Hi, 

In a recent post  (Monte Carlo Integration) Radaar shared its work about the numerical integration, with the Monte Carlo method, of a function defined in polar coordinates.
Radaar used a raw strategy based on a sampling in cartesian coordinates plus an ad hoc transformation.
Radaar obtained reasonably good results, but I posted a comment to show how Monte Carlo summation in polar coordinates can be done in a much simpler way. Behind this is the choice of a "good" sampling distribution which makes the integration problem as simple as Monte Carlo integration over a 2D rectangle with sides parallel to the co-ordinate axis.

This comment I sent pushed me to share the present work on Monte Carlo integration over simple polygons ("simple" means that two sides do not intersect).
Here again one can use raw Monte Carlo integration on the rectangle this polygon is inscribed in. But as in Radaar's post, a specific sampling distribution can be used that makes the summation method more elegant.

This work relies on three main ingredients:

1. The Dirichlet distribution, whose one form enables sampling the 2D simplex in a uniform way.
2. The construction of a 1-to-1 mapping from this simplex into any non degenerated triangle (a mapping whose jacobian is a constant equal to the ratio of the areas of the two triangles).
3. A tesselation into triangles of the polygon to integrate over.

This work has been carried out in Maple 2015, which required the development of a module to do the tesselation. Maybe more recent Maple's versions contain internal procedures to do that.
 

Monte_Carlo_Integration.mw

Hi. My name is Eugenio and I’m a Professor at the Departamento de Didáctica de las Ciencias Experimentales, Sociales y Matemáticas at the Facultad de Educación of the Universidad Complutense de Madrid (UCM) and a member of the Instituto de Matemática Interdisciplinar (IMI) of the UCM.

I have a 14-year-old son. In the beginning of the pandemic, a confinement was ordered in Spain. It is not easy to make a kid understand that we shouldn't meet our friends and relatives for some time and that we should all stay at home in the first stage. So, I developed a simplified explanation of virus propagation for kids, firstly in Scratch and later in Maple, the latter using an implementation of turtle geometry that we developed long ago and has a much better graphic resolution (E. Roanes-Lozano and E. Roanes-Macías. An Implementation of “Turtle Graphics” in Maple V. MapleTech. Special Issue, 1994, 82-85). A video (in Spanish) of the Scratch version is available from the Instituto de Matemática Interdisciplinar (IMI) web page: https://www.ucm.es/imi/other-activities

Introduction

Surely you are uncomfortable being locked up at home, so I will try to justify that, although we are all looking forward going out, it is good not to meet your friends and family with whom you do not live.

I firstly need to mention a fractal is. A fractal is a geometric object whose structure is repeated at any scale. An example in nature is Romanesco broccoli, that you perhaps have eaten (you can search for images on the Internet). You can find a simple fractal in the following image (drawn with Maple):

Notice that each branch is divided into two branches, always forming the same angle and decreasing in size in the same proportion.

We can say that the tree in the previous image is of “depth 7” because there are 7 levels of branches.

It is quite easy to create this kind of drawing with the so called “turtle geometry” (with a recursive procedure, that is, a procedure that calls itself). Perhaps you have used Scratch programming language at school (or Logo, if you are older), which graphics are based in turtle geometry.

All drawings along these pages have been created with Maple. We can easily reform the code that generated the previous tree so that it has three, four, five,… branches at each level, instead of two.

But let’s begin with a tale that explains in a much simplified way how the spread of a disease works.

- o O o -

Let's suppose that a cat returns sick to Catland suffering from a very contagious disease and he meets his friends and family, since he has missed them so much.

We do not know very well how many cats each sick cat infects in average (before the order to STAY AT HOME arrives, as cats in Catland are very obedient and obey right away). Therefore, we’ll analyze different scenarios:

1. Each sick cat infects two other cats.
2. Each sick cat infects three other cats.
3. Each sick cat infects five other cats

1. Each Sick Cat Infects Two Cats

In all the figures that follow, the cat initially sick is in the center of the image. The infected cats are represented by a red square.

· Before everyone gets confined at home, it only takes time for that first sick cat to see his friends, but then confinement is ordered (depth 1)

As you can see, with the cat meeting his friends and family, we already have 3 sick cats.

· Before all cats confine themselves at home, the first cat meets his friends, and these in turn have time to meet their friends (depth 2)

In this case, the number of sick cats is 7.

