Applications, Examples and Libraries

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Are you teaching a calculus course? Then use Maple Learn, Maplesoft’s free online product, to do so.

Below are some examples of calculus documents you can create in Maple Learn.

 

1. Documents Explaining Concepts with Interactive Visuals

Example: Visualizing the Formal Definition of the Derivative

 

2. Interactive Quizzes

Example: The Product Rule: Practice Questions

 

3. Documents Using Maple to Perform Complex Operations

Example: Taylor Series Approximation Calculator

 

Maplesoft’s learn content team has already created about 200 Maple Learn calculus documents! The full list is here. You can modify these documents easily, and use them to teach your calculus class as well.

A user wonders if there is a straightforward way to show US states with names using the WorldMap Data Set in Maple

We suggest something like the attached: map-of-us-with-states.mw

 

restart; with(DataSets:-Builtin); r := Reference("GeoNames"); states := r[[Country = "United States", Type = "first-order administrative division"]]; w := WorldMap(); w:-AddPoints(w, states); Display(w, mapdata = fine, style = polygonoutline, size = [2000, 1500])

 

 

I've said it before, and I'll say it again, at Maplesoft, I have the privilege of working with some of the most talented and creative minds around. My colleagues are constantly pushing the boundaries of what we can build and what our products can do.Christmas Wreath in Maple Learn

So to close out 2021, I wanted to share a video that one of our brilliant developers, Marek, sent the company. Marek emails a greeting every year wishing his Maplesoft colleagues a Happy Holiday.  Well, this year, he stepped it up a notch and created this superb video explaining "How to decorate for Christmas using Math", where he created a wreath using Maple Learn.

Watching the video brought a smile to my face, and I know it did the same for others.

I hope this video warms your heart as it did mine. On behalf of all of us at Maplesoft, Happy Holidays!

Recently, the Maple Learn team hosted an internal Maple Learn day. The team encouraged Maplesoft employees to create Maple Learn content. A lot of art was created.

Below is a link to an example of Maple Learn art, and a picture relating to it. The document is interactive, so open it to see what it does.

Christmas Art, by Marek Krzeminski - Senior Architect at Maplesoft

If you too like to combine math and art, use Maple Learn here to create artwork yourself, and share it with us in the comments.


 

Recently I decided to compare continuity, related notions, and differentiability. Can a function be differentiable, but not continuous? What about uniformly continuous, but not differentiable? I used Maplesoft's new online product, Maple Learn (free to use at learn.maplesoft.com), to explore.

Here is a Maple Learn document I created. It is an organizational diagram, as shown below. Each rectangle in the diagram corresponds to a different property that a function may satisfy. Within each rectangle, examples are provided of functions satisfying the appropriate properties.

If you click on an example, it will be selected, and the corresponding function will be plotted in Maple Learn's context panel. Try it!

I've also created companion documents to explain certain concepts in greater detail. For instance, below is a snapshot of a document explaining uniform continuity, which you can access here.

By using sliders in the document, you can move and resize the rectangle drawn in the graph. You should notice when doing this that the green function never touches the horizontal sides of the rectangle. This turns out to be the "reason" why the function is uniformly continuous.

You can find a companion document on Lipschitz continuity here.

I’ve learnt a lot about continuity in creating the documents shown. I hope that you too have learnt something from them!

Although not mentioned in the documentation, the flexible beam component of MapleSim allows for simulation of large deflections.  

In the animation, a flexible beam is loaded with a moment (red arrow) at its free end. Assuming an Euler-Bernoulli beam and slow loading (i.e., no dynamic forces), the beam should deform to an arc of constant radiusNot only the deformation of the beam can be described analytically, also the path (red trace) of the free end follows an analytical curve.

