Applications, Examples and Libraries

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My interface has frozen, but above is a screen shot of what is by far the most unusual response from the CAS in the i guess 8 or so years ive been using it in total

The development of the calculation of moments using force vectors is clearly observed by taking a point and also a line. Different exercises are solved with the help of Maple syntax. We can also visualize the vector behavior in the different configurations of the position vector. Applications designed exclusively for engineering students. In Spanish.

Moment_of_a_force_using_vectors.mw

Lenin Araujo Castillo

Ambassador of Maple

A project that I have been working on is adding some functionality for Cluster Analysis to Maple (a small part of a much bigger project to increase Maple’s toolkit for exploratory data mining and data analysis). The launch of the MapleCloud package manager gave me a way to share my code for the project as it evolves, providing others with some useful new tools and hopefully gathering feedback (and collaborators) along the way.

At this point, there aren’t a lot of commands in the ClusterAnalysis package, but I have already hit upon several interesting applications. For example, while working on a command for plotting clusters of points, one problem I encountered was how to draw the minimal volume enclosing ellipsoid around a group (or cluster) of points. After doing some research, I stumbled upon Khachiyan’s Algorithm, which related to solving linear programming problems with rational data. The math behind this is definitely interesting, but I’m not going to spend any time on it here. For further reading, you can explore the following:

Khachiyan’s Algorithm had previously been applied in some other languages, but to the best of my knowledge, did not have any Maple implementations. As such, the following code is an implementation of Khachiyan’s Algorithm in 2-D, which could be extended to N-dimensional space rather easily.

This routine accepts an Nx2 dataset and outputs either a plot of the minimum volume enclosing ellipsoid (MVEE) or a list of results as described in the details for the ‘output’ option below.

MVEE( X :: DataSet, optional arguments, additional arguments passed to the plotting command );

The optional arguments are as follows:

  • tolerance : realcons;  specifies the convergence criterion
  • maxiterations : posint; specifies the maximum number of iterations
  • output : {identical(data,plot),list(identical(data,plot))}; specifies the output. If output includes plot, then a plot of the enclosing ellipsoid is returned. If output includes data, then the return includes is a list containing the matrix A, which defines the ellipsoid, the center of the ellipse, and the eigenvalues and eigenvectors that can be used to find the semi-axis coordinates and the angle of rotation, alpha, for the ellipse.
  • filled : truefalse; specifies if the returned plot should be filled or not

Code:

#Minimum Volume Enclosing Ellipsoid
MVEE := proc(XY, 
              {tolerance::positive:= 1e-4}, #Convergence Criterion
              {maxiterations::posint := 100},
              {output::{identical(data,plot),list(identical(data,plot))} := data},
              {filled::truefalse := false} 
            )

    local alpha, evalues, evectors, i, l_error, ldata, ldataext, M, maxvalindex, n, ncols, nrows, p1, semiaxes, stepsize, U, U1, x, X, y;
    local A, center, l_output; #Output

    if hastype(output, 'list') then
        l_output := output;
    else
        l_output := [output];
    end if;

    kernelopts(opaquemodules=false):

    ldata := Statistics:-PreProcessData(XY, 2, 'copy');

    nrows, ncols := upperbound(ldata);
    ldataext := Matrix([ldata, Vector[column](nrows, ':-fill' = 1)], 'datatype = float');

    if ncols <> 2 then
        error "expected 2 columns of data, got %1", ncols;
    end if;

    l_error := 1;

    U := Vector[column](1..nrows, 'fill' = 1/nrows);

    ##Khachiyan Algorithm##
    for n to maxiterations while l_error >= tolerance do

        X := LinearAlgebra:-Transpose(ldataext) . LinearAlgebra:-DiagonalMatrix(U) . ldataext;
        M := LinearAlgebra:-Diagonal(ldataext . LinearAlgebra:-MatrixInverse(X) . LinearAlgebra:-Transpose(ldataext));
        maxvalindex := max[index](map['evalhf', 'inplace'](abs, M));
        stepsize := (M[maxvalindex] - ncols - 1)/((ncols + 1) * (M[maxvalindex] - 1));
        U1 := (1 - stepsize) * U;
        U1[maxvalindex] := U1[maxvalindex] + stepsize;
        l_error := LinearAlgebra:-Norm(LinearAlgebra:-DiagonalMatrix(U1 - U));
        U := U1;

    end do;