· Before every cat is confined at home, there is time for the initially sick cat to meet his friends, for these to meet their friends, and for the latter (friends of the friends of the first sick cat) to meet their friends (depth 3).

There are already 15 sick cats...

· Depth 4: 31 sick cats.

· Depth 5: 63 sick cats.

Next we’ll see what would happen if each sick cat infected three cats, instead of two.

2. Every Sick Cat Infects Three Cats

· Now we speed up, as you’ve got the idea.

The first sick cat has infected three friends or family before confining himself at home. So there are 4 infected cats.

· If each of the recently infected cats in the previous figure have in turn contact with their friends and family, we move on to the following situation, with 13 sick cats:

· And if each of these 13 infected cats lives a normal life, the disease spreads even more, and we already have 40!

· At the next step we have 121 sick cats:

· And, if they keep seeing friends and family, there will be 364 sick cats (the image reminds of what is called a Sierpinski triangle):

4. Every Sick Cat Infects Five Cats

· In this case already at depth 2 we already have 31 sick cats.

5. Conclusion

This is an example of exponential growth. And the higher the number of cats infected by each sick cat, the worse the situation is.

Therefore, avoiding meeting friends and relatives that do not live with you is hard, but good for stopping the infection. So, it is hard, but I stay at home at the first stage too!

Monte Carlo integration uses random sampling unlike classical techniques like the trapezoidal or Simpson's rule in evaluating the integration numerically.

Download MONTE_CARLO_INTEGRATION1.mw

The purpose of this document is:

a) to correct the physics that was used in the document "Minimal Road Radius for Highway Superelevation" recently submitted to the Maple Applications Center;

b) to confirm the values found in the manual for the American Association of State Highway and Transportation Officials (AASHTO) that engineers use to design and build these banked curves are physically sound. 

c) to highlight the pedagogical value inherent in the Maple language to distinguish between assignment ( := )  and equivalence (  =  );

d) but most importantly, to demonstrate the pedagogical value Maple has in thinking about solving a problem involving a physical process. Given Maple's symbolic mathematics capabilities, one can implement a top-down approach to the physics and the mathematics, working from the general principle to the specific example. This allows one to avoid the types of errors that occur when translating the problem into a bottom up approach, from specific values of the example to the general principle, an approach that is required by most other computational systems.

I hope that others are willing to continue to engage in discussions related to the pedagogical value of Maple beyond mathematics.

I was asked to post this document to both here and the Maple Applications Center

[Document edited for typos.]

Minimum_Road_Radius.mw

Maple's pdsolve() is quite capable of solving the PDE that describes the motion of a single-span Euler beam.  As far as I have been able to ascertain, there is no obvious way of applying pdsolve() to solve multi-span beams.  The worksheet attached to this post provides tools for solving multi-span Euler beams.  Shown below are a few demos.  The worksheet contains more demos.

Download worksheet: euler-beam-with-method-of-lines.mw

This is my attempt to produce a subplot within a larger plot for magnifying data in a small region, and putting that subplot into the white space of the figure.
Based on the questions: How to insert a plot into another plot? and Inset figure in Maple, I wrote a couple of procedures that create sub-plots and allow the user to place the subplot window as he/she chooses. This avoids the graininess issues mentioned by acer in the second link (and experienced by me).

So far, I only have this completed for point plots, but using acer's method of piecewise functions posted in the plotin2b.mw of the second article, with the subplot function being defined only if it satisfies your conditions, would allow the subplot generating procedure to be generalized easily enough. But the data I'm working with all point plots, so that's the example here.

The basic idea  is to use one procedure to create boxes, make tickmarks on the expanded region, and make tickmark labels, combine all of those into one graph. Then create scaled and shifted versions of the data series, then make graphs of those. Lastly, combine them all into one picture.

Hope this helps someone who has to do the same.

Mapleprimes isn't inserting the contents, but here is the download of the file: SubPlotBoxesandVectorDataSeries.mw

Here is a little cute demo that shows how a cube may be paritioned into three congruent pyramids.  This was inspired by a Mathematica demo that I found in the web but I think this one's better :-)

Download square-partitioned-into-pyramids.mw

Download The_equations_of_motion_in_curvilinear_coordinates.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Research work

The fractal structure’s researching.

Modeling of the fractal sets in the Maple program.

