User:SamHB/new talk page

This is the talk page from my second life, January 2010 to April 2010. Many links are of course incorrect. The "Archive" was a link to the "old" talk page.

Archive 1

MVCalc Course Structure

I was thinking of working on lecture 3 tonight and I noticed some oddities in the order of topics. For example, I had planned to cover tangent planes in a lecture or two after the gradient and extrema? I don't know what I was thinking, but I just wanted to say, if you find you would like to refer in a lecture piece to info not yet presented, or if you think one topic naturally flows into another which is for some reason in a different lecture or something, please, feel free to switch up the course order. It may well be necessary, actually! Just remember to change the order both on the pages and on the outline given in lecture 1. JacobB 01:07, 7 January 2010 (EST)

PS: have you considered archiving your talk page? just a suggestion JacobB 01:08, 7 January 2010 (EST)

Earlier you asked about arching your page. I do it just by cutting and pasting everything to User talk:SamHB/Archive 1, and then putting at the top of my talk page For older discussions, see the archives:<1>. JacobB 18:20, 7 January 2010 (EST)

Done. Great minds think alike. I had believed there was some complicated procedure involving renaming files, that one had to do in order to get the history correct. But I looked around at how existing sysops do it, and they don't bother with any of that. Good enough for me. SamHB 19:55, 7 January 2010 (EST)

A rambling philosophical comment: While reading your wave equation stuff, I was appalled at the way you were approaching it, fiddling around (get it?) with the physics, in advance of doing the mathematics. Then I had an epiphany: You and I have very different ways of approaching these problems. We are each very good at what we do. You approach it in terms of "What is the pedagogically right way to present the material in a sequence of lectures, in a fixed order, with quizzes and exercises and such?" I approach it in terms of "What is the pedagogically right way to present the material in terms of separate pages, that the student can follow in whatever order they want, by clicking whatever hyperlinks interest them?" I recently sort of messed up one of your lectures by moving around the concepts of "vector functions" vs. "vector fields". Feel free to move it back. I will put explanation of these concepts in the separate pages, and defer to you on the order in the lectures. Meanwhile, I need to improve the wave equation page, explaining what sorts of mathematical functions satisfy the equation, and then showing how it arises in a large number of physical problems.

About your specific questions about lecture 3, I need to look more closely at your overall order. You seem to have a more "geometric" approach rather than a "pure math" approach. For example, I would be inclined to cover local maxima/minima in arbitrary dimensions early on, just after talking about partial derivatives, so that all that remains in "extrema" problems is the issue of checking the boundaries. What is the correct order of gradients, directional derivatives, and parametric surfaces? Hard to say. I'd have to write up a draft of these topics first. Maybe I'll do that, but not now. I'm too psyched about the wave equation at the moment.

I hope at least some of this makes sense. SamHB 22:59, 7 January 2010 (EST)

I think you're absolutely right about our styles, and I don't want you to defer to me, at all! I think we can create a blend of styles which will yield the best of both worlds. I like what you did with the vector functions/fields, no need to change it back! I'm going to peruse our articles for a bit now, and I'll get back to you in a bit.

BTW, wave equation wasn't done by a long shot. I hadn't talked about boundary conditions, damping effects, etc., let alone solutions. Feel free to add; just know that my version wasn't what I considered a finished product. JacobB 00:19, 8 January 2010 (EST)

Jacob:

As usual, I've put my foot in my mouth. There is nothing wrong with your wave equation section. It just had a style different from mine, that led to the insight of why our styles are different. I didn't even read it all the way through. I just looked at the first few sentences and extrapolated the trajectory from the initial velocity vector. Of course we will check each other's work very carefully when the time is right.

I've gone through the recent email from you and Andy, and noticed that Andy unblocked me largely for the purpose of assisting with the "calc 3" class. Therefore it would be kind of unsociable for me to run off and only write whatever individual pages strike my fancy at any given instant. So here's my first serious comment, based on your outline. But it's on another page. It seems to me that discussing this stuff here, rather than on the actual course talk page seems a little cabal-like and antisocial. So, having gotten personal stuff out of the way, I'm going to continue on the other page.

SamHB 22:11, 8 January 2010 (EST)

image uploaded

the image you emailed me is available here. JacobB 17:59, 9 January 2010 (EST)

course structure and existence proofs

As promised, I'm doing a whole bunch of math material tonight, but I wanted your thoughts on something, and there's also a favor I'd like to ask.

First up, as part of the restructuring we discussed to cover more PDE material, I'm thinking of combining all the integral definitions and calculations into a single lecture, and then the integral changes under coordinate transforms + applications into another lecture. That'll free up one lecture for some more PDE stuff. What do you think of this? I may or may not have done it by the end of tonight, so you can check it out on the lecture 1 page and see what you think.

