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Calculus for mathematicians (1997) [pdf]

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Re: Calculus for mathematicians (1997) [pdf]

#51

Post by threatofrain »

replicant wrote: post #45
threatofrain wrote: post #40

Earlier quoted context omitted.

How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio? If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they…

I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it? When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we co…

I think I wasn't clear in saying that I believe the current pedagogical value of trigonometry is in giving students a brief familiarity with the trig functions when they see it again in the context of physics or engineering. Or standardized testing. I think those are the likely scenarios where students are going to be seeing relevance in trigonometry.

What other foundation or learning pathway do you see trigonometry serving as? Somebody else mentioned that it gives students a sense of applications, so they know that calculus is not for nothing. So then I question: what applications? And I pose, how about statistics?

Re: Calculus for mathematicians (1997) [pdf]

#52

Post by stephencanon »

rndn wrote: post #36
bandrami wrote: post #11

The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…

> An integral is the inner product of a function and a suitably-dimensioned unit. Does that mean that ∫f(x)dx is f.(dx, dx, …) = f(x_0)·dx + f(x_1)·dx + f(x_2)·dx + … for all x in the domain?

Integral[f(x)g(x)dx] defines an inner product on a space of functions (glossing over exactly what functions) on an interval. So Integral[f(x) dx] is the inner product (f, 1), where "1" is the constant function g(x) = 1 on the interval.

Re: Calculus for mathematicians (1997) [pdf]

#53

Post by jerf »

bandrami wrote: post #11

The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…

"(Side rant: why do they teach trig before calculus in high school? That's completely backwards. Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.)"

I'm fairly convinced the answer is that That Is What The Curriculum Does, and Do Not Question It.

None of the subsequent 9 hours of debate since you posted this convinces me otherwise. It's not possible that there's a better way, and we can marshal all sorts of rationalizations about how this is the best way, and the possibility that it might not be simply can not be conceived. If you think that there might be a better way, you just must not be aware of how what we already have is perfect.

Our math curriculum could be significantly improved in many ways, except that this is the general societal attitude towards it that I see, and it turns out that "Fix the math curriculum" becomes an unsolvable problem when you add the constraint "But don't make any changes to it of any kind, not even to merely reorder a few topics". Feh.

Re: Calculus for mathematicians (1997) [pdf]

#54

Post by threatofrain »

AnimalMuppet wrote: post #44
threatofrain wrote: post #40

Earlier quoted context omitted.

How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio? If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they…

> How did you, in a non-magical way, go from information about an angle to information about a ratio? By having a right triangle? The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles. Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kind…

What kind of pedagogical or pragmatic relevance do you see trig as a building block for? I would answer that question by saying that it most likely comes up again either in physics or engineering contexts, or in a standardized exam like MCAT. And only in the sense of familiarity with the unit circle and trig functions.

What other foundation or learning pathway do you see trig serving as? Somebody else mentioned that trig serves use by teaching students that calculus has rich applications. So then I question, what kind of applications are students learning in trig? And if students are to learn rich examples of calculus applications, then why not statistics, which is also relevant to the bio / social sciences? Also, couldn't we mash trig inside calculus?

Re: Calculus for mathematicians (1997) [pdf]

#55

Post by dragonwriter »

bandrami wrote: post #11

The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…

> Side rant: why do they teach trig before calculus in high school? That's completely backwards. Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.

Math is hardly the only place where arbitrary facts/formulas are taught, and people taught to apply them, before learning the underlying math/reasoning behind the arbitrary facts/formulas.

And trig is useful in lots of places in the science curriculum without the backing calculus, so teaching it in the math curriculum early to support the broader curriculum makes sense from that perspective.

Re: Calculus for mathematicians (1997) [pdf]

#57

Post by btilly »

AnimalMuppet wrote: post #50
btilly wrote: post #48

Earlier quoted context omitted.

The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a wa…

Well... you can use the half-angle formulas and the angle addition formulas to calculate sin and cos for angles that are arbitrarily close to the ones that you want. Add to that the idea that sin and cos must be continuous (I consider that intuitively obvious from a unit circle, but I don't know how to make that argument rigorous), and you can start to interpolate. You can in fact use these methods to calculate sin a…

I do not believe that there is an argument for continuity without starting with Calculus. Certainly starting from ruler and compass constructions it is not obvious.

That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on.

It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.

Re: Calculus for mathematicians (1997) [pdf]

#58

Post by bumbledraven »

ww2 wrote: post #56

Definition 5.1/5.2 is interesting. It defines derivative at point c, not the derivative function of f. Note that f1(x) is not equal to f'(x) for all x, but f1(c) = f'(c).

Yeah. The paper says: The derivative of f at c is written f'(c). The derivative of f, written f', is the function c -> f'(c).

So the derivative (f') is the result of substituting c for x in f1. For example, if f1 = (x -> x + c) then we would have f' = (c -> c + c) = (c -> 2c).

Re: Calculus for mathematicians (1997) [pdf]

#59

Post by mafribe »

tel wrote: post #21
mafribe wrote: post #20

Earlier quoted context omitted.

I'm not familiar with constructive Cauchy sequences. The usual way of using Cauchy sequences is to quotient them by the ideal of Cauchy sequences that converge to 0.

I've seen an impl somewhere where the (int => rat) bit from the Isabelle/HOL design was (stream rat). That's about the best I have at the moment.

BTW, in what sense is that coinductive? For what functor is it a final coalgebra?

Re: Calculus for mathematicians (1997) [pdf]

#60

Post by Tloewald »

adamtj wrote: post #41
Tloewald wrote: post #35

Earlier quoted context omitted.

Trig is the math of triangles and circles — it can be fully understood in geometric terms. Calculus requires far more foundation. The linked article is interesting but the definition of differentiability looks wrong to me — maybe my brain needs more coffee but it looks like only linear functions are differentiable as defined. I learned my calculus the pure math way — axioms and analysis. Epsilon delta arguments make…

You can find your epsilons and deltas in the definition of an open ball. They are just two different ways of writing the same idea. For example, in Definition 2.1, the 'h' in the open ball F is your epsilon and the 'h' in B is the delta. In an epsilon-delta proof, you show that for all |x - c| Any epsilon or delta that you choose implies a set of numbers satisfying those conditions. Those sets are open balls. By usin…

I understand what an open ball is (and surely if you read my original post carefully this would be clear), I just don't think it actually makes the discussion clearer. In particular, when explaining something to people, starting with creating unfamiliar concepts is a Bad Idea if those concepts don't have a significant payoff.

In particular, because this is not a discussion about arbitrary spaces, the use of the word "Ball" is counter-intuitive. (But I admit I am probably biased by my own experience.)

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