Municipal Budget Educational Establishment “School # 57” of Kirov district of Kazan

    Author:  Ibragimova Evelina

    Scientific advisor:  Alsu Gibadullina - mathematics teacher

    Translator:  Aigul Gibadullina

In Russian

ИбрагимоваЭ_Фракталы.docx

In English

Fractals_researching.doc

     ( Images - in attached files )

Table of contents:

Introduction

I. Studying of principles of fractals construction

II. Applied meaning of fractals

III. Researching of computer programs of fractals construction

Conclusion

Introduction

We don’t usually think about main point of things, which we have to do with every day. Environmental systems are many-sided, ever–changing and compli­cated, but they are formed by a little number of rules. Fractals are apt example of this – they are complicated, but based on simple regulations. Self – similarity is the main attribute of them.  Just one fractal element contains genetically information about all system.  This information have a forming role for all system. But some­times self – similarity is partial.

Hypothesis of the research. Fractals and various elements of the Universe have general principles of structural organization. It is a reason why the theory of frac­tals is instrument for cognition of the world.

Purpose of the research. Studying  of genetic analogy  between fractals and alive and non-living Universe systems with computer-based mathematical mod­eling in the Mapel’s computer space.

Problems of the research. 

1. studying of principles of fractal’s construction;
2. Detection of  general fractal content of physical, biological and artificial sys­tems;
3. Researching of applied meaning of fractals;
4. Searching of computer programs which can generate all of known fractals;
5. Researching of fractals witch was assigned by complex variables;
6. Formation of innovative ideas of using of fractals in different spaces;

The object of research.  Fractal structures, nature and society objects.

The subject of research. Manifestation of fractality in different objects of the Universe.

Methods of the researching.

1. Studying and analysis of literature of research’s problem;
2. Searching of computer programs which can generate fractals and experimenta­tion with them;
3. Comparative analysis of principles of generating of fractals and structural or­ganizations of physical, biological and artificial systems;
4. Generation and formulation of innovation ways to applied significance of fractals.

Applied significance.

Researching of universality of fractals gives general academic way of cognition of nature and society.

I. Studying of principles of fractal construction

We can see fractal constructions everywhere – from crystals and different accu­mulations (clouds, rivers, mountains, stars etc.) to complex ecosystems and bio­logical objects like fern leafs or human brain. Actually, the idea that frac­tal principles are genetic code of our Universe has been discussed for about fifteen years. The first attempt of modeling of the process of the Universe construction was done by A.D. Linde. We also know that young Andrey Saharov had solved “fractal” calculation problem – it was already half a century ago.

Now therefore, fractal picture of the world was intuitively anticipated by human genius and it inevitably manifested in its activity.

Fractals are divided into four groups in the traditional way: geometric (constructive), algebraical (dynamical), stochastical and natural.

The first group of fractals is geometric. It is the most demonstrative type of fractals, because we can instantly observe the self-similarity in it. This type of fractals is constructed in the basis of original figure by her fragmentation and real­izing of different transformations. Geometrical fractals ensue on repeating of this procedure. They are using in computer-generated graphics for generating the pic­tures of leafs, bush, dimensional structures, etc.

The second large group of fractals -  algebraical. This fractals are constructed by iteration of nonlinear displays, which set by simple formulas. There are two types of algebraical fractals – linear and nonlinear. The first of them are determined by first order equates (linear equates), and the second by nonlinear equates, their na­ture significantly brighter, richer and more diverse than first order equates.

The third known group of fractals – stochastical. It is generated by method of random modification of options in iterative process. Therefore, we get an objects which is similar to nature fractals – asymmetrical trees, rugged coasts, mountain scenery etc. Such fractals are useful in modeling of land topography, sea–surface and electrolysis process etc.

The fourth group of fractals is nature, they are dominate in our life. The main difference of such fractals is that they can’t demonstrate infinite self-similarity. There is “physical fractals” term in the classification concept for nature fractals, this term notes their naturalness. These fractals are created with two simple opera­tions: copy and scaling. We can indefinitely list examples of nature fractals: hu­man’s circulatory system, crowns and leafs of trees, lungs, etc.  It is impossible to show all diversity of nature fractals.

II. Applied meaning of fractals

Fractals are having incredibly widespread application nowadays.

