Applications, Examples and Libraries

Directional derivatives of complex functions

by: _Maxim_ 724

September 06 2017

1 3

It can be interesting to consider a directional derivative of f(z) in the direction w:

%ddiff(f(z), z, w) = %limit((f(z + w*h) - f(z))/(w*h), h = 0);

ddiff := proc (fz0, z, dir) local rule, fz, dfz, ans;
  dfz := %ddiff(fz, z, dir)*dir;
  rule :=
   [abs(1, fz::anything) = (conjugate(fz)*dfz + fz*conjugate(dfz))/(2*abs(fz)*dfz),
    signum(1, fz::anything) = (conjugate(fz)*dfz - fz*conjugate(dfz))/(2*abs(fz)*conjugate(fz)*dfz)];
  ans := applyrule(rule, diff(fz0, z));
  ans := value(ans);
  ans := [ans, op(convert~(ans, [abs, argument, Re, Im, signum, conjugate]))];
  op(1, sort(simplify(ans, size), length)) end proc;

For analytic functions the derivative is the same in any direction:

ddiff(sqrt(Re(f(z))^2+Im(f(z))^2)*exp(I*argument(f(z))), z, w); # f(z) in disguise
                             d      
                            --- f(z)
                             dz

For non-analytic functions that's no longer the case:

ddiff(conjugate(z), z, w);
                               1     
                           ----------
                                    2
                           signum(w) 

ddiff(conjugate(f(z)), z, 1+I);
                             ________
                              d      
                          -I --- f(z)
                              dz     

ddiff(abs(z), z, z);
                               1    
                           ---------
                           signum(z)

ddiff(ln(abs(z)), z, z);
                               1
                               -
                               z

Some of those derivates have simple geometric interpretations: the derivative of argument(z) in the direction z is zero, since argument(z) doesn't change when moving in the direction z from the point z; the derivative of abs(z) in the direction I*z is zero, since the direction is tangent to the circle abs(z)=constant; since signum(z) is a function of argument(z) only, its derivative in the direction z is zero as well.

Interestingly, the derivative taken twice in the direction z is zero for each of the six basic functions:

map(fz -> ddiff(ddiff(fz, z, z), z, z), [abs, argument, Re, Im, signum, conjugate](z));
                       [0, 0, 0, 0, 0, 0]

Does that have some simple geometric interpretation as well?

Displacement and distance traveled with vectors

by: Lenin Araujo Castillo 1790 Product: Maple 18

August 28 2017

2 2

In this app you can use from the creation of curve, birth of the position vector and finally applied to the displacement and the distance traveled. All this application revolves around the creation of a path and the path of a particle over this generated by vectors. You will only have to insert the vector components and the times to evaluate. Designed for engineering students guided through Maple. In Spanish.

Displacement_and_distance_traveled_with_vectors.mw Updated

Displacement_and_distance_traveled_with_vectors_updated_2020.mw

Video

https://www.youtube.com/watch?v=jOcKYZ5EEM0

Lenin Araujo C

Ambassador of Maple

Shortcut in Vectors for Engineering

by: Lenin Araujo Castillo 1790 Product: Maple 18

August 25 2017

2 0

Here in this video you can observe the correct insertion of vectors; Making use of the keyboard, ascii code and tool palette of our Maple program. As our worksheet is very large, I made the explanation in two parts; I recommend that you observe this first part of performing any execution on your Maple worksheet. You can contrast your results with the apps also made in this software. In Spanish.

Shortcut_in_Vectors_for_Engineering.mw

Movie # 01

https://www.youtube.com/watch?v=EJtAli54q_A

Movie # 02

https://www.youtube.com/watch?v=m-JUmhkbWI8

Lenin Araujo Castillo

Ambassador of Maple

Kitonum's IntOverDomain Deconstructed

by: rlopez 2985 Product: Maple

August 24 2017

8 6

On 5/July/2017, Kitonum responded to the 3/July/2017 MaplePrimes question "How to perform double integration over subdomain" by providing code for a procedure IntOverDomain that implements Green's theorem applied to a planar region whose boundary is a simple, closed, rectifiable, oriented curve (SCROC by some authors).