Second, I'm just going to state the conditions under which the integrals exist, and that they can be evaluated as iterations of single integrals yadda yadda yadda. Would you like to make some existence proofs and link to them from the lectures? JacobB 23:26, 12 January 2010 (EST)

I reorganized the course a little bit, freeing up lectures 7&8. I'm also thinking of deleting lecture 4 and moving a discussion of velocity and acceleration earlier in the course, and skipping completely or downsizing the intended coverage of Frenet frame and curvature.

So, as of right now, that's two lectures freed up, and possibly another, which is tons of space for the extra PDE stuff you wanted to add (which I think is a great idea!). The course structure STILL needs work - any further idea you have would be welcome. JacobB 04:27, 13 January 2010 (EST)

Well, I seem to be falling farther and farther behind your ambitious pace of writing. I really don't have a lot of time to devote to this, so I need to make the most of my time. I haven't even had time to do more than a cursory reading of the CLEP website that you gave.

My particular areas of interest are integration in its various forms, and curvilinear coordinate systems. That means, for example, the various forms of Stokes' theorem, and line integrals / surface integrals, etc. All this stuff is hard to cover properly; my bookshelves are filled (exaggeration) with books that do it badly, and don't really explain what a "differential form" is, or what it means to integrate it. Doing this really right (e.g. Spivack Calculus on Manifolds) requires some mathematical machinery (exterior forms, or, if you like, alternating covariant tensor fields) that goes well beyond what is appropriate here. But I'd like to give it an intuitive but not mathematically rigorous treatment.

The starting point should be the change of variable theorem. Students already know the basics of that from Calc 1 and Calc 2, of course. But we can take it to the next step, showing how it ties together the material on parametric curves/surfaces, and integrals over same. How? You change to the coordinate system of the parameter(s), so that the integral becomes a standard (Cartesian) Riemann integral. Of course we know that, but we need to make it central to the presentation.

About theorems and "rigor": Of course I like my math rigorous, but there's one area of math (well, maybe more than one, but humor me) where the "Oh, I see how that's true" insight is very useful in advance of all the rigorous "Let S be an open subset of a smooth submanifold of ..." theorems. And that area just happens to be the area we are in. So, for example, the geometric insight of the 2-dimensional Stokes' theorem is useful, and the rigorous proof is just unnecessary tedium. (They'll see it again if they major in math.)

So I'd like to see:

• Review of integration, change-of-variable theorem.
• Integration over 2-dimensional regions to find, for example, areas. Of course, plain 1-dimensional integration can find the area of a region bounded by a function, the x-axis, and 2 vertical lines, but we want to break out of that.
  • Use of the change-of-variable theorem to deal with such integrals.
• Path integration, with the change-of-variable theorem changing to a coordinate system for which this is natural, and the connection with parametric curves and surfaces. (A parametric curve of surface is just a coordinate change that makes the curve or surface trivial.)
• Continue to higher dimensions.
• Easy proof of Stokes' theorem (well, not easy, and maybe not a rigorous proof) by choosing a coordinate system that "flattens whatever the curvy surface was.
• Similarly for divergence and Green's theorem; show that Green's theorem is just a trivial case of Stokes' theorem, and the change of coordinate system converts the latter into the former.

I find that the pages on this stuff (not your lectures, the article pages) are scattered all over the place, with Iterated integral, and Double integral, and Line integral and Surface integral and Vector integration. They would benefit from being consolidated into a smaller number of more comprehensive pages.

Other subjects I'm interested in are Laplace's equation, the physical significance, and simple applications of separation of variables (Fourier series over a circular membrane) to solve it. This is probably way too ambitious. Please stop me from doing it. :-)

About your question about proofs, what do you have in mind for existence proofs? Do mean prove Fubini's theorem? And the conditions under which a multiple integral, in which the individual slices all converge, doesn't converge? That theorem, if I recall correctly, is very hairy. And those conditions are mostly of interest to upperclass topology and analysis majors. Do I misremember? If you say that I do, and you really want those proofs, I will read up on them. On the other hand, a sort of intuitive "hand-wavey" not-really rigorous presentation is something I'm all in favor of.

SamHB 20:57, 13 January 2010 (EST)

I'd like a proof of some basic conditions under which a Riemann integral is defined. We don't need to cover all cases in which it is defined, and we don't need to explore more advanced forms of integration, but right now we have nothing.

While I'm all in favor of a brief discussion of Laplace's equation, using Fourier series at this level is probably too advanced.