In the medicine. Human’s organism is consists from fractal structures: circulatory system, bronchus, muscle, neuron system, etc. So it’s naturally that fractal algorithms are useful in the medicine. For example, assessment of rhythm of fractal dimension while electric diagrams analyzing allows to make more infor­mative and accurate view on the beginning of specific illnesses. Also fractals are using for high–quality processing of  X–ray images (in the experimental way). There are designing of new methods in the gastroenterology which allows to ex­plore gastrointestinal tract organs qualitative and painlessly. Actually, there are discoveries of application of fractal methods for diagnosis and treatment of cancer.

In the science. There are no scientific and technical areas without fractal calcu­lations nowadays. It happens due to the fact that majority of nature objects have fractal structures and dimension: coasts of the continents; natural resources alloca­tion; magnetic field anomaly; dissemination of surges and vibrations in an elastic environments; porous, solid and fungal bodies; crystals; turbulence; dynamic of complicated systems in general, etc. Fractals are useful in geology, geophysics, in the oil sciences… It’s impossible to list all the spaces of adaptability.

Modeling of chaotic processes, particularly, in description of population models.

In telecommunications. It’s naturally that fractals are popular in this area too. Natan Coen is person, who had started to use fractal antennas. Fractal antenna has very compact form which provides high productivity. Due to this, such antennas are used in marine and air transport, in personal devises. The theory of fractal an­tennas has become an independent, well-developed apparatus of synthesis and analysis of electric small antenna (ESA) nowadays. There are developments of possibility of fractal compression of the traffic which is transmitted through the web. The goal of this is more effective transfer of information.

In the visual effects. The theory of fractals has penetrated area of formation of different kinds of visualizations and creation of special effects in the computer graphics soon. This theory are very useful in modeling of nature landscapes in computer games. The film industry also has not been without fractal geometry. All the special effects are based in fractal structure: mountain landscape, lava, flame, fog, large flows and the same. In the modern level of the cinema creation of the special effects is impossible without modeling of fractals.

In the economics. The Veirshtrass’s function is famous example of stochastic fractals. Analysis of graph of the function in interactive mathematic environment Maple allows to make sure in fractal structure of function by way of entry of dif­ferent ranges of graphic visualization. In any indefinitely small area of the part graph of the function absolutely looks like area of this part in the all . The property of function is used in analysis of stock markets.

In the architect. Notably, fractal structures have become useful in the architect more earlier than B. Mandelbrode had discovered them. S.B. Pomorov, Doctor of Architecture, Professor, member of Russian Architect Union, talks about applica­tion of fractal theory in the architect in his article. Let’s see on the part of this arti­cle:

“Fractal structures were found in configuration of African tribal villages, in an­cient Vavilon’s ziggurats, in iconic buildings of ancient India and China, in gothic temples of ancient Russia .

We can see the high fractal level in Malevich’s Architectons. But they were cre­ated long before emergence of the notion of fractals in the architect. People started to use fractal algorithms on the architect morphogenesis consciously after Mandel­brot’s publications. It was made possible to use fractal geometry for analyzing of architectural forms.

Fractals had become available to the majority of specialists due to the comput­erization.  They had been incredibly attractive for architectors, designers and town planners in aesthetic, philosophical and psychological way. Fractal theory was per­ceived on emotional, sensual level in the first phase. The constant repression lead­ing to loss of sensuality.

Application of fractal structures is effective on the microenvironment designing level: interior, household items and their elements. Fractal structures introduction allows creating a new surroundings for people with fractal properties on all levels. It corresponds to nesting spaces.

Fractal formations are not a panacea or a new era in the architect history. But it’s a new way to design architect forms which enriches the architectural theory and practice language. The understanding of na­ture fractal impacts on architectural view of urban environment. An attempt to de­velop the method of architectural designing which will base in an in-depth fractal forms is especially interesting. Will this method base only on mathematics? Will it be different methods and features symbiosis? The practice experiments and re­searches will show us. It’s safe to say that modern fractal approach can be useful not only for analysis, but also for harmonic order and nature’s chaos, architect which may be semantic dominant in nature and historic context.”

Computer systems. Fractal data compression is the most useful fractal applica­tion in the computer science. This kind of compression is based on the fact that it’s easy to describe the real world by fractal geometry. Nevertheless, pictures are compressed better than by other methods (like jpeg or gif). Another one advantage is that picture isn’t pixelateing while compressing. Often picture looks better after increase in fractal compressing.

