Riemann indexing

[Alex Jordan]

I am light-heartedly complaining that Riemann sum indexing of tick marks for an integral over [a,b] runs from index 1 (for a) to index (n+1) for b.

https://opentext.uleth.ca/apex-video/sec_riemann.html#idea_riemann

Greg, can I sweet talk you into indexing from 0 to n instead?

[Sean Fitzpatrick]

I tried taking Greg into this a year or two ago! 

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[gregory...@gmail.com]

Cash payments will go a lot further than sweet-talking.

I remember (vaguely) thinking about the indexing and acknowledging that I was doing something different than what most/all other texts had. I also thought it made sense that the i^th interval starts at x_i. What's the argument for starting with i=0? Familiarity is a fine argument: all other texts do it, along with similarities to programming in C & Python, etc. (In Mathematica & Matlab, where I spend my time, indexing starts naturally at 1, not 0. So part of why I did what I did was personal familiarity.)

I found the old conversation with Sean about the indexing. (Link below.) My biggest concern then & now is the adjustment faculty will have to make to adapt to the new indexing.

https://groups.google.com/g/apexactive-calculus-mbx-conversion/c/o6yhcBdBbwM/m/4oGkS7QiCgAJ

[Sean Fitzpatrick]

Cash payments and volunteering to track down all the follow-on edits that result?

I find the indexing from 0 easier/more natural -- it's easier to write down some of the expressions.

In particular, I think the main simplification comes from the fact that x_i is then given by adding Delta x i times.

[Alex Jordan]

> My biggest concern then & now is the adjustment faculty will have to make to adapt to the new indexing.

That is important, no doubt.

Sean has a good one:

x_i  =  a + i \Delta x

instead of

x_i  =  a + (i-1) \Delta x

For me it starts with when I draw the picture with these tick marks (when starting from 0):

x_0  x_1  x_2          x_n

and I don't need to put more than one character in a subscript. It's a little thing, but a student can turn x_{n+1} into x_n + 1 and get more confused from there.

It's not a big deal. I am not offering money, nor time spent chasing down consequences of an edit. :)

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[Rob Beezer]

My two cents.

x_0 = a, x_n = b

\Delta x_i = x_i - x_{i-1}

That's how I was raised, and how it has always been. ;-)

Rob

On 4/15/21 4:35 PM, Alex Jordan wrote:
> > My biggest concern then & now is the adjustment faculty will have to make to
> adapt to the new indexing.
>
> That is important, no doubt.
>
> Sean has a good one:
> x_i  =  a + i \Delta x
> instead of
> x_i  =  a + (i-1) \Delta x
>
> For me it starts with when I draw the picture with these tick marks (when
> starting from 0):
> x_0  x_1  x_2          x_n
> and I don't need to put more than one character in a subscript. It's a little
> thing, but a student can turn x_{n+1} into x_n + 1 and get more confused from there.
>
> It's not a big deal. I am not offering money, nor time spent chasing down
> consequences of an edit. :)
>
>
>
>
> On Thu, Apr 15, 2021 at 2:10 PM Sean Fitzpatrick <dsfitz...@gmail.com
> <mailto:dsfitz...@gmail.com>> wrote:
>
> Cash payments and volunteering to track down all the follow-on edits that
> result?
> I find the indexing from 0 easier/more natural -- it's easier to write down
> some of the expressions.
>
> In particular, I think the main simplification comes from the fact that x_i
> is then given by adding Delta x i times.
>
> On Wednesday, April 14, 2021 at 9:10:12 AM UTC-6 gregory...@gmail.com

> <https://groups.google.com/d/msgid/apexactive-calculus-mbx-conversion/CA%2BR-jrdjRv%2Bcaj8o4NGkKxeGuyeK3NbnGH_Yv3tPqUYq_nL22w%40mail.gmail.com?utm_medium=email&utm_source=footer>.

>
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[Alex Jordan]

One reason to consider the change: my students are reporting some
confusion as they look at APEX but then also look at other online
resources. I have not done a survey, but I would guess the other way
is more common. Documenting for posterity, not pestering, I hope.

I wrote out a Riemann sum for the right-endpoint approach in class the
other day, and it was weird that the first index was 2.

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[Sean Fitzpatrick]

I'm willing to take this on this summer.
Maybe we should employ the new versioning technology, and then the
original would be available if someone really wants it?

I'll have to go see what I did with the videos for this chapter.
For some time I resisted starting the index at 1 but last year I gave in,
since students were learning online and I wanted to keep things consistent.

[gregory...@gmail.com]

Yes, students should be able to not be confused by indices when looking at outside sources. (Though ... apex calc is _so good_ who needs outside sources?) 

Sigh.

Ok. Long term, it's probably best to to just make the change and then we/I won't be bugged about it for the next 20 years. 

If Sean is willing to make the changes, great. FWIW, I have planned to revisit the concavity and spherical coordinate issues we raised earlier this summer, too.

[Sean Fitzpatrick]

I might have time to work on this next week. Greg, if you send me a summary of the changes for spherical coordinates and concavity, I can do those as well.

(I could grab the details off of the other threads, but I know we went back and forth on things a bit, so it would be good to have confirmation on the version you want to see.)

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[gregory...@gmail.com]

Sean,

Spherical

Here's a summary from a thread about the spherical. I think this is something we can implement without breaking too much within the text - subsequent sections depend on this minimally, etc. 

<><><><>

I'd much rather rather use what think is "natural" (which is certainly subjective):

Polar:

(r,theta) as currently done. (When drawn on the board, theta is an "angle of elevation".) Would still call this the "polar angle."

