It's simple. Calculus only applies to changing curvature.

The circle has no derivative.

Mitch Raemsch; Falling light changes colour

The proof is self evident. The circle has no tangent. Its chord would be an infinitely small arc and the derivative doesn't apply there.
A tangent line is a chord of two points infinitely close or more accurately as close as you can calculate with differentiation but it doesn't apply to a circle.


Mitch Raemsch

Last edited by Nicholas (2008-05-05 02:03:44)

The arc of the circle is round and round is unchanging curvature that the calculus cannot define as a derivative.

Only chaning curvature has a derivative.

Mitch Raemsch

I am not arguing against the straight line mr M; just the circle.

It is perfectly round.

The derivative of anything that is geomtrically round does not exist.

I have no delusions that further evidence will incline you to reason, but I think this picture is helpful.

Hmm... for whatever reason, I can't delete the first two images (which are too small to see well).

Last edited by M@Man (2008-05-08 19:15:50)

This whole thread would be humorous (like an  Abbott & Costello routine) were it not such a tail-chasing waste of time.

If read quickly, the question "If the circle is a curve how do we calculate its derivative?" seems reasonable enough. But it's not. In fact, it's meaningless (or at best, ill formed). But the thread immediately ran off along a nonsensical "tangent" (pun obviously intended). Chris almost got it back on course when he asked "BTW, the derivative with respect to what?" Unfortunately, the metaphorical train continued on its "Who's on First?" track.

A derivative is the instantaneous change of one variable relative to another. Perhaps Chris would've had better luck getting things back on course had he asked "BTW: the derivative of what with respect to what?"

The circle is a curve described by an equation involving two variables—in rectangular coordinates, x and y. If you write down the equation of a circle of radius R whose center is located at the point x=a, y=b,



you can then ask the meaningful question, "What is the derivative of y with respect to x?" and quickly determine

.

The value of this derivative at any particular point is indeed the slope of the line tangent to the circle at the point .

Finding a tangent line of a changing curve requires calculus. You must choose two points as close together as you can get to create a chord that is a tangent line. Since you cannot define two points infinitely close to one another the tangent line or chord is just an estimate.

It is the same with finding slope or area under a real world curve. These have no corresponding functions such as binomials. The two principles of calculus are only estimations if they are applied to real world curves.

Differentiation requires two points infinitely close. You may be able to calculate two points very close but not infinitely close therefor your derivative will always be an estimate.

Intergration requires summing infinite amount of infinitely small boxes of area to make an exact calculation. Since infinite approximations are out of question intergration is also just an estimate for a real world curve.

Mitch Raemsch

Nicholas wrote:

Finding a tangent line of a changing curve requires calculus. You must choose two points as close together as you can get to create a chord that is a tangent line. Since you cannot define two points infinitely close to one another the tangent line or chord is just an estimate.

It is the same with finding slope or area under a real world curve. These have no corresponding functions such as binomials. The two principles of calculus are only estimations if they are applied to real world curves.

Differentiation requires two points infinitely close. You may be able to calculate two points very close but not infinitely close therefor your derivative will always be an estimate.

Intergration requires summing infinite amount of infinitely small boxes of area to make an exact calculation. Since infinite approximations are out of question intergration is also just an estimate for a real world curve.

Mitch Raemsch

This is utter nonsense. While it's true that computational methods often are used (e.g., the algorithm a computer uses to "number crunch" a solution) to calculate derivatives that are, strictly speaking, "approximations," every continuous, smooth curve (with some notable exceptions) has a derivative defined at every point along the curve. (Integrability has less-stringent requirements.) If the equation of the curve can be expressed in closed form, its derivative function can be determined in closed form. In those instances where the equation of the curve cannot be expressed in closed form, iterative methods can be used to arrive at a convergent sequence at each point whose limit is indeed the derivative at that point.

I will grant you that there are "approximations" that can't be expressed as rational numbers. But your "broad brush" statements in this thread (as well as in virtually every other thread you've created or contributed to on this site) suggest that either (a) your understanding of the fundamentals is seriously lacking, or (b) you simply enjoy being contrary. I suspect that it's both. In any event, if you continue to behave in this manner, you will succeed in irritating and alienating every member of ilovephysics.com to the point that you will be completely ignored, or even banned.

M@Man wrote:

It's not that I disagree with you, Martin, but I can't help but take offense at the degree of complicity this assigns to me.  I only wanted to have enough factual information on record to assure that anyone reading the thread would not be misled by Nicholas' claims.

No offense intended, Matt, none whatsoever. Quite the contrary, from my perspective, you -- and to a certain degree, Chris as well -- were "victims" of Nicholas, frustrated in your repeated efforts to offer your expertise in an intelligent dialogue.

I concluded a very long time ago that Nicholas is unwilling -- or unable -- to participate in such a dialogue. Consequently, to avoid getting sucked in to an instructional "black hole," I simply don't respond to any of his posts; for the most part, I don't even read them! But the old professor in me does sometimes (rarely, however) get the best of my better judgment, and occasionally I do jump into the fray, hoping against hope that I won't end up feeling like Lou Costello.

My unsolicited advice is to save yourself the aggravation of banging your head against a wall. Reserve your well-intentioned efforts for someone who appreciates (and can benefit from) your valuable assistance.

Martin wrote:

Nicholas wrote:

Finding a tangent line of a changing curve requires calculus. You must choose two points as close together as you can get to create a chord that is a tangent line. Since you cannot define two points infinitely close to one another the tangent line or chord is just an estimate.

It is the same with finding slope or area under a real world curve. These have no corresponding functions such as binomials. The two principles of calculus are only estimations if they are applied to real world curves.

Differentiation requires two points infinitely close. You may be able to calculate two points very close but not infinitely close therefor your derivative will always be an estimate.

Intergration requires summing infinite amount of infinitely small boxes of area to make an exact calculation. Since infinite approximations are out of question intergration is also just an estimate for a real world curve.

Mitch Raemsch

This is utter nonsense. While it's true that computational methods often are used (e.g., the algorithm a computer uses to "number crunch" a solution) to calculate derivatives that are, strictly speaking, "approximations," every continuous, smooth curve (with some notable exceptions) has a derivative defined at every point along the curve.

A zero dimensional point has no slope. It requires two points to find a slope. This is obvious.

Mitch Raemsch

In algebra two points on a line are required to find a slope; the same with calculus and a curve. Since we cannot calculate 2 points infinitely close to each other on a real world curve the calculus is just an approximation there.

Mitch Raemsch