Basic concept for fractal computer graphics is “Fractal triangle”. Also there are “Fractal figure”, “Fractal object”, “Fractal line”, “Fractal composition”, “Parent object” and “Heir object”. However, it should be noted that fractal computer graphics has recently received as a kind of computer graphics of 21th century.

 The opportunities of fractal computer graphics cannot be overemphasized. It allows creating abstract compositions where we can realize a lot of moves: hori­zontal and vertical, diagonal directions, symmetry and asymmetry etc. Only a few programmers from all over the world know about fractal graphics today.  To what can we compare fractal picture? For example, with complex structure of crystal or with snowflake, the elements of which line up in the one complex composition. This property of fractal object can be useful in ornament creating or designing of decorative composition. Algorithms of synthesis of fractal rates which allows to reproduce copy of any picture too close to the original are developed today.

III. Researching of computer programs of fractal construction

Strict algorithms of fractals are really good for programming. There are a lot of computer programs which introduce fractals nowadays. Computer mathematic systems are stand out from over programs, especially, Maple. Computer mathe­matics is mathematic modeling tool. So programming represents genetic structure of fractal in these systems and we can see precise submission of fractal structure in the picture while we enter a number of iterations . This is the reason why mathematic fractals should be studied with computer mathematics.  The last dis­covery in fractal geometry has been made possible by powerful, modern com­puters. Fractal property researching is almost completely based on computer cal­culations. It allows making computer experiments which reproduce processes and phenomenon which we can’t experiment in the real world with.

Our school has been worked with computer mathematics Maple package more than 10 years. So we have unique opportunity to experiment with mathematic fractals, thanks to that we can understand how initial values impact on outcome   (it is stochastic fractal). For example, we have understood the meaning of the fact that color is the fourth dimension: color changing leads to changing of physical char­acteristics. That is what astrophysics mean talking about “multicolored” of the Universe. While fractal constructing in interactive mathematic environment we re­ceived graphic models which was like A. D. Linde’s model of the Universe. Perhaps, it demonstrates that Universe has fractal structure.

Conclusion

Scientists and philosophers argue, can we talk about universality of fractals in recent years. There are two groups of two opposite positions. We agree with the fact that fractals are universal. Due to the fact that movement is inherent property of material also we always have fractals wherever we have movement.  

We are convinced that fractal is genetic property of the Universe, but it is not mean that all the Universe elements to the one fractal organization. In deployment process fractal structure is undergoing a lot of fluctuations (deviations) and a lot of points of bifurcation (branching) lead grate number of fractal development varie­ties.  

Therefore, we think that fractals are general academic method of real world re­searching. Fractals give the methodology of nature and community researching.

In transitional, chaotic period of society development social life become harder. Different social systems clash. Ancient values are exchanged for new values literally in all spaces. So it’s vitally important for science to develop behavior strategies which allow to avoid tragic mistakes. We think that fractals play important role in developing of such technologies. And – synergy is theory of evolving systems self- organization. But evolution happens on fractal principles, as we know now.

P.S.  Images - in attached files

This is still a work in progress, but might be of use to anybody interested in Maxwell's Equations :-)

Examining_Maxwells_Equations.mw

Since we are getting many questions on how to create Math apps to add to the Maple Cloud. I wanted to go over the different GUI aspects of how you go about creating a Math App in Maple. The following Document also includes some code examples that are used in the the Math App but doesn't go into them in detail. For more details on the type of coding you do in a Math App see the DocumentTools package help page.

Some of the graphical features of the Math app don't display on Maple Primes so I'd recommend downloading this worksheet from here: HowToMathApp.mw to follow along.

