hurkyl]

"Is dy/dx a fraction?" is answered by "Is dy/dx the tangent gradient?"

But you've changed the question! It dawns on me that I should do a translation of this into pure geometry:

"Is the tangent line a secant line?" is answered by "is it a secant of itself?"​

This is a rather silly objection, don't you think? But that's exactly how you're responding to "Is dy/dx a fraction?".

The other two ways of phrasing it would be

1. \left.\frac{df}{dx}\right|_{x=a} is a difference quotient of f(a) + f'(a) (x-a)​

and

2. The tangent line at (a,f(a)) is the secant to the graph of f(a) + f'(a) (x-a).​

All of these are factually correct statements, but have nothing to do with what is meant by the question "is dy/dx a fraction?" or the question "is the tangent line a secant?"

 

[Ja4Coltrane]

I didn't make a a "personal theory." That's why I put little stars around the word "notation" in my post. I was stressing that thinking of it as a "sort of generalization of a tangent line" was only a heuristic and not a mathematical idea.

Didn't this whole thing start because you wanted to say that a derivative is sort of like a fraction? (Actually, your first post said derivatives ARE fractions).

 

[chrisr999]

i didnt say anything of the sort JA4,
as you continually misquote me that's the end of our conversation,
i only have time to consider something that's been thought through clearly, sorry,
or a genuine exploration,
sorry we have to draw a line somewhere.

No Hurkyl,
that is not how i was responding.
what is a secant and what is a tangent.
a secant cuts through two points of a curve.
the tangent skims across one.
this is how we can differentiate between them.
that is clear geometrically and the mathematics follows through on it.
there's no change in the question, just a clarification, an exploration to check that the definition is clear first of what a tangent and secant is.
the tangent is "the" line whose gradient gives the instantaneous rate of change at a particular single point of the curve.
the secant is the line (of which there are countless) that starts the mathematics going,
it also touches the point of tangency, but it touches another point also.
the secant gets the ball rolling and it's gradient gives us a false reading for the derivative,
in eliminating the error introduced by the secant (mathematically with all the clever techniques), we end up with the tangent gradient, the exact derivative.

Initially, we have a curve for which we want the instantaneous rate of change dy/dx.
it is given by the tangent gradient at the point of interest.
we can write the secant gradient from the function equation but not the tangent gradient.
This unfortunately introduces error.
The error is eliminated through mathematical techniques.
When it is, you have the tangent gradient, the exact value of dy/dx.
the function does not come with a secant and tangent, we use them as tools to get a geometric understanding of the entire situation from which we can go to any mathematical complexity for highly involved calculus.

All of this pertains to functions. When all of that is clear, then what dy/dx is when it comes to non-functions and so on will not introduce ambiguous complexity to the student.

 

[AUMathTutor]

Wow. After having read this thread in its entirety, I can seriously say that I feel substantially stupider for having read it. I condemn everybody who posted to it. Shame on you all.

P.S. If you want to do math with infinitesimals, you should major in physics. Every physics professor I had (a) held mathematics and those who practiced it in general contempt and (b) treated everything as a differential all the time.

 

[Ja4Coltrane]

Yeah, AUMathTutor, you're basically a savior. I am embarrassed for posting here and I accept your condemnation.

I feel so bad for the original poster. His innocent little question should not have led to the least clear, least productive discussion in Physics Forums' history.

 

[Dunkle]

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

I think I found an answer: http://en.wikipedia.org/wiki/Troll_(Internet)
Although chrisr999 may not intentionally fit this definition, he indubitably does.

 

[snipez90]

Wow. After having read this thread in its entirety, I can seriously say that I feel substantially stupider for having read it. I condemn everybody who posted to it. Shame on you all.

Huh what no, I think I made a valid point somewhere in this thread. I actually forget now, so whatever I guess? Anyways it was tempting to join the discussion but I had to keep reminding myself that I knew what a derivative was and I was pretty sure some of the best explanations took less than 2 pages of a textbook. And for the record, I prefer to think of the derivative as conceptualizing the general notion of the rate of change of a function with respect to a variable. I found some of the geometrical arguments pretty annoying. There is nothing difficult about the notion of the limit of "secant lines". Either way, I'll stick to the precise definitions, since calculus is kind of junk compared to functional analysis.

 

[Phrak]

Umm. I guess someone has to point it out. It was nicely subtle, snipez.

AUMathTutor was being colorful--saying the opposite of what he meant. I'm impressed as well by all of your posts. And I thank you all, as well, guys.

 

[snipez90]

Are you sure you're not being colorful? Actually, in retrospect, I was not the least bit annoyed by any of chrisr999's pedagogy.

 

[Hurkyl]

No Hurkyl,
that is not how i was responding.

Yes, yes it is. If you want to justify "dy/dx is a fraction" by computing difference quotients on the tangent line instead of on the curve itself, then I get to justify "the tangent line is a secant line" by computing secant lines of the tangent line.

 

[chrisr999]

This is a long post, and to be honest it's worth it,
it doesn't matter that it is a very small topic,
atoms are small but not insignificant and look at the trouble there has been about these in terms of nuclear energy and weapons and so on, God help us!
at least this is safe.

of course there will be all kinds of input from all kinds of characters, some not polite but what the heck, if something's worth following, it can be followed to the end.

 

[chrisr999]

Can you imagine finding the geometry annoying!
why is it so difficult to recognise that the math in this case is expressing the geometry in symbolic terms?
Annoyance is an intolerance.
No one is forcing anyone to change their point of view or accept anything they are not prepared to.
When you put things simply that a little kid would understand, it's amazing what responses you can get.

Try some examples.
Differentiate y=(sqrt)(9-x(squared)).
Is your answer recogniseable due to a circle having two tangents for each x or y except at two particular points. Maybe it's better to write the maths using an angle for the variable.

Another one... differentiate x(squared) + y(squared)=0.
why are you dealing with complex numbers?
does it have anything to do with trying to draw a tangent to a circle with a radius of zero?

i don't know lads, you can only flog a dead horse for so long, or can you?

 

[chrisr999]

I might come back to this, but I'm not sure,
to be honest, there's no benefit in operating at the level of some of those last few posts,
this stuff is really simple, basic.
you see the same stuff over and over, there are guys that seem totally intent on making something simple totally confusing for a student trying to learn,
i don't know why they get involved if they can't be contributive.
?

enjoy the pub, lads.

 

[Hurkyl]

Well, I've given this thread far more chances than it deserves, and now it's even degraded to personal attacks with a smattering of political ideology -- there is no point in letting this continue.