 

I used this test case to get a better understanding of nonlinearities observed in an oscillating system using flexible beams (https://www.mapleprimes.com/posts/215718-Mode-Coupling-With-Flexible-Beams-). The system required tuning of the structure to develop mode coupling. This could not be explained by linear theory. It was unclear whether the large deflections (nonlinear kinematics of the beam) themselves or the implementation of the flexible beam component were responsible for that.  

 

What I have learned so far with the test case using only default settings: 

  • For moderate deflections there is no difference to textbook formulas.
  • Up to 15 degrees rotation of the end frame, the difference between observed displacement and the Bernoulli beam stays bellow 5%.  
  • Up to 30 degrees rotation of the end frame (as in the mode coupling example) the trace of the end frame conforms well with the analytical path.
  • To simulate very large deflections beyond 45 degrees rotation, the beam needs to be segmented to closely follow the analytical path.  

For those that are unsure about the fidelity of their models, I can suggest increasing the numbers of flexible beam components and to compare. I did this in the case of the mode coupling example and noticed no difference. So, the component was not responsible for the nonlinearities. It were the kinematics.

It's unclear whether this good performance in large deflections was intended or is a byproduct of the sophisticated multibody dynamics under the hood.  Maybe an expert can tell more.

Overall, to what I have seen the (static) performance was very satisfying. Judging dynamic performance is much more difficult. Has anyone experience to share with that?

Flexible_beams_const_moment_curve_MP.mw

Flexible_beams_const_Moment_single_beam_MP.msim

Flexible_beams_const_Moment_7_beams_MP.msim

is what I have used.

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.


 

waterfall := proc (data, colorinc := green, colordec := red) local i, r1; r || 1 := plots:-display(plottools:-rectangle([0, 0], [1, data[1]]), color = colorinc); for i from 2 to nops(data) do if data[i-1] < data[i] then r || i := plots:-display(plottools:-rectangle([i-1, data[i-1]], [i, data[i]]), color = colorinc) elif data[i] < data[i-1] then r || i := plots:-display(plottools:-rectangle([i-1, data[i-1]], [i, data[i]]), color = colordec) else r || i := plots:-display(plottools:-rectangle([i-1, data[i-1]], [i, data[i]])) end if end do; plots:-display(seq(r || i, i = 1 .. nops(data))) end proc
``

data := [6, 4, 4, 4, 7, 9, 12, 16, 25, 100, 105, 95, 90, 55, 45, 30]

[6, 4, 4, 4, 7, 9, 12, 16, 25, 100, 105, 95, 90, 55, 45, 30]

(1)

waterfall(data)

 

waterfall(data, purple, yellow)

 

``


 

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

 

As a student I came across an amazing lab experimentA T-type structure with two masses attached to it showed a sudden change in oscillation mode.  

 

With MapleSim I was able to reproduce the experiment.

At the time I was told that this perplexing phenome happens because there are always imperfections. 

 

Today we would probably say that the symmetry has to be broken. The attached example has two parameter sets that a) break symmetry of boundary conditions and b) by structural asymmetry (i.e imperfection). Asymmetry in the initial conditions should also be possible (but I could make work with flexible beams). 

Compared to coupled oscillators that exchange energy via a coupling spring, this example exchanges energy via masses. In fact in its simplest implementation only one mass and two elastic structures are required for this type of mode coupling. MapleSim multibody library offers plenty of possibilities to demonstrate thisFlexible beams are not required. However, flexible beams show mode coupling beautifully and allow a simple reproduction in real life. For that the worksheet contains a parameter set to build a real model with steel wires. Tuning by adjusting the length of the vertical post is required since nonlinearities already shift frequencies in the model. 

 

I would be interested in other cool examples of mode coupling. I am also interested in solutions for flexible beams that impose asymmetry in the initial conditions. To keep it realistic at the start, the T should be bend as one would bend it with a fingertip in x direction. It would be even more realistic if the arms are flexed by gravity with zero velocity at the start of the simulation. How can this be done? 

 

Flexible_beam_mode_coupling.msim

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