I was intrigued. First, this is a significant extension of existing Maple functionalities. Second, the implementation admits boundaries defined piecewise with sections defined parametrically; or sections that are polygonal lines defined by a list of nodes.

But how was the line integral around such boundaries coded? In the worksheet "IntOverDomain_Deconstructed," I summarize the existing Maple functionality for implementing iterated double integrals over specified domains, then analyze how Kitonum coded Green's theorem as an extension of Maple's capabilities. After recognizing the great coding skills of Kitonum, I conclude with a short wishlist of related extensions that I would like to see added to Maple in the future.

Download the worksheet: IntOverDomain_Deconstructed.mw

A new and efficient code in Maple for Boundary...

by: Dr. Venkat Subramanian 386 Product: Maple

August 23 2017

4 10

A new code based on higher derivative method has been implemented in Maple. A sample code is given below and explained. Because of the symbolic nature of Maple, this method works very well for a wide range of BVP problems.

The code solves BVPs written in the first order form dy/dx = f (Maple’s dsolve numeric converts general BVPs to this form and solves).

The code can handle unknown parameters in the model if sufficient boundary conditions are provided.

This code has been tested from Maple 8 to Maple 2017. For Digits:=15 or less, this code works in all of the Maple versions tested.

Most problems can be solved with Digits:=15 with atol = 1e-10 or so. This code can be used to get a tolerance value of 1e-20 or any high precision as needed by changing the number of Digits accordingly. This may be needed if the original variables are not properly sacled. With arbitrarily high Digits, the code fails in Maple 18 or later version, etc because Maple does not support SparseDirect Solver at high precision in some of the versions (hopefuly this bug can be removed in the future versions).

For simple problems, Maple’s dsolve/numeric is superior to the code developed as it is implemented in hardware floats. For large scale problems and stiff problems, the method developed is much more superior to Maple and comparable to (and often times better than) state of the art codes for BVPs - bvp4c (MATLAB), COLSYS,TWPBVP, etc.

The code, as written, cannot be used for problems with a singularity at end points (doable in the future). In addition, mixed boundary conditions are not supported in this version of the code (for example, y1(1)=y2(0)). Future updates will include the application of this approach for DAE-BVPs, currently not supported by Maple’s dsolve/numeric command.

A paper has been submitted to JCAM. I welcome feedback on the code and solicit input from Mapleprimes members if they are able to test (and break this code) for any BVP.

PDF of the paper submitted, example maple code and the solver as a text file needed are uploaded here. Additional examples are hosted on my website at http://depts.washington.edu/maple/HDM.html

>

##################################################################################

Troesch's problem
This is an inherently unstable, difficult, nonlinear, two-point BVP formulated by Weibel and Troesch that describes the confinement of a place column by radiation pressure. Increasing epsilon increases the stiffness of the ODE.
1. E.S. Weibel, On the confinement of a plasma by magnetostatic fields, Phys. Fluids. 2 (1959) 52-56.
2. B. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys. 21 (1976) 279-290.

Introduction
The package HDM solves boundary value problems (BVPs) using higher derivative methods (HDM) in Maple®. We explain how to solve BVPs using this package. HDM can numerically solve BVPs of ordinary differential equations (ODEs) of the form shown is the fowllowing example.

###################################################################################

>

Reset the program to clear the memory from previous execution command.

>

restart:

>

Read the txt file which contains the HDM solver for BVPs.

>	read("HDM.txt");

>

Declare the precision for the entire Maple® sheet.

>	Digits:=15;

(1)

>

Enter the first-order ODEs into EqODEs list.

>	EqODEs:=[diff(y1(x),x)=y2(x),diff(y2(x),x)=epsilonsinh(epsilony1(x))];

(2)

>

Define the left boundary condition (bc1), and the right boundary condition (bc2). One should collect all the terms in one side.

>	bc1:=evalf([y1(x)]);

(3)

>	bc2:=evalf([y1(x)-1]);

(4)

>

Define the range (bc1 to bc2) of this BVP.

