Some years ago I taught a calculus course for especially talented students. I made up the following problem as an interesting challenge.

Take a circular disk made of paper. Cut out a sector of some angle α from the disk. Roll each of the resulting two pieces into cones. Let V(α) be the sum of the volumes of the two cones. Find the α that maximizes V(α).

Here is an animated statement of the problem, produced in Maple.

One of the things I love most about my job is working and collaborating with math teachers across the globe. Every discussion leads to additional insights into the challenges facing teachers today, and new ideas on how to make Maple and Maple Learn better. And sometimes, I even learn some math I thought I already knew!

A few months ago, I introduced Maple Learn to a friend of mine who teaches high school math in Kingston, Ontario. I showed her how she could use Maple Learn to teach many concepts during our call, including Completing the Square. I walked her through Maple Learn’s free-form canvas and explained how her students could work through a problem line-by-line just as they would in their notebooks. I highlighted the live plot window and showed how her students could graphically verify that their solution was equivalent to the initial expression. And, I demonstrated the power of Maple Learn’s intelligent context panel and how her students could check their answers algebraically. I thought I had done a good job, until she said: “Karishma, that’s not how we teach Completing the Square anymore!”. Huh! I was floored. What I had shown was the way I had learned the concept so many years ago. I was surprised to learn that there was a new way.

My friend then introduced me to Algebra Tiles and how she used it to teach Completing the Square. Once we went through a few examples, I realized that I had never fully appreciated what I was doing when I completed the square. I had memorized a series of steps without really understanding what I was trying to do. The progression of our discussion naturally led to the inevitable question: “Karishma, does Maple Learn include Algebra tiles? Because that would be a game-changer for my students. Currently, we use physical tiles, but with remote learning, we need something digital.” At that time, my answer was ‘not yet’; however, with the introduction of image support last week, I’m happy to announce that Maple Learn can support algebra tiles and other interactive supports.

Here is the Maple Learn document I created on Completing the Square using Algebra Tiles.

Feel free to change the expressions listed in the document and share it with your students. To see algebra tiles in action inside Maple Learn, take a look at the short video that I created.  If you have any suggestions for improving this application, please feel free to let me know.

In the plotting guide I didn't see a waterfall chart so I created a procedure. 
If anyone has a more efficent, better or alternate way please feel free to add.

Download Waterfall.mw

Universidad Metropolitana de Ciencias de la Educación
Santiago de Chile

Derivative operator on vectors of real variable (R3): applied to curvilinear motion with Maple and MapleSim

In the present work it will be demonstrated how the derivative operator acts in functions of real variable in the movement of a particle that performs a curvilinear trajectory; using the scientific software of the Maplesoft company known by the names Maple and MapleSim, because nowadays most university teachers (higher education) do not visualize the movement of the particle in real time as well as the results of the calculations of speed and acceleration simultaneously. The objectives achieved are to use the vector operator with the help of these programs. As a theoretical tool we will use the three-dimensional vector spaces of real variable with Newton's notation. The methodology we have used was native syntax and embedded components using block diagrams. For the case of particle motion we use the graphical programming proposed by MapleSim. Viable results were achieved for motivational effects and time reduction in complex calculations without neglecting innovation in physical sciences, for teachers in higher education and university students. This work is self-sustaining via Maple Cloud.

Lenin Araujo Castillo

Ambassador of Maple

This is my second try---my previous post about the Maple Conference  https://www.maplesoft.com/mapleconference/2021/ seems to have vanished into thin electrons.

Anyway!  The conference opens tomorrow!  There are many really interesting prerecorded talks, three live plenaries, two excellent panels, and registration is free!  See the above link.

I look forward to "seeing" you tomorrow.

Rob Corless, co-Chair of the Program Committee

on behalf of the organizers

As many of you are aware, the Maple Application Center is a very important resource for Maple users. It is a place for authors to share their Maple work, and for users to have access to a rich collection of over 2,500 curated Maple documents covering a wide array of topics and disciplines.

I am very pleased to announce that we have been hard at work on a new version of the Application Center, and it’s at a state where we’re ready to open it up to the public for testing. You can access the new site here: https://www.maplesoft.com/applications_beta . We are looking for feedback, so please give it a try, and let us know what you think!

Here are a few of my favorite features of the new site:

Updated Look & Feel
The interface of the current version of the Application Center has not changed in many years, and it was time for a new paint job. I think you’ll find that the new site is cleaner, modern, and more enjoyable to use.

Easier to Find the Documents you Want
The updated Application Center provides multiple new ways to find content that is relevant for you. Browse user-made collections of documents or use tags (the same tags used in MaplePrimes) to find documents for the topics you are interested in. Alternatively, you can use the search bar to quickly find documents, tags or authors.

