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Calculus for mathematicians (1997) [pdf]

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Re: Calculus for mathematicians (1997) [pdf]

#41

Post by adamtj »

Tloewald wrote: post #35
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…

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 using them, you don't have to say things like "all 'x' such that ...". Which method you prefer probably depends on what you're more familiar with and how you tend to think. Open balls can be easier to visualize, if that's how you think.

Note that not just any ball will do. Closed balls are open balls that also include their boundary. That is, they use a less-than-or-equal-to instead of less-than. Which you use can make a big difference. An open ball on the real number line is just an open interval, an interval excluding its endpoints. It's easy to generalize: an open ball in a Cartesian plane is a circle excluding its border. In three dimensions, it's a sphere excluding its surface . . . and that's why it's called a ball.

Re: Calculus for mathematicians (1997) [pdf]

#42

Post by j2kun »

As much as I respect djb, the title is a huge misnomer. If you're a mathematician you want analysis and topology. You want definitions of continuity and differentiability that generalize nicely to arbitrary dimension, to manifolds, to Lie groups, to whatever. You want to marvel at the connections between certain special integrals and infinite sums, and you want to see the full construction of the real numbers for its own sake. You want a list of equivalent definitions of differentiability so you can get a better intuition and use whichever one is the nicest for what you're trying to do.

In a very strong sense, writing a document that "focuses purely on calculus" is antithetical to mathematics.

Re: Calculus for mathematicians (1997) [pdf]

#43

Post by blahblah3 »

I remember the "concept" part of my high school calculus class. The teacher had us recite out loud the epsilon-delta definition of continuity until everybody memorized it so they could regurgitate it on the test.

I always wondered why analogies and pictures weren't used more often:

examples:

A 100m sprint is a continuous function of time (f(t) = distance from starting line) because sprinters cant teleport. In fact it is uniformly continuous because people have a maximum speed.

Beating usain bolt's record is a discontinuous function of completion time because f(world_record + epsilon) = 0 while f(world_record - epsilon) = 1.

Fundamental theorem of calculus: If you want to know how fast a guy is running at time t, look at how much ground he covered in 1 second. To get more and more accurate, look at how far he traveled in 0.5 seconds and so on...

Re: Calculus for mathematicians (1997) [pdf]

#44

Post by AnimalMuppet »

threatofrain wrote: post #40
tarentel wrote: post #33

Earlier quoted context omitted.

Ratios in a right triangle perhaps?

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 kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.

Re: Calculus for mathematicians (1997) [pdf]

#45

Post by replicant »

threatofrain wrote: post #40
tarentel wrote: post #33

Earlier quoted context omitted.

Ratios in a right triangle perhaps?

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 could show that this ratio was a constant.

I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...

Re: Calculus for mathematicians (1997) [pdf]

#46

Post by mhartl »

anon4 wrote: post #9
hudibras wrote: post #4

Definitely stashing this away in my time machine for when I travel back to the 17th century. Make both Leibniz and Newton cry...

If you're taking things like that back in time, make sure to translate them to French. Also, learn French. Not knowing French in the 17th century is like not knowing English today.

Did Newton speak French? I'd guess not (or not very well). His principal scientific works were written in Latin, which was the academic lingua franca of the day.

Re: Calculus for mathematicians (1997) [pdf]

#47

Post by tokenrove »

impendia wrote: post #29
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…

> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications. As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich exa…

I don't know, I agree with the parent. I hated trig in my first encounters with it; the thing that made it practical was writing videogames and graphics demos, and the thing that made it interesting was calculus. For many people, it's the last math they're taught, they don't get the applications, and it's no wonder they don't hunger for more.

Re: Calculus for mathematicians (1997) [pdf]

#48

Post by btilly »

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…

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 way that makes sense for all angles.

In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps.

But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that.

(Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.)

(I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)

Re: Calculus for mathematicians (1997) [pdf]

#49

Post by Someone »

threatofrain wrote: post #40
tarentel wrote: post #33

Earlier quoted context omitted.

Ratios in a right triangle perhaps?

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…

"So, what applications are students taught in typical trigonometric texts?"

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?" (http://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Py...)

Well, maybe not that typical, but it is an example without any periodicity in sight.

Re: Calculus for mathematicians (1997) [pdf]

#50

Post by AnimalMuppet »

btilly wrote: post #48
AnimalMuppet wrote: post #44

Earlier quoted context omitted.

> 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…

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 and cos for an arbitrary angle to any desired degree of precision... if you have the patience. It will be shorter to use the series derived from calculus, I'll admit.
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