How to make a Math App (An example of using the Document Tools).   This Document will provide a beginners guide on one way to make a Math app in Maple. It will contain some coding examples as well as where to find different options in the user interface. Step 1 Insert a Table     •  When making a Math App in Maple I often start with a table. You can enter a table by going to Insert > Table...      •  I often make the table 1 x 2 to start with as this gives an area for input and an area for the output (such as plots).   Add a plot component to one of the cells of the table     •  From the Components  Palette you can add a Plot Component . Add it to the table by clicking and dragging it over.     Add another table inside the other cell     •  In the other cell of the table I'll add another table to organize my use of buttons, sliders, and other components. Add some components to the new table     •  From the Components Palette I'll add a slider, or dial, or something else for interaction.   •  You may also want a Math region for an area to enter functions and a button to tell Maple to do something with it.   Arrange the Components to look nice     •  You can change how the components are placed either by resizing the tables or changing the text orientation of the contents of the cells.   Write some code for the interaction of the buttons.     •  Using the DocumentTools  package there are lots of ways you can use the components. I often will start writing my code using a code edit region  as it provides better visualization for syntax. On MaplePrimes these display as collapsed so I will also include code blocks for the code.   Let's write something that takes the value of the slider and applies it to the dial     •  Note that the names of the components will change in each section as they are copies of the previous section.   with(DocumentTools): with(DocumentTools): sv:=GetProperty('Slider2',value); SetProperty('Dial2',value,sv); •  This code will only execute when run using the  button. Change the value of the slider below then run the code above to see what happens.   Move the code 'inside' the slider     •  Instead of putting the code inside the code edit region where it needs to be executed, we'll next add the code to the value changed code of the slider.   •  Right click the Slider then select "Edit Value Changed Code".     •  This will open the code editor for the Slider     •  Enter your code (ensuring you're using the correct name for the slider and dial).   •  Notice that you don't need to use the with(DocumentTools): command as "use DocumentTools in ... end use;" is already filled in for you.   •  Save the code in the Slider and hit the  button inside it once. •  Now move the slider.   •  On future uses of the App you won't need to hit  as the code will be run on startup. Add some more details to your App     •  Let's make this app do something a bit more interesting than change the contents of a dial when a slider moves.   •  The plan in the next few steps is to make this app allow a user to explore parameters changing in a sinusoidal expression.   •  I'm going to add a second Math Component, put the expression into both then uncheck the box in the context panel that says "Editable".   •  To make the Math containers fit nicely I'll check the Auto-fit container box and set the Minimum Width Pixels to 200.   Add code to change the value of phi in the second Math Container when the Slider changes     Note: Maple uses Radians for trigonometric functions so we should convert the value of phi to Radians. use DocumentTools in   use DocumentTools in phi_s:=GetProperty(Slider5,value); expr:= GetProperty(MathContainer6,expression); new_expr:=algsubs(phi=phi_s*Pi/180,expr); SetProperty(MathContainer7,expression,new_expr); end use: Make the Dial go from 0 to 360°     •  Click the Dial and look at the options in the context panel on the right.   •  Update the values in the Dial so that the highest position is 360 and the spacing makes sense for the app.   Have the Dial update the theta value of the expression     •  Add the following code to the Dial   use DocumentTools in use DocumentTools in theta_d:=GetProperty(Dial7,value); phi_s:=GetProperty(Slider7,value); #This is added so that phi also has the value updated expr:= GetProperty(MathContainer10,expression); new_expr0:=algsubs(theta=theta_d*Pi/180,expr); new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0); #This is added so that phi also has the value updated SetProperty(MathContainer11,expression,new_expr); end use:   •  Update the value in the slider to include the value from the dial   use DocumentTools in   use DocumentTools in theta_d:=GetProperty(Dial7,value); #This is added so that theta also has the value updated phi_s:=GetProperty(Slider7,value); expr:= GetProperty(MathContainer10,expression); new_expr0:=algsubs(theta=theta_d*Pi/180,expr); #This is added so that theta also has the value updated new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0); SetProperty(MathContainer11,expression,new_expr); end use:   Notice that the code in the Dial and Slider are the same     •  Since the code in the Dial and Slider are the same it makes sense to put the code into a procedure that can be called from multiple places.   Note: The changes in the code such as local and the single quotes are not needed but make the code easier to read and less likely to run into errors if edited in the future (for example if you create a variable called dial8 it won't interfere now that the names are in quotes).     