Personalize your Experience
If you are logged-in when using the Application Center, you will be able to customize what you see by pinning your favorite collections, authors or tags to your home page.

Community Moderation & Reputation
As with MaplePrimes, the strength of the Application Center comes from the amazing community of individuals who contribute to it. In addition to submitting your own content to the Application Center, users can now edit tags and create collections of content that others can use. Similar to MaplePrimes, community moderation is restricted to members who have a sufficient reputation score. Speaking of reputation, quality contributions to MaplePrimes will now be reflected in your reputation score. When someone likes one of your submissions, your reputation will increase by 5.

There are many other great new features as well, and we have a roadmap of future updates planned that will make it even better.

I invite you to take a look at the new site and play with it. Browse some content, search, look through tags, and create some collections. Most importantly, I’m really hopeful that you will then use the comments section below to let us know what you think. Did you discover any bugs or issues? What do you like? What do you dislike? What other features would you like to see?

We are hoping to run the Beta for a period of a few weeks, and I’m looking forward to hearing and reading your thoughts. Hope you enjoy it!

https://www.maplesoft.com/applications_beta

Bryon

Dear all,

The November issue of Maple Transactions is now up (we will be adding a few more items to that issue over the course of the month).  See https://mapletransactions.org/index.php/maple/index for the articles.

More importantly, Maple Primes seems to have a great many interesting posts, some of which could well be worked up into a paper (or a video).  Maple Transactions accepts worksheets (documents, workbooks) for publication, as well, although we want a high standard of readability for that.  I invite you to contribute.

The next issue of Maple Transactions will be the Special Issue that is the Proceedings of the Maple Conference 2021 (see my previous post :)

-r

From a tweet by Tamás Görbe : plotting Chebyshev polynomials in polar coordinates leads to some interesting pictures.  Screenshot here, link to the worksheet (and some perhaps interesting puzzles) at the end.

ChebyshevRose.mw

Dear all,

Reversion of series---computing a series for the functional inverse of a function---has been in Maple since forever, but many people are not aware of how easy it is.  Here's an example, where we are looking for "self-reverting" series---which I called "ambiverts".  Anyway have fun.

https://maple.cloud/app/5974582695821312/Series+Reversion%3A+Looking+for+ambiverts

PS There looks to be some "code rot" in the branch point series for Lambert W in Maple, which we encounter in that worksheet.  Or, I may simply have not coded it very well in the first place (yeah, that was mine, once upon a time).  Checking now.  But there is a workaround (albeit an ugly one) shown in that worksheet.

Dear all,

Recently I discovered the noncommuting variables in the Physics package due to Edgardo Cheb-Terrab; doubtless there are many posts here on Maple Primes describing them.  Here is one more, which shows how to use this package to prove the Schur complement formula.

https://maple.cloud/app/6080387763929088/Schur+Complement+Proof+in+Maple

I guess I have a newbie's question: how well-integrated are Maple Primes and the Maple Cloud?  Anyway that seemed the easiest way to share this.

-r

Dear all;

Some of you will have heard of the new open access (and free of page charges) journal Maple Transactions https://mapletransactions.org which is intended to publish expositions on topics of interest to the Maple community. What you might not have noticed is that it is possible to publish your papers as Maple documents or as Maple workbooks.  The actual publication is on Maple Cloud, so that even people who don't have Maple can read the papers.

Two examples: one by Jürgen Gerhard, https://mapletransactions.org/index.php/maple/article/view/14038 on Fibonacci numbers

and one by me, https://mapletransactions.org/index.php/maple/article/view/14039 on Bohemian Matrices (my profile picture here is a Bohemian matrix eigenvalue image).

I invite you to read those papers (and the others in the journal) and to think about contributing.  You can also contribute a video, if you'd rather.

I look forward to seeing your submissions.

Rob Corless, Editor-in-Chief, Maple Transactions

Dear all,

Recently we learned that the idea of "anti-secularity" in perturbation methods was known to Mathieu already by 1868, predating Lindstedt by several years.  The Maple worksheet linked below recapitulates Mathieu's computations:

https://github.com/rcorless/MathieuPerturbationMethod

Nic Fillion and I wrote a more general introduction to perturbation methods using Maple and you can find that paper at 

https://arxiv.org/abs/1609.01321

and the supporting Maple code in a workbook at 

https://github.com/rcorless/Perturbation-Methods-in-Maple

For instance, one of the problems solved is the lengthening pendulum and when we do so taking proper account of anti-secularity (we use renormalization for that one, I seem to remember) we get an error curve that is bounded over time.

Hope that some of you find this useful.