Cylindrical:

(r,theta,z)    Using these letters makes sense, and it's a right handed system. Theta is still the "polar angle".

Note: (r,theta,0) ~ (r,theta)

Spherical:

(rho, theta, phi), where theta is the same as in both polar and cylindrical; rho is the radius (but as it doesn't directly embed into polar/cylindrical, we use a different symbol), and phi is *another* angle of elevation, using the "natural" inclination to measure angles "up from the horizontal."

Note: (5, theta, 0) means the same thing in both cylindrical and spherical, and corresponds to (5,theta) in polar.

Theta is still the "polar angle." And we'd have to get a good term for phi. Not azimuth. "Angle of elevation"? "Elevation angle"?

<><><><>

Concavity: 

This one seems harder to me to implement. At the moment, I like Alex's non-calculus definition (below) with an extension of incr/decr and concave up/down to include endpoints. (But allowing WW to accept open intervals.) However, this may break a good many things, so implementation needs to be done with care, imo. I'd prefer to give this a shot, though I don't want to slow progress unnecessrily.

In general, I like the idea of presenting a non-calc definition of incr/decr and concavity, then showing how derivatives provide sufficient conditions for knowing incr/decr and up/down, then spending all of our time using these calc definitions. APEX does this for incr/decr.

(FWIW, Alex's definition uses " > ", not " >= ", which is important. I had forgotten he used ">" and for a while I didn't like his def. much.)

<><><><>

Personally I think of concave up in terms of chords: that the midpoint of a chord within the interval is higher than the function's value at the midpoint of the x-values:  [f(a) + f(b)] / 2  >  f([a + b] / 2) for all a!=b in the interval. Is there any interest in moving to this sort of pre-calculus definition for concave up?

[Sean Fitzpatrick]

Aside from catching a few things I missed on the first pass,
I'm almost done with shifting all the Riemann indexing.

But it now occurs to me to ask a question, about this, and spherical coordinates:

Do we want to preserve the "old version"?
PreTeXt now supports customization, as detailed here:
https://pretextbook.org/doc/guide/html/publisher-customizations.html

I think it wouldn't be too much work to set things up so that we can toggle between versions using the publisher file.

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[Alex Jordan]

For Riemann indexing, I don't see value in preserving the old version.
But then again, I am not Greg who will field the complaints that may
come in.

You are maintaining 3 versions at Lethbridge, is that right? Does that
give you a way to play with the customizations feature to see how you
feel about it? Maybe it would be cool to be able to switch the
spherical coordinate convention being used.

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[Sean Fitzpatrick]

Right now I have 4 versions, sort of.
But I am managing that with different top level files, because they only
really differ in the order of the <xi:include>s.

Versions are:

"Official" APEX Calculus -- this has all of Greg's sections, in the
original order. Also, there are no inline answers to exercises, and only
odd-numbered answers in the back.
Only major difference compared to V4: I included the 'appendix' on
Differential Equations as a chapter.

"Official" APEX Calculus, but with videos. (Inclusion of videos is
controlled using a thin XSL stylesheet.) As an aside: starting later
this week I'm going to resume making videos. I'll do the two chapters
not yet done, then reshoot a few that have audio problems, and maybe
update the Riemann sum videos to have the right indexing.

APEX for U of L "Standard" Calculus: material is ordered according to
our Arts & Science calculus sequence, and there is a division into 4
parts, corresponding to the 4 terms of calculus in this sequence.

APEX for U of L "Accelerated" Calculus: material is ordered according to
our (pre) Engineering calculus sequence, and is divided into 3 parts,
corresponding to a 3 term sequence.

Both U of L versions have extra sections on multivariable calculus that
I added:
- derivative as a matrix transformation (and chain rule as matrix
multiplication)
- Lagrange multipliers
- classification of critical points for 3 or more variables
- general change of variables and Jacobians

There is one figure for a video that I need to decide what to do with.
It's in the section on the chain rule for functions of several
variables, and references the matrix approach.

I'm considering putting this into a <custom> tag so I can control its
existence in the publisher file, rather than commenting and uncommenting
it depending on which version I'm compiling.

[gregory...@gmail.com]

I'm not too worried about keeping the old versions around, though it seems that it wouldn't be that hard to keep them *just in case*. Of course, that all depends on how the current changes have been made and how many sections they affect. Riemann sum indexing is only in 1 section, right? And spherical show up in one section of chapter 13, though probably make a (very) few appearances in chapter 14. 

As Sean says, his versions mainly deal with the order of the content. Here we are talking about a different kind of change. Would certain sections have two versions, such as the standard version and another with a suffix "_old"? And in a few places, the publisher file just uses the _old version?

[Sean Fitzpatrick]

There are a couple of ways to handle versioning. But when it's a change on the level of a sentence/equation, 

you take the thing with two versions, and replace it with something like <custom ref="some-thing-with-two-defs"/>.

Then there is a separate file that defines what "some-thing-with-two-defs" is, and you can have two versions of that file, with the two definitions.

In the publisher file, you specify which version of that file to use.

You can have a look at the changes for indexing here: https://github.com/APEXCalculus/APEXCalculusPTX/pull/134

(and if it looks OK, approve the pull request!)

There are 11 files where a partition is used/mentioned at least once.

Biggest ones are Riemann Sums and Numerical Integration, but it also shows up in things like arc length, polar coordinates, etc.

Spherical will primarily change one section, but it will also impact some of the examples and exercises in the last chapter.