>	Range:=[0.,1.];

(5)

>

List any known parameters in the list.

>	pars:=[epsilon=2];

(6)

>

List any unknown parameters in the list. When there is no unknown parameter, use [ ].

>	unknownpars:=[];

(7)

>

Define the initial derivative in nder (default is 5 for 10th order) and the number of the nodes in nele (default is 10 and distributed evenly across the range provided by the user). The code adapts to increase the order. For many problems, 10th order method with 10 elements are sufficient.

>	nder:=5;nele:=10;

(8)

>

Define the absolute and relative tolerance for the local error. The error calculation is done based on the norm of both the 9th and 10th order simulation results.

>	atol:=1e-6;rtol:=atol/100;

(9)

>

Call HDMadapt procedure, input all the information entered above and save the solution in sol. HDMadapt procedure does not need the initial guess for the mesh.

>	sol:= HDMadapt(EqODEs,bc1,bc2,pars,unknownpars,nder,nele,Range,atol,rtol):

>

Present some details of the solution.

>	sol[4]; # final derivative

(10)

>	sol[5]; # Maximum local RMSE

(11)

>

Store the dimension of the solution (after adjusting the mesh) to NN.

>	NN:=nops(sol[3])+1;

(12)

>

Plot the interested variable (the ath ODE variable will be sol[1][i+NN*(a-1)] )

>	node:=nops(EqODEs); odevars:=select(type,map(op,map(lhs,EqODEs)),'function');

(13)

>	xx:=Vector(NN):

>	xx[1]:=Range[1]:

>	for i from 1 to nops(sol[3]) do xx[i+1]:=xx[i]+sol[3][i]: od:

>	for j from 1 to node do plot([seq([xx[i],rhs(sol[1][i+NN*(j-1)])],i=1..NN)],axes=boxed,labels=[x,odevars[j]],style=point); end do;

>

Download Example_3_Troesch.mws

The patch package project

by: Christopher2222 5785 Product: Maple

August 19 2017

0 4

Was just pondering this idea and posted this in the post topic for discussion.

Each Maple finished version of Maple may still have certain bugs that will not be updated for that version, so I am suggesting (I think anyone could implement it) that if there is a workaround, one could wrap it up in something I would call a patch package updateable by us users we could update here on mapleprimes. It would be good for people who haven't upgraded or can't upgrade due to costs etc...

For example, there was recent issue with pdsolve that was fixed quite quickly in the seperate updateable Physics package. Things could be done similarily that might work with other workarounds using this patch package idea.

If anyone thinks this is good or even viable idea then lets implement it. I envisioned it with just this one rule to follow - the name of the patch package would reflect the version we are patching (ie. with(patch12) or with(patch2016) for Maple 12 and Maple 2016 respectively etc...) We could make these patch packages available in this post or start another.

As I said, I'm just throwing the idea out there. Thoughts?

Plot of Position Vector in Maple

by: Lenin Araujo Castillo 1790 Product: Maple 18

August 07 2017

2 0

As you can see this app performs the trace of a given path r (t), then locate the position vector in a specific time. It also graphs the velocity vector, acceleration, Tangential and Normal unit vectors, along with the Binormal. Very good app developed entirely in Maple for our engineering students.

Plot_of_Position_Vector_UPDATED.mw

https://youtu.be/OzAwShHHXq8

Lenin Araujo Castillo

Ambassador of Maple

ACA 2017 - The Appell doubly hypegeometric Functio...

by: ecterrab 13431 Product: Maple

August 02 2017

3 0

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fifth presentation, about "The Appell doubly hypergeometric functions", describes a very recent project I've been working at Maple, i.e. the very first complete computational implementation of the Appell doubly hypergeometric functions. This work appeared in Maple 2017. These functions have a tremendous potential in that, at the same time, they have a myriad of properties, and include as particular cases most of the existing mathematical language, and so they have obvious applications in integration, differential equations, and applied mathematics all around. I think these will be the functions of this XXI century, analogously to what happened with hypergeometric functions in the previous century.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

The four double-hypergeometric Appell functions,

a complete implementation in a computer algebra system

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:
The four multi-parameter Appell functions, AppellF1 , AppellF2 , AppellF3 and AppellF4 are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric and some cases of the MeijerG function, and with them most of the known functions of mathematical physics. Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity. In this talk, a full implementation of these functions in the Maple computer algebra system, including, for the first time, their numerical evaluation over the whole complex plane, is presented, with details about the symbolic and numerical strategies used.