As most Maple Primes readers have hopefully seen, Maplesoft is having our Maple Conference again this fall. This year we decided to add a space to the conference to showcase creative and artistic work that would be interesting to our Maple Community. The conference organizers asked me if I would coordinate and curate this exhibition of creative uses of Math and Maple, and I agreed. So now, I am asking the Maple community to send us your most creative work related to or using Maple.

The obvious thing to submit would be a beautiful digital plot or animation with an interesting mathematical story and of course, we are really interested to see those. But, we would are especially excited to see some art created with physcial media. I would love to see your knitting or needle point project that is inspired by a mathematical theme or was created with the help of Maple.

The full announcement can be found at the Maple Conference Art Gallery page. We would like to have all submissions by October 12th so that can review and finalize the gallery before the conference begins November 1st.

Oh yeah, there will also be prizes.

I can't wait to see what everyone sends in!

Hi everyone! It's been a remarkably long time since I posted on MaplePrimes -- I should probably briefly reintroduce myself to the community here. My name is Erik Postma. I manage the mathematical software group at Maplesoft: the team that writes most of the Maple-language code in the Maple product, also known as the math library. You can find a longer introduction at this link.

One of my tasks at Maplesoft is the following. When a request for tech support comes in, our tech support team can usually answer the request by themselves. But no single person can know everything, and when specialized knowledge of Maple's mathematical library is needed, they ask my team for help. I screen such requests, answer what I can by myself, and send the even more specialized requests to the experts responsible for the appropriate part of the library.

Yesterday I received a request from a user asking how to unwrap angles occurring in an expression. This is the general idea of taking the fact that , and similarly for the other trig functions; and using it to modify an expression of the form  to make it look "nicer" by adding or subtracting a multiple of  to the angle. For a constant, real value of  you would simply make the result be as close to 0 as possible; this is discussed in e.g. this MaplePrimes question, but the expressions that this user was interested in had arguments for the trig functions that involved variables, too.

In such cases, the easiest solution is usually to write a small piece of custom code that the user can use. You might think that we should just add all these bits and pieces to the Maple product, so that everyone can use them -- but there are several reasons why that's not usually a good idea:

• Such code is often too specialized for general use.
• Sometimes it is reliable enough to use if we can communicate a particular caveat to the user -- "this will not work if condition XYZ occurs" -- but if it's part of the Maple library, an unsuspecting user might try it under condition XYZ and maybe get a wrong answer.
• This type of code code generally doesn't undergo the careful interface design, the testing process, and the documentation effort that we apply to the code that we ship as part of the product; to bring it up to the standards required for shipping it as part of Maple might increase the time spent from, say, 15 minutes, to several days.

That said, I thought this case was interesting enough to post on MaplePrimes, so that the community can take a look - maybe there is something here that can help you with your own code.

So here is the concrete question from the user. They have expressions coming from an inverse Laplace transform, such as:

with(inttrans):
F := -0.3000*(-1 + exp(-s))*s/(0.0500*s^2 + 0.1*s + 125);
f := invlaplace(F, s, t)*u(t);
# result: (.1680672269e-1*exp(1.-1.*t)*Heaviside(t-1.)*(7.141428429*sin(49.98999900*t-
#         49.98999900)-357.*cos(49.98999900*t-49.98999900))+.1680672269e-1*(-7.141428429*sin
#         (49.98999900*t)+357.*cos(49.98999900*t))*exp(-1.*t))*u(t)

I interpreted their request for unwrapping these angles as replacing the expressions of the form  with versions where the constant term was unwrapped. Thinking a bit about how to be safe if unexpected expressions show up, I came up with the following solution:

unwrap_trig_functions := module()
local ModuleApply := proc(expr :: algebraic, $)
  return evalindets(expr, ':-trig', process_trig);
end proc;

local process_trig := proc(expr :: trig, $)
  local terms := convert(op(expr), ':-list', ':-`+`');
  local const, nonconst;
  const, nonconst := selectremove(type, terms, ':-complexcons');
  const := add(const);
  local result := add(nonconst) + (
    if is(const = 0) then
      0;
    else
      const := evalf(const);
      if type(const, ':-float') then
        frem(const, 2.*Pi);
      else
        frem(Re(const), 2.*Pi) + I*Im(const);
      end if;
    end if);
  return op(0, expr)(result);
end proc;
end module;

# To use this, with f defined as above:
f2 := unwrap_trig_functions(f);
# result: (.1680672269e-1*exp(1.-1.*t)*Heaviside(t-1.)*(7.141428429*sin(49.98999900*t+
#         .27548346)-357.*cos(49.98999900*t+.27548346))+.1680672269e-1*(-7.141428429*sin(
#         49.98999900*t)+357.*cos(49.98999900*t))*exp(-1.*t))*u(t)

Exercise for the reader, in case you expect to encounter very large constant terms: replace the calls to frem above with the code that Alec Mihailovs wrote in the question linked above!