Maple's fsolve command can quickly solve expressions involving large floating point numbers where the (symbolic) solve command can take much longer attempting to solve the equivalent rational expression. For example, consider the following worksheet:

Download solve-fsolve-primes.mw

We have just released updates to Maple and MapleSim.

Maple 2023.1 includes improvements to the math engine, Plot Builder, import/export, and more.  We recommend that all Maple 2023 users install this update.

This update is available through Tools>Check for Updates in Maple, and is also available from the Maple 2023.1 download page, where you can find more details.

In particular, please note that this update includes fixes to problems with Quantifier Elimination and Group  Theory, and improves performance after a period of inactivity, all of which were reported on MaplePrimes. Thanks for the feedback!

At the same time, we have also released an update to MapleSim, which contains a variety of improvements to MapleSim and its add-ons. You can find more information on the MapleSim 2023.1 download page.

Sometimes, it’s the little things. Those little improvements that make a good tool even better. Sometimes, it’s as simple as an easy shorthand notation that allows you to create and label points on a graph with a single command. Just to pick a totally random example.

Okay, maybe it’s not totally random. Maybe this new point notation is one of our newest features in Maple Learn, and maybe it’s now easy and quick for you to create labeled points to your heart’s content. Maybe you could learn more about all the ins and outs of this new feature by checking out the how-to document.

But I can’t make any guarantees, of course.

That said, if this hypothetical scenario were true, you would also be able to see it in action in our new document on the proof of the triangle inequality.

With this document, you can explore a detailed (and interactive!) visualization of the proof using Euclidean geometry. You can adjust the triangles to see for yourself that the sum of the lengths of any two sides must be greater than the third side, read through the explanation to see the mathematical proof, and challenge yourself with the questions it leaves you to answer. And those points on those triangles? Labeled. Smoothly and easily. I wonder how they might have done it?

We hope you enjoy the new update! Let us know what other features you want to see in Maple Learn, and we’ll do our best to turn those dreams into reality.

We all know that math is beautiful in and of itself—but sometimes students might need a little convincing. What better way to do that then sprucing up your math with a little colour? With Maple Learn, plot colours are fully customizable. We have several colour palettes to choose from—want your document to evoke the delicate tones of springtime? Looking for a palette that’s colourblind friendly? Or maybe you’re just nostalgic for the colours of Maple V? All these options and more are available for making your graphs colourful and coordinated. But maybe you’re the kind of person who wants to go against the grain, and you laugh in the face of predetermined colour coordination. Don’t worry, we’ve got you covered too! With our colour selector, you can also choose your own custom colours. The full colour spectrum is right at your fingertips. To learn more about how to customize the colours on your document, check out this How-To guide.

And of course, the potential for colour inevitably leads to the potential for art. Our Maple Learn Art Gallery has plenty of fun and colourful works you can admire and contemplate (and maybe even draw inspiration from!). One of our most recent and most colourful additions is this document showcasing the history of the rainbow pride flag, in honour of June being Pride Month. You can use the slider to move through time, letting you see how the colours on the flag evolved and read about the meanings behind them. And, thanks to the colour selector, the colours match the precise shades used for the original flags! That’s the magic of hexadecimal colours for you.

Hold on—the magic of hexadecimal colours, I hear you ask? What an enticing concept. If only we had some kind of document, perhaps one made in Maple Learn, that explained how hexadecimal colours worked and included an interactive example so that you could easily see how the red, green, and blue colour values blend together to create any given colour… Too bad we don’t!

Just kidding. Of course we do.

If all these colours have inspired you, be sure to check out our Call for Creative Works for the upcoming Maple Conference! Maybe your colourful creation could be this year’s winner.

Ever wonder how to show progress updates from your executing code without printing new lines each time?

One way to do this is to use a TextArea component and the DocumentTools package. The TextArea could be inserted from the Components Palette in Maple, or programmatically like so:

Download text-area-update-progress.mw

This post discusses a solution for modeling a traveling load on Maplesim's Flexible Beam component and provides an example of a bouncing load.

The idea for the above example came from an attempt to reproduce a model of a mass sliding on a beam from MapleSim's model gallery. However, reproducing it using contact components in combination with the Flexible Beam component turned out to be not straightforward, and this will be discussed in the following.

To simulate a traveling load on the Flexible Beam component, one could apply forces at discrete locations for a certain duration. However, the fidelity of this approach is limited by the number of discrete locations, which must be defined using the Flexible Beam Frame component, as well as the way in which the forces are activated.