Appell Functions (symbolic)

The main references:

•	P. Appel, J.Kamke de Feriet, "Fonctions hypergeometriques et Hyperspheriques", 1926

•	H. Srivastava, P.W. Karlsson, "Multiple Gaussian Hypergeometric Series", 1985

•	24 papers in the literature, ranging from 1882 to 2015

	Definition and Symmetries

	Polynomial and Singular Cases

	Single Power Series with Hypergeometric Coefficients

	Analytic Extension from the Appell Series to the Appell Functions

	Euler-Type and Contiguity Identities

	Appell Differential Equations

	Putting all together

	Problem: some formulas in the literature are wrong or miss the conditions indicating when are they valid (exchange with the Mathematics director of the DLMF - NIST)

Appell Functions (numeric)

Goals

•	Compute these Appell functions over the whole complex plane

•	Considering that this is a research problem, implement different methods and flexible optional arguments to allow for: a) comparison between methods (both performance and correctness), b) investigation of a single method in different circumstances.

•	Develop a computational structure that can be reused with other special functions (abstract code and provide the main options), and that could also be translated to C (so: only one numerical implementation, not 100 special function numerical implementations)

Limitation: the Maple original evalf command does not accept optional arguments

The cost of numerically evaluating an Appell function

•	If it is a special hypergeometric case, then between 1 to 2 hypergeometric functions

•	Next simplest case (series/recurrence below) 3 to 4 hypergeometric functions plus adding somewhat large formulas that involve only arithmetic operations up to 20,000 times (frequently less than 100 times)

•	Next simplest case: the formulas themselves are power series with hypergeometric function coefficients; these cases frequently converge rapidly but may involve the numerical evaluation of up to hundreds of hypergeometric functions to get the value of a single Appell function.

Strategy for the numerical evaluation of Appell functions (or other functions ...)

The numerical evaluation flows orderly according to:

1) check whether it is a singular case

2) check whether it is a special value

3) compute the value using a series derived from a recurrence related to the underlying ODE

4) perform an sum using an infinite sum formula, checking for convergence

5) perform the numerical integration of the ODE underlying the given Appell function

6) perform a sequence of concatenated Taylor series expansions

	Examples

	Series/recurrence

	Numerical integration of an underlying differential equation (ODEs and dsolve/numeric)

	Concatenated Taylor series expansions covering the whole complex plane

Subproducts

	Improvements in the numerical evaluation of hypergeometric functions

	Evalf: an organized structure to implement the numerical evaluation of special functions in general

	To be done

Download Appell_Functions.mw
Download Appell_Functions.pdf

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

ACA 2017 - The FunctionAdvisor, beyond a database...

by: ecterrab 13431 Product: Maple

August 02 2017

2 1

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fourth presentation, about "The FunctionAdvisor: extending information on mathematical functions with computer algebra algorithms", describes the FunctionAdvisor project at Maple, a project I started working during 1998, where the key idea I am trying to explore is that we do not need to collect a gazillion of formulas but just core blocks of mathematical information surrounded by clouds of algorithms able to derive extended information from them. In this sense this is also unique piece of software: it can derive properties for rather general algebraic expressions, not just well known tabulated functions. The examples illustrate the idea.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

The FunctionAdvisor: extending information on mathematical functions

with computer algebra algorithms

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:

A shift in paradigm is happening, from: encoding information into a database, to: encoding essential blocks of information together with algorithms within a computer algebra system. Then, the information is not only searchable but can also be recreated in many different ways and actually used to compute. This talk focuses on this shift in paradigm over a real case example: the digitizing of information regarding mathematical functions as the FunctionAdvisor project of the Maple computer algebra system.