Working on the various special cases of Stokes theorem is great, and I encourage you to. We have some unassigned lectures, so feel free to explore those areas in those open lectures. JacobB 17:05, 14 January 2010 (EST)

OK, that's much easier than Fubini's theorem. I believe the official statement is that is is defined if the set of discontinuities has measure zero and the integral is finite. Or we could dispense with the finiteness and say "Yeah, it's defined, but it's defined to be infinite." That's better than saying "I have no idea." In Lebesgue integration theory, I believe that it is considered to be nonintegrable if the integral is infinite; I've never liked that. But never mind. I'll look up stuff and try (for once) not to re-invent the wheel this time. Anyway, more to the point, for our purposes we can use a weaker condition: the set of discontinuities is finite. Getting the students involved in Lebesgue measure is probably not what we want to do. And, of course, we are doing the Riemann integral, not the Lebesgue integral. This stuff (showing that the limit exists) will belong on the Riemann integral page. It will be necessary to get that unlocked. SamHB 23:04, 14 January 2010 (EST)

We also need some material on coordinate transforms, which isn't accounted for at all in the current course structure. Care to help me with that? Also, Riemann Integral has been unlocked by TK. JacobB 20:52, 16 January 2010 (EST)

riemann sums.gif

Gifs often don't work very well when we try to resize them, I'm not sure why. I've uploaded a jpg version of the file here. JacobB 10:46, 19 January 2010 (EST)

More about course structure

A few thoughts -- I don't think we should try to do exterior derivatives; that's just too advanced. That is, I can't picture having one course go all the way from talking about polar coordinates up to doing exterior derivatives. Of course, we need it for the general Stokes' theorem, so we may need to do some hand-waving. I'll look at the Stokes' article.

Should we be keeping any of the exercises/problems secret for use in a final exam? I thought of a very doable but interesting (read: diabolical if you haven't followed the lecture!) problem for integration of parametric surfaces.

I've moved the stuff earlier. Lecture 2 is now bloated, while some others are emaciated.

SamHB 22:22, 22 January 2010 (EST)

my recent edits

See your email - my removal of this content was not a commentary on the quality of your contributions at all, but just trying to keep Conservapedia on-mission. JacobB 21:45, 24 January 2010 (EST)

great work!

nice stuff on coordinate changes JacobB 23:36, 3 February 2010 (EST)

Wow!

Wow, terrific effort!--Andy Schlafly 23:19, 7 February 2010 (EST)

some help?

I'm trying to formulate a parametrization of one side of a hyperbola ${\vec {r}}(t)$ so that ${\vec {r}}\ ''=a{\vec {r}}\left|{\vec {r}}\ \right|^{-3}$ for a constant $a\$. I've been doing all kinds of research on hyperbolic trajectories, but can't find what I'm looking for. JacobB 03:31, 10 February 2010 (EST)

You haven't been shooting alpha particles at gold foil lately, have you? :-)

It looks as though you want the actual trajectory, parameterized by time, of a particle in an inverse square repulsive field. That is, x(t) and y(t) in closed form. You may be out of luck on that, though I can't say for sure.

This is, of course the famous Kepler problem, which has the famous and elegant conic section solution for r in terms of theta. Let $\mu$ be the attractive/repulsive acceleration, so that

$\mu =GM\,$ in the gravitational case, or $\mu ={\frac {Q_{1}Q_{2}}{4\pi \epsilon _{0}}}$ in the electrostatic case. And L is the angular momentum divided by the mass. ($\mu$ is part of the problem statement; L and e are constants of the integration.)

For the attractive case,

$r={\frac {L^{2}}{\mu (1+e\cos \theta )}}$

${\frac {d\theta }{dt}}={\frac {L}{r^{2}}}$

${\frac {dr}{dt}}={\frac {\mu e\sin \theta }{L}}$

which is the neat and elegant conic that we know and love. But it doesn't track the actual passage of time.

For the repulsive case, I'm still going to have $\mu >0$, so the solution is

$r={\frac {L^{2}}{\mu (-1-e\cos \theta )}}$

${\frac {d\theta }{dt}}={\frac {L}{r^{2}}}$

${\frac {dr}{dt}}={\frac {-\mu e\sin \theta }{L}}$

How to get the time dependence? Warning: I haven't checked this stuff personally yet.

