You cannot differentiate a geometric figure?

[talk2dream]

I was reading a different post, https://www.physicsforums.com/threads/differntiating-a-circle.279719/#post-2003287 and was doing great... guys, thanks so much for this forum!

HallsofIvy posted "1: You titled this "differentiation of a circle" which makes no sense. You cannot differentiate a geometric figure!"

I'm dearly sorry for needing to ask this. It's been years and years that I'm out of school. The terms are all new for me again, but I should hope that I haven't gotten any stupider with time! Can we explain it to me like a kid? With really easy English?

 

[verty]

I think he meant a circle isn't a function. A function has only one value for each x.

 

[fresh_42]

I was reading a different post, https://www.physicsforums.com/threads/differntiating-a-circle.279719/#post-2003287 and was doing great... guys, thanks so much for this forum!

HallsofIvy posted "1: You titled this "differentiation of a circle" which makes no sense. You cannot differentiate a geometric figure!"

I'm dearly sorry for needing to ask this. It's been years and years that I'm out of school. The terms are all new for me again, but I should hope that I haven't gotten any stupider with time! Can we explain it to me like a kid? With really easy English?

It's meant, that the the function which describes the circle is differentiated. And although a circle cannot be described by a function globally, locally it can. Since differentiation is a local procedure, that's all what is needed. So the differentiation is about the functions, e.g. $x \longmapsto \sqrt{r^2-x^2}$ which gives parts of the circle.

 

[talk2dream]

I think he meant a circle isn't a function. A function has only one value for each x.

Right. Thanks for jogging my memory. Ok... so... next question... theoretically speaking, it is possible but I'd have to cut the circle in half, and do the postive y and negative y values separately. Can we remind me how to differentiate a function f(x,z)=x^2 + z^2, or what the word is for what I'm trying to ask but don't know to call to google at the moment?

 

[fresh_42]

Even if you cut it in half, then the points where you performed the cut will still be a problem. Therefore in mathematics, such a variety of points like a circle is first covered by sets in such a way, that differentiation is possible on every single of such sets and that the results are the same for points which are in two sets.

In case of $f(x)=\sqrt{r^2-x^2}$ we have $\left. \dfrac{d}{dx}\right|_{x=a}f(x)=\left. \dfrac{d}{dx}\right|_{x=a}\sqrt{r^2-x^2}=-\dfrac{a}{\sqrt{r^2-a^2}}$

For differentiation in general and certain functions in detail, you could look up Wikipedia:
https://en.wikipedia.org/wiki/Differential_calculus
https://en.wikipedia.org/wiki/Differentiation_rules

 

[talk2dream]

It's meant, that the the function which describes the circle is differentiated. And although a circle cannot be described by a function globally, locally it can. Since differentiation is a local procedure, that's all what is needed. So the differentiation is about the functions, e.g. $x \longmapsto \sqrt{r^2-x^2}$ which gives parts of the circle.

I'm not understanding something. First, I can't picture what I'm trying to do, and that's just irksome. So I'm trying to understand that process of how to picture it, and that requires me understanding differentiation a bit better. "the function which describes the circle is differentiated" means not that the original function is itself a differentiation, but that the differentiating is happening to the function... but it is possible to consider the original function the differentiated state of its integral, yes? And the integral of a circle is not possible because the circle itself isn't a function, but we can take the integral of two halves to produce something that does represent the whole (or is there something lost in the integration of the parts that prevent them from appearing to represent the one same object that together represent?)

Why can't a circle be described by a function globally, if its global sphere is a local event of a larger complex / unimagined state? Is it just our inability to represent the function, so that it's possible but no one's done it yet?

Still actually stuck on the first question. I'm not seeing the inability. Guidance for how to understand the whole of what I need is appreciated... really just anything that pops to mind that you think could distantly relate, that's probably what I need to hear. Sincerely, however, this is entirely sincere.

 

[talk2dream]

Even if you cut it in half, then the points where you performed the cut will still be a problem. Therefore in mathematics, such a variety of points like a circle is first covered by sets in such a way, that differentiation is possible on every single of such sets and that the results are the same for points which are in two sets.

In case of $f(x)=\sqrt{r^2-x^2}$ we have $\left. \dfrac{d}{dx}\right|_{x=a}f(x)=\left. \dfrac{d}{dx}\right|_{x=a}\sqrt{r^2-x^2}=-\dfrac{a}{\sqrt{r^2-a^2}}$

For differentiation in general and certain functions in detail, you could look up Wikipedia:
https://en.wikipedia.org/wiki/Differential_calculus
https://en.wikipedia.org/wiki/Differentiation_rules

I did read those first.

 

[fresh_42]

We have a circle (black) and a tangent (red) at some point (green).

The idea is, that we can approximate the circle by the linear tangent, since linear functions are easier. It's immediately clear, that this approximation gets worse the farther away we get from the touching point, but locally, it is a good approximation.

What we actually do, is to determine the slope of our tangent at this point, because point and slope together already determine the tangent. In order to calculate this, we need some coordinate system, in which we can express what is done in the picture.

 

[talk2dream]

thanks for that, I forgot that local meant relative distance from the tangent to the curve... the picture explanation definitely helps

 

[talk2dream]

What I'm really trying to do is figure out what was going through Fermat's head when he came up with his last theorem. I tried reading the Wiles proof, and was lost pretty much right in the first paragraph. It didn't read like English at all, and if Fermat was going to be so brilliant to jot down "Here's a theory, I can write a proof, but there's not enough room in the margin", I'm going to assume he had a flash of insight that he could later revisit to elaborate his theorem, and which certainly wouldn't have taken him 129 pages of modern day math to do.