	The FunctionAdvisor (basic)

Beyond the concept of a database

" Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

	Formal power series for algebraic expressions

	Differential polynomial forms for algebraic expressions

	Branch cuts for algebraic expressions

	The n^th derivative problem for algebraic expressions

	Conversion network for mathematical and algebraic expressions

	References

Download FunctionAdvisor.mw

Download FunctionAdvisor.pdf

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

ACA 2017 - Computer Algebra in Theoretical Physics...

by: ecterrab 13431 Product: Maple

August 02 2017

5 1

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This third presentation, about "Computer Algebra in Theoretical Physics", describes the Physics project at Maplesoft, also my first research project at University, that evolved into the now well-known Maple Physics package. This is a unique piece of software and perhaps the project I most enjoy working.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

Computer Algebra in Theoretical Physics

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:

Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system have implemented most of the mathematical objects and mathematics used in theoretical physics computations, and have dramatically approximated the notation used in the computer to the one used with paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum.

In this talk, the Physics project at Maplesoft is presented and the resulting Physics package is illustrated by tackling problems in classical and quantum mechanics, using tensor and Dirac's Bra-Ket notation, general relativity, including the equivalence problem, and classical field theory, deriving field equations using variational principles.

... and why computer algebra?

We can concentrate more on the ideas instead of on the algebraic manipulations

We can extend results with ease

We can explore the mathematics surrounding a problem

We can share results in a reproducible way

	Representation issues that were preventing the use of computer algebra in Physics

Classical Mechanics

	*Inertia tensor for a triatomic molecule

Quantum mechanics

	*The quantum operator components of satisfy

*Unitary Operators in Quantum Mechanics

	*Eigenvalues of an unitary operator and exponential of Hermitian operators

*Properties of unitary operators

Consider two set of kets and , each of them constituting a complete orthonormal basis of the same space.

	*Verify that , maps one basis to the other, i.e.:

	*Show that is unitary

	*Show that the matrix elements of in the and basis are equal

	Show that and have the same spectrum (eigenvalues)

	Schrödinger equation and unitary transform

	Translation operators using Dirac notation

	*Quantization of the energy of a particle in a magnetic field

Classical Field Theory

	The field equations for the model

	*Maxwell equations departing from the 4-dimensional Action for Electrodynamics

	*The Gross-Pitaevskii field equations for a quantum system of identical particles

General Relativity

	Exact Solutions to Einstein's Equations

	*"Physical Review D" 87, 044053 (2013)

	The Equivalence problem between two metrics

	*On the 3+1 split of the 4D Einstein equations

	Tetrads and Weyl scalars in canonical form

Download Physics.mw

Download Physics.pdf

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

ACA 2017 - Differential Algebra for an extended...

by: ecterrab 13431

August 02 2017

3 0

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This second presentation, about "Differential algebra with mathematical functions, symbolic powers and anticommutative variables", describes a project I started working in 1997 and that is at the root of Maple's dsolve and pdsolve performance with systems of equations. It is a unique approach. Not yet emulated in any other computer algebra system.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

Differential algebra with mathematical functions,

symbolic powers and anticommutative variables

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:
Computer algebra implementations of Differential Algebra typically require that the systems of equations to be tackled be rational in the independent and dependent variables and their partial derivatives, and of course that , everything is commutative.

It is possible, however, to extend this computational domain and apply Differential Algebra techniques to systems of equations that involve arbitrary compositions of mathematical functions (elementary or special), fractional and symbolic powers, as well as anticommutative variables and functions. This is the subject of this presentation, with examples of the implementation of these ideas in the Maple computer algebra system and its ODE and PDE solvers.