According to wikipedia, we can set

$a={\frac {L^{2}}{\mu (e^{2}-1)}}$

$b={\frac {L^{2}}{\mu {\sqrt {e^{2}-1}}}}$

Now introduce a new parameter E (why the heck did they call it that?) in place of t, and we can get x and y in closed form:

$x=a(e-\cosh E)\,$

$y=b\sinh E\,$

${\frac {dx}{dt}}=-a\sinh E{\frac {dE}{dt}}$

${\frac {dy}{dt}}=b\cosh E{\frac {dE}{dt}}$

$t=a{\sqrt {\frac {a}{\mu }}}(e\sinh E-E)$

The first two of those equations get x and y in closed form as functions of E, so we need E as a function of t. The last equation gives t as a function of E. But I don't think that can be inverted in closed form!

SamHB 21:16, 10 February 2010 (EST)

glad to see you're on

can you see Calc3.10? i think i wrote this WAY too advanced, but i also feel like fourier analysis is really the only way to understand what's happening there. any thoughts? JacobB 23:15, 17 February 2010 (EST)

Yeah. I saw what you wrote a couple of days ago (how's that for a dangling participle? :-) Too advanced. I'm not familiar with this particular technique, and will have to brush up on it. But: Delta? Is that the Dirac delta function? Or some representation of a kernel? They won't follow it. And the subscript *. Fourier transform? I'm not familiar with that notation.

The way I would present it, and Laplace's equation too, is to show some simple examples of solutions. You and I know that as the first few eigenfunctions. The student can't learn how to solve PDE's until they have seen some examples of solutions. PDE's are way too hard otherwise. (It's like teaching integral calculus. We start by showing some lucky guesses -- "Hey, the sine function has the derivative that we want, let's talk about it for a bit." -- and then we get down to techniques for solving them for real.

This gets into the method of separation of variables. It's simplest form is with Laplace's equation on a disk. (I know much more about Laplace than the heat equation, so maybe what I'm saying doesn't apply to the heat equation.) We work out the standard solutions by guessing. They are of the form r^k cos (2\pi\k (\theta +M)) or something like that. We show that those work. And that any linear combination works. We're almost there! If the problem is to find the equilibrium temperature everywhere, given the temperature at the periphery, we just have to figure out how to represent that as sines and cosines. I'll be darned! Fourier series will do it!

Then we show how separation of variables works in general -- "We look for solutions of the form X(x) Y(y), such that they are each subject to their own differential equation.

But it's probably still too complicated.

SamHB 00:10, 18 February 2010 (EST)

Agreed. We may need to consider not including this material in the course at all. On a seperate note, I was hoping you might be able to add some exercises for your Jacobian stuff in Lec. 2? JacobB 22:09, 25 February 2010 (EST)

Yes. I'm working on putting together the presentation of vectors in curvilinear coordinates. In case you hadn't noticed, I'm using a somewhat nontraditional definition of basis vectors -- they aren't normalized. The formulas for dot, cross, div, and curl are more natural, and they can handle any coordinate system, not just orthogonal ones. Still needs more work. A lot more. And I can't put in as much time on this as I would like. I will work out exercises for them.

By the way, I thought of a really cool problem. "You are designing a universe. You want Maxwell's equations to be true, because they are so elegant. You want the electric field around a point charge to be a radial vector, proportional to the radius raised to some power. What does that power have to be, so that the divergence will be zero?" The answer is, of course, -2. And, when things are expressed in spherical coordinates and the correct divergence formula is used, it is very easy.

SamHB 23:53, 25 February 2010 (EST)

3.3

Can you take your line/surface integral stuff from lec 3.3 and merge it with the material on these topics presented in 3.5? JacobB 17:00, 27 February 2010 (EST)

OK. But not right now. I finally put together the divergence/curl stuff, so a huge delivery is about to happen. It may be that, in addition to your request, the div/curl stuff will need to move to a later section. SamHB 23:07, 27 February 2010 (EST)

Done. It's all in a pretty messed-up state, but at least it's in one place. And the right place--lecture 5. There's nothing about integration in earlier lectures. Well, a little bit--the arc length discussion requires an integral, but it's a completely vanilla integral. Lecture 5 is where the fancy-shmancy integrals on manifolds occur.

In fact, a thing to think about is the relationship between the lecture 4 material--tangents, binormals, torsion, etc. (that is, the stuff that you are interested in :-) and the generalized coordinate/manifold stuff (the stuff I'm interested in :-). They are both part of multivariable calculus, but sort of at opposite poles of the subject. Maybe we could think about switching lecture 3 and lecture 4. Not saying we should; I haven't thought about in any detail. But it's a possibility.

Forget that. I've looked again. The lectures are in the right order. SamHB 22:54, 1 March 2010 (EST)

Hope this makes at least some sense.

SamHB 22:51, 1 March 2010 (EST)

New Musical Examples up at Mannheim School

Enjoy! JDWpianist 15:02, 5 March 2010 (EST)