UpdateMath:=proc()  UpdateMath:=proc() local theta_d, phi_s, expr, new_expr, new_expr0; use DocumentTools in theta_d:=GetProperty('Dial8','value'); #Get value of theta from Dial phi_s:=GetProperty('Slider8','value'); #Get value of phi from slider expr:= GetProperty('MathContainer12','expression'); new_expr0:=algsubs('theta'=theta_d*Pi/180,expr); # Put value of theta in expression new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0); # Put value of phi in expression SetProperty('MathContainer13','expression',new_expr); # Update expression end use: end proc:   •  Now change the code in the components to call the function using .   •  Since the code above is only defined there it will need to be run once (but only once) before moving the components. Instead of leaving it here you can add it to the Startup code by clicking  or going to Edit > Startup code.  This code will run every time you open the Math App ensuring that it works right away.   •  The startup code isn't defined in this document to allow progression of these steps.   Make the button initialize the app     •  Since the startup code isn't defined in this document we are going to move this function into the button.   UpdateMath:=proc()   UpdateMath:=proc() local theta_d, phi_s, expr, new_expr, new_expr0; use DocumentTools in theta_d:=GetProperty('Dial9','value'); #Get value of theta from Dial phi_s:=GetProperty('Slider9','value'); #Get value of phi from slider expr:= GetProperty('MathContainer14','expression'); new_expr0:=algsubs('theta'=theta_d*Pi/180,expr); # Put value of theta in expression new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0); # Put value of phi in expression SetProperty('MathContainer15','expression',new_expr); # Update expression end use: end proc: •  First click the button to rename it, you'll see the  option in the context panel on the right. Then add the code above to the button in the same way as the Slider an Dial (Right click and select Edit Click Code).   Now it is easy to add new components     •  Now if we want to add new components we just have to change the one procedure.  Let's add a Volume Gauge to change the value of A.   •  Click in the cell containing the Dial, the context panel will show the option to Insert a row below the Dial. •  Now drag a Volume Gauge into the new cell.   •  Click in the cell and choose the alignment (from the context panel) that looks best to you. In this case I chose center:     Update the procedure code for the Gauge     •  Add two lines for the volume gauge to get the value and sub it into the expression UpdateMath:=proc() UpdateMath:=proc() local theta_d, phi_s, expr, new_expr, new_expr0; use DocumentTools in theta_d:=GetProperty('Dial11','value'); #Get value of theta from the Dial phi_s:=GetProperty('Slider11','value'); #Get value of phi from the Slider A_g:=GetProperty('VolumeGauge1','value'); #Get value of A from the Guage expr:= GetProperty('MathContainer18','expression'); new_expr0:=algsubs('theta'=theta_d*Pi/180,expr); # Put value of theta in expression new_expr1:=algsubs('phi'=phi_s*Pi/180,new_expr0); # Put value of phi in expression new_expr:=algsubs('A'=A_g,new_expr1); # Put value of A in expression SetProperty('MathContainer19','expression',new_expr); # Update expression end use: end proc: •  Now add UpdateMath();   to the Gauge.   Plot the changing expression     •  Make a procedure to get the value in the second Math Container and plot it   PlotMath:=proc() PlotMath:=proc() local expr, p; use DocumentTools in expr:=GetProperty('MathContainer21','expression'); p:=plot(expr,'t'=-Pi/2..Pi/2,'view'=[-Pi/2..Pi/2,-100..100]): SetProperty('Plot14','value',p) end use: end proc: •  Put this procedure in the Initialize button and the call to it in the components.   Tidy up the app     •  Now that we have an interactive app let's tidy it up a bit.   •  The first thing I'd recommend in your own app is moving the code from the initialize button to startup code. In this document we choose to use the button instead to preserve earlier versions.   •  You can also remove the borders around the components by clicking in the table and selecting "Interior Borders" > "None" and "Exterior Borders" > "None" from the context panel. Now you have a Math App     •  You can upload your Math App to the Maple Cloud to share with others by going to "File" > "Save to Cloud".   •  I'd recommend also including a description of what your app does. You can do this nicely using another table and Text mode.

HowToMathApp.mw

Download log_misinterpretation2.mw

One of the forums asked a question: what is the maximum area of a triangle inscribed in a given ellipse x^2/16 + y^2/3 - 1 = 0? It turned out to be 9, but there are infinitely many such triangles. There was a desire to show them in one of the possible ways. This is a complete (as far as possible) set of such triangles.
(This is not an example of Maple programming; it is just an implementation of a Maple-based algorithm and the work of the Optimization package).
MAX_S_TRIAN_ANINATION.mw

This app shows the modeling and simulation of DNA carried out entirely in Maple. The mathematical model is inserted through the combination of trigonometric functions. It shows the graphs of the curvature vs time for its interpretation. Made for engineering and health science students.

MV_AC_R3_UNI_2020.mw

Lenin Araujo Castillo

Ambassador of Maple