>

(1)

>

(2)

>

(3)

>

	Differential polynomial forms for mathematical functions (basic)

	Differential polynomial forms for compositions of mathematical functions

	Generalization to many variables

	Arbitrary functions of algebraic expressions

	Examples of the use of this extension to include mathematical functions

	Differential Algebra with anticommutative variables

Download DifferentialAlgebra.mw

Download DifferentialAlgebra.pdf

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

ACA 2017 - Computer Algebra in High-School Educati...

by: ecterrab 13431 Product: Maple

August 02 2017

3 0

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This first presentation, about "Active Learning in High-School Mathematics using Interactive Interfaces", describes a project I started working 23 years ago, which I believe will be part of the future in one or another form. This is work actually not related to my work at Maplesoft.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

Active learning in High-School mathematics using Interactive Interfaces

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:

The key idea in this project is to learn through exploration using a web of user-friendly Highly Interactive Graphical Interfaces (HIGI). The HIGIs, structured as trees of interlinked windows, present concepts using a minimal amount of text while maximizing the possibility of visual and analytic exploration. These interfaces run computer algebra software in the background. Assessment tools are integrated into the learning experience within the general conceptual map, the Navigator. This Navigator offers students self-assessment tools and full access to the logical sequencing of course concepts, helping them to identify any gaps in their knowledge and to launch the corresponding learning interfaces. An interactive online set of HIGIS of this kind can be used at school, at home, in distance education, and both individually and in a group.

	Computer algebra interfaces for High-School students of "Colegio de Aplicação" (UERJ/1994)

Motivation

When we are the average high-school student facing mathematics, we tend to feel

•	Bored, fragmentarily taking notes, listening to a teacher for 50 or more minutes

•	Anguished because we do not understand some math topics (too many gaps accumulated)

•	Powerless because we don't know what to do to understand (don't have any instant-tutor to ask questions and without being judged for having accumulated gaps)

•	Stressed by the upcoming exams where the lack of understanding may become evident

Computer algebra environments can help in addressing these issues.

•	Be as active as it can get while learning at our own pace.

•	Explore at high speed and without feeling judged. There is space for curiosity with no computational cost.

•	Feel empowered by success. That leads to understanding.

•	Possibility for making of learning a social experience.

Interactive interfaces

Interactive interfaces do not replace the teacher - human learning is an emotional process. A good teacher leading good active learning is a positive experience a student will never forget

Not every computer interface is a valuable resource, at all. It is the set of pedagogical ideas implemented that makes an interface valuable (the same happens with textbooks)

A course on high school mathematics using interactive interfaces - the Edukanet project

–	Brazilian and Canadian students/programmers were invited to participate - 7 people worked in the project.

–	Some funding provided by the Brazilian Research agency CNPq.

Tasks:

-Develop a framework to develop the interfaces covering the last 3 years of high school mathematics (following the main math textbook used in public schools in Brazil)

- Design documents for the interfaces according to given pedagogical guidelines.

- Create prototypes of Interactive interfaces, running Maple on background, according to design document and specified layout (allow for everybody's input/changes).

	The pedagogical guidelines for interactive interfaces

The Math-contents design documents for each chapter

	Example: complex numbers

Each math topic: a interactive interrelated interfaces (windows)

For each topic of high-school mathematics (chapter of a textbook), develop a tree of interactive interfaces (applets) related to the topic (main) and subtopics

Example: Functions

•	Main window

•	Analysis window

•

•	Parity window

•	Visualization of function's parity

•	Step-by-Step solution window

The Navigator: a window with a tile per math topic

•	Click the topic-tile to launch a smaller window, topic-specific, map of interrelated sub-topic tiles, that indicates the logical sequence for the sub-topics, and from where one could launch the corresponding sub-topic interactive interface.

•	This topic-specific smaller window allows for identifying the pre-requisites and gaps in understanding, launching the corresponding interfaces to fill the gaps, and tracking the level of familiarity with a topic.

	The framework to create the interfaces: a version of NetBeans on steroids ...

Complementary classroom activity on a computer algebra worksheet

This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do with paper and pencil in high school and 1st year of undergraduate science courses.

Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the learning of mathematics into interesting understanding, success and fun.

