Algebraic Operations on Differentials in Liebniz Notation: An Abuse?

[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.