One potential solution to address the issue of temporal activation of forces is to attach contact elements (such as Rectangle components) at distinct locations along the beam, which are defined by Flexible Beam Frame components, and make contact using a spherical or toroidal contact element. However, this approach also introduces two new problems:

• An additional bending moment is generated when the load is not applied at the center of the contact element's attachment point, the Flexible Beam Frame component. Depending on the length of the contact element, deformations caused by this moment can be greater than the deformation caused by the force itself when the force is applied at the ends of the contact element. Overall, this unwanted moment makes the simulation unrealistic and must be avoided.
• When the beam bends, a gap (see below) or an overlap is created between adjacent Rectangle components. If there is a gap, the object exerting a force on the beam can fall through it. Overlaps can create differences in dynamic behavior when the radius of curvature of the beam is on the opposite side of the point of contact.

To avoid these problems, the solution presented here uses an intermediate kinematic chain (encircled in yellow below) that redistributes the contact force on the Rectangle component on two support points (ports to attach Flexible Beam Frame components) in a linear fashion.

To address gaps, the contact element (Rectangle) attached to the kinematic chain has the same width as the chain and connects to the adjacent contact elements via multibody frames. In the image below, 10 contact elements are laid on top of a single Flexible Beam component, like a belt made out of tiles. The belt has to be pinned to the flexible beam at one location (highlighted in yellow). The location of this fixed point determines how the flexible beam is loaded by tangential contact forces (friction forces) and should be selected carefully.

Some observations on the attached model:

• Low damping and high initial potential energy of the ball can result in a failed simulation (due to constraint projection failure). Increasing the number of elastic coordinates has a similar effect. Constraint projection can be turned off in the simulation settings to continue simulation.
• The bouncing ball excites several eigenmodes at once, causing the beam to wiggle chaotically in combination with the varying bouncing frequency of the ball. A similar looking effect can also be achieved with special initial conditions, as demonstrated with Maple in this excellent post on Euler beams and partial differential equations.
• Repeated simulations with low damping lead to different results (an indication of chaotic behavior; see three successive simulations below (gold) compared to a saved solution(red)). The moment in the animation when the ball travels backward represents a metastable equilibrium point of the simulation. This makes predictions beyond this point difficult, as the behavior of the system is highly dependent on the model parameters. Whether the reversal is a simulation artifact or can happen in reality remains to be seen. Overall, this example could evolve into a nice experimental fun project for students.

• Setting the gravitational constant for Mars, everything is different. I could not reproduce the fun factor on Earth. A reason more to stay ,-)

Ball_bouncing_on_a_flexible_beam.msim

# countourplot3d piggybacks on top of plot3d.
# For the "coloring=[lowColor, highColor]", the "filledregions=true" option must be present.
# If "filledregions=true" is not present, plot3d will throw an error.
# This code shows the three cases, only one of which will work.
Note to support. I cannot add a new tag. contourplot3d should be a tag.

restart;
with(plots);
with(ColorTools);
cGr4s := Color([0.50, 0.50, 0.50]);
contourplot3d(-5*d/(d^2 + y^2 + 1), d = -3 .. 3, y = -3 .. 3, color = black, thickness = 3, coloring = [cGr4s, cGr4s], contours = [-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2]);
contourplot3d(-5*d/(d^2 + y^2 + 1), d = -3 .. 3, y = -3 .. 3, filledregions = false, color = black, thickness = 3, coloring = [cGr4s, cGr4s], contours = [-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2]);
contourplot3d(-5*d/(d^2 + y^2 + 1), d = -3 .. 3, y = -3 .. 3, filledregions = true, color = black, thickness = 3, coloring = [cGr4s, cGr4s], contours = [-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2]);

The recent Maple 2023 release comes with a multitude of new features, including a new Canvas Scripting Gallery full of templates for creating interactive Maple Learn documents.

The Maple Learn Scripting Gallery can be accessed through Maple, by searching “BuildInteractiveContent Maple2023” in the search bar at the top of the application and clicking on the only result that appears. This will bring you to the help page titled “Build and Share Interactive Content”, which can also be found by searching “scripting gallery” in the search bar of a Maple help page window. The link to the Maple Learn Scripting Gallery is found under the “Canvas Scripting” section on this help page and clicking on it will open a Maple workbook full of examples and templates for you to explore.

The interactive content in the Scripting Gallery is organized into five main categories – Graphing, Visualization, Quiz, Add-ons and Options, and Applications Optimized for Maple Learn – each with its own sub-categories, templates, and examples.

One of the example scripts that I find particularly interesting is the “Normal Distribution” script, under the Visualizations category.

All of the code for each of the examples and templates in the gallery is provided, so we can see exactly how the Normal Distribution script creates a Maple Learn canvas. It displays a list of grades, a plot for the grade distribution to later appear on, math groups for the data’s mean and variance, and finally a “Calculate” button that runs a function called UpdateStats.

The initial grades loaded into the document result in the below plot, created using Maple’s DensityPlot and Histogram functions, from the Statistics package. 

The UpdateStats function takes the data provided in the list of grades and uses a helper function, getDist, to generate the new plot to display the data, the distribution, the mean, and the variance. Then, the function uses a Script object to update the Maple Learn canvas with the new plot and information.