Since he was cited as fathering differential calculus, I figure he was probably playing with concepts learned from basic calc that got him to realize what he saw, and that he could probably explain the proof in five minutes to a kindergarten kid with basic English, or Latin or Greek as the case may be.

So I'm really trying to picture what a^n + b^n = c^n is even able to look like in terms of escalating 'n', and, to do that, I'm trying to use derivatives? I so don't know where to start with this, but I assure you that I'm not the dumbest among us so there has to be a way to get me started!

Maybe throw a few questions at me that I don't know to ask myself? :/

 

[fresh_42]

There's a nice book about it. I wouldn't try to read the proof, unless you're an expert in algebraic geometry, K-theory and modular forms. As Wiles has published his proof, it was said, that at most a handful of people in the world would be able to understand it. Well, this might have changed, but it aren't many still.

What Fermat 's thoughts were? We suspect that he had an idea of how to prove it for $n=3$ and he might had believed, that this could be generalized. Also at this time, mathematicians communicated via letters, and often they wrote each other puzzles in order to annoy the other one; certainly not the standard, but sometimes they did. And Fermat was a big one in doing so. However, the remark in his book has been published posthum IIRC, so we can rule out that he tried to be mean - whereas factually he was in the ultimate sense.

Since he was cited as fathering differential calculus

What? Father of modern number theory, yes, but the fathers of modern Calculus have been Leibniz and Newton - plus the many who came afterwards: Gauß, Cauchy, Lagrange, Weierstraß, Bernoulli, d'Alembert etc. etc.

 

[WWGD]

Right. Thanks for jogging my memory. Ok... so... next question... theoretically speaking, it is possible but I'd have to cut the circle in half, and do the postive y and negative y values separately. Can we remind me how to differentiate a function f(x,z)=x^2 + z^2, or what the word is for what I'm trying to ask but don't know to call to google at the moment?

For the sake of redundancy ( it helps makes things stick, at least to me ) : Differentiaton is an attribute of formulas, not of graphs. Sure, a graph may be the depiction/implementation of a formula, but, to differentiate, you need a formula.

 

[WWGD]

I'm not understanding something. First, I can't picture what I'm trying to do, and that's just irksome. So I'm trying to understand that process of how to picture it, and that requires me understanding differentiation a bit better. "the function which describes the circle is differentiated" means not that the original function is itself a differentiation, but that the differentiating is happening to the function... but it is possible to consider the original function the differentiated state of its integral, yes? And the integral of a circle is not possible because the circle itself isn't a function, but we can take the integral of two halves to produce something that does represent the whole (or is there something lost in the integration of the parts that prevent them from appearing to represent the one same object that together represent?)

Why can't a circle be described by a function globally, if its global sphere is a local event of a larger complex / unimagined state? Is it just our inability to represent the function, so that it's possible but no one's done it yet?

Still actually stuck on the first question. I'm not seeing the inability. Guidance for how to understand the whole of what I need is appreciated... really just anything that pops to mind that you think could distantly relate, that's probably what I need to hear. Sincerely, however, this is entirely sincere.

You may want to differentiate ( ha-ha) between the total derivative and partial derivatives here. So, when you differentiate , obtaining a local linear approximation, it may be done with respect to different variables.
EDIT: I hope these comments are helping. Also, just to warn you about the confusion you may run into between the concepts of derivative and differential.

 

[talk2dream]

You may want to differentiate ( ha-ha) between the total derivative and partial derivatives here. So, when you differentiate , obtaining a local linear approximation, it may be done with respect to different variables.
EDIT: I hope these comments are helping. Also, just to warn you about the confusion you may run into between the concepts of derivative and differential.

I'll be sure to read up on the difference between the concepts of/for derivative and differential. I think I can imagine it. One is specific, the other is for scope. If you could elaborate, that would be helpful of course!

I do stand by my assertion that Fermat would have been sincere in his scribing of the original problem (theorem). The inspiration itself would have been a simple mental picture, moving, moved from a recognized state to produce understanding of something previously unthought. The result is an "Ah ha" that can be summed up in a simple sentence, like a shortest line between two concepts, or, like the Wiles proof, an infinite road traveling through every known station from one recognized concept to another. While Wiles did it, it is certainly agreed that his road isn't nearly the neatest possible nor the most easily understood. There exists a better road to get us to the same conclusion, with a lot less words, a lot fewer newer math concepts, and certainly much more applicable in the true sense of any pan-applicable idea. I'm just after the plain English to describe the rest of it, which must come if, like the quacks who profess "one spirit", or "one origin" for us all ensure must exist and remain equally available for any who truly seek it. So long as the intent remains pure, the answer must present itself in time. (I'm meeting the obscene with a completely ridiculous idea, and putting my own lived history in play to lead the way to it--which is something possibly also new).

Again, elaboration of what you'd thought would be handy for me to know about the difference is greatly appreciated :)

 

[talk2dream]

I am unbelievably thankful to the administrators for this forum. I haven't the learned mathematical vocabulary to more quickly trace a veritable logical route.

I did see a picture of what I believe led Fermat to his inspirational theorem: https://drive.google.com/file/d/1zglreBBEq2TRfUhx1KetXbwGfuhhbW8d/view

To be sure sorces remain cited, I've clipped the picture from: https://drive.google.com/file/d/1R4nDybWsilLVbBzNfWHQLkfANmdXsN4G/view

I also understand this quote from this wiki file, https://drive.google.com/file/d/1g2iiUmBxM1JSQZI3QEtyofcJfz1_0VHT/view

One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle.

To give a better idea of what Fermat had at his disposal, formally describable knowledge-wise, to permit the theorem (and the resulting language base that should be needed to describe the proof more eloquently)

We're collectively working this one out :)