	1. Arithmetic operations and elementary functions

	2. Algebraic Expressions, Equations and Functions

	3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations

	4. Algebraic manipulation: simplify, factorize, expand

	5. Matrices (Linear Algebra)

	Advanced students: guiding them to program mathematical concepts on a computer algebra worksheet

Status of the project

Prototypes of interfaces built cover:

•	Natural numbers

•

Functions

•	Integer numbers

•	Rational numbers

•	Absolute value

•	Logarithms

•	Numerical sequences

•	Trigonometry

•

Matrices

•	Determinants

•	Linear systems

•

Limits

•	Derivatives

•	Derivative of the inverse function

•	The point in Cartesian coordinates

•

The line

•	The circle

•	The ellipse

•	The parabole

•	The hyperbole

•	The conics

	More recent computer algebra frameworks: Maple Mobius for online courses and automated evaluation

Download Computer_Algebra_in_Education.mw

Download Computer_Algebra_in_Education.pdf

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

A geometric construction for the Summer Holiday

by: vv 12453 Product: Maple

August 02 2017

1 2

A geometric construction for the Summer Holiday

Does every plane simple closed curve contain all four vertices of some square?
This is an old classical conjecture. See:
https://en.wikipedia.org/wiki/Inscribed_square_problem

Maybe someone finds a counterexample (for non-analytic curves) using the next procedure and becomes famous!

>

SQ:=proc(X::procedure, Y::procedure, rng::range(realcons), r:=0.49)
local t1:=lhs(rng), t2:=rhs(rng), a,b,c,d,s;
s:=fsolve({ X(a)+X(c) = X(b)+X(d),
            Y(a)+Y(c) = Y(b)+Y(d),
            (X(a)-X(c))^2+(Y(a)-Y(c))^2 = (X(b)-X(d))^2+(Y(b)-Y(d))^2,
            (X(a)-X(c))*(X(b)-X(d)) + (Y(a)-Y(c))*(Y(b)-Y(d)) = 0},
          {a=t1..t1+r*(t2-t1),b=rng,c=rng,d=t2-r*(t2-t1)..t2}); #lprint(s);
if type(s,set) then s:=rhs~(s)[];[s,s[1]] else WARNING("No solution found"); {} fi;
end:

Example

>	X := t->(10-sin(7t)exp(-t))cos(t); Y := t->(10+sin(6t))sin(t); rng := 0..2Pi;

(1)

>	s:=SQ(X, Y, rng): plots:-display( plot([X,Y,rng], scaling=constrained), plot([seq( eval([X(t),Y(t)],t=u),u=s)], color=blue, thickness=2));

Google Maps and Geocoding for Maple

by: Samir Khan 1854 Product: Maple

July 28 2017

10 9

As a momentary diversion, I threw together a package that downloads map images into Maple using the Google Static Maps API.

If you have Maple 2017, you can install the package using the MapleCloud Package Manager or by executing PackageTools:-Install("5769608062566400").

This worksheet has several examples, but I thought I'd share a few below .

Here's the Maplesoft office

Let's view a roadmap of Waterloo, Ontario.

The package features over 80 styles for roadmaps. These are examples of two styles (the second is inspired by the art of Piet Mondrian and the De Stijl movement)

You can also find the longitude and latitude of a location (courtesy of Google's Geocoding API). Maple returns a nested list if it finds multiple locations.

The geocoding feature can also be used to add points to Maple 2017's built-in world maps.

Let me know what you think!

Tangent plane as the square at any point of a...

by: one man 1140 Product: Maple 15

July 28 2017

3 18

The representation of the tangent plane in the form of a square with a given length of the side at any point on the surface.

The equation of the tangent plane to the surface at a given point is obtained from the condition that the tangent plane is perpendicular to the normal vector. With the aid of any auxiliary point not lying on this normal to the surface, we define the direction on the tangent plane. From the given point in this direction, we lay off segments equal to half the length of the side of our square and with the help of these segments we construct the square itself, lying on the tangent plane with the center at a given point.

An examples of constructing tangent planes at points of the same intersection line for two surfaces.
Tangent_plane.mw

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