    A := (1/ncols) * LinearAlgebra:-MatrixInverse(LinearAlgebra:-Transpose(ldata) . LinearAlgebra:-DiagonalMatrix(U) . ldata - (LinearAlgebra:-Transpose(ldata) . U) . LinearAlgebra:-Transpose((LinearAlgebra:-Transpose(ldata) . U)));
    center := LinearAlgebra:-Transpose(ldata) . U;
    evalues, evectors := LinearAlgebra:-Eigenvectors(A);
    evectors := evectors(.., sort[index](1 /~ (sqrt~(Re~(evalues))), `>`, ':-output' = ':-permutation'));
    semiaxes := sort(1 /~ (sqrt~(Re~(evalues))), `>`);
    alpha := arctan(Re(evectors[2,1]) / Re(evectors[1,1]));

    if l_output = [':-data'] then
        return A, center, evectors, evalues;
    elif has( l_output, ':-plot' ) then
            x := t -> center[1] + semiaxes[1] * cos(t) * cos(alpha) - semiaxes[2] * sin(t) * sin(alpha);
            y := t -> center[2] + semiaxes[1] * cos(t) * sin(alpha) + semiaxes[2] * sin(t) * cos(alpha);
            if filled then
                p1 := plots:-display(subs(CURVES=POLYGONS, plot([x(t), y(t), t = 0..2*Pi], ':-transparency' = 0.95, _rest)));
            else
                p1 := plot([x(t), y(t), t = 0..2*Pi], _rest);
            end if;
        return p1, `if`( has(l_output, ':-data'), op([A, center, evectors, evalues]), NULL );
    end if;

end proc:

 

You can run this as follows:

M:=Matrix(10,2,rand(0..3)):

plots:-display([MVEE(M,output=plot,filled,transparency=.3),
                plots:-pointplot(M, symbol=solidcircle,symbolsize=15)],
size=[0.5,"golden"]);

 

 

As it stands, this is not an export from the “work in progress” ClusterAnalysis package – it’s actually just a local procedure used by the ClusterPlot command. However, it seemed like an interesting enough application that it deserved its own post (and potentially even some consideration for inclusion in some future more geometry-specific package). Here’s an example of how this routine is used from ClusterAnalysis:

with(ClusterAnalysis);

X := Import(FileTools:-JoinPath(["datasets/iris.csv"], base = datadir));

kmeans_results := KMeans(X[[`Sepal Length`, `Sepal Width`]],
    clusters = 3, epsilon = 1.*10^(-7), initializationmethod = Forgy);

ClusterPlot(kmeans_results, style = ellipse);

 

 

The source code for this is stored on GitHub, here:

https://github.com/dskoog/Maple-ClusterAnalysis/blob/master/src/MVEE.mm

Comments and suggestions are welcomed.

 

If you don’t have a copy of the ClusterAnalysis package, you can install it from the MapleCloud window, or by running:

PackageTools:-Install(5629844458045440);

 

These worksheets provide the volume calculations  of a small causal diamond near the tip of the past light cone, using dimensional analysis and particular test metrics.

I recommend them for anyone working in causet theory on the problem of finding higher order corrections.

2D.mw

4D.mw

4Dflat.mw

 

 

 

 

With this app you will be able to interpret the curvatures generated by two position vectors, either in the plane or in space. Just enter the position vectors and drag the slider to calculate the curvature at different times and you will of course be able to observe its respective graph. At first I show you how it is developed using the natural syntax of Maple and then optimize our
 app with the use of buttons. App made in Maple for engineering students. In spanish.

Plot_of_Curvature.mw

Videotutorial:

https://www.youtube.com/watch?v=SbXFgr_5JDE

Lenin Araujo Castillo

Ambassador of Maple

This is Maple:

These are some primes:

22424170499, 106507053661, 193139816479, 210936428939, 329844591829, 386408307611,
395718860549, 396412723027, 412286285849, 427552056871, 454744396991, 694607189303,
730616292977, 736602622363, 750072072203, 773012980121, 800187484471, 842622684461

This is a Maple prime:


In plain text (so you can check it in Maple!) that number is:

111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111116000808880608061111111111111111111111111111111
111111111111111111111111111866880886008008088868888011111111111111111111111111
111111111111111111111116838888888801111111188006080011111111111111111111111111
111111111111111111110808080811111111111111111111111118860111111111111111111111
111111111111111110086688511111111111111111111111116688888108881111111111111111
111111111111111868338111111111111111111111111111880806086100808811111111111111
111111111111183880811111111111111111100111111888580808086111008881111111111111
111111111111888081111111111111111111885811188805860686088111118338011111111111
111111111188008111111111111111111111888888538888800806506111111158500111111111
111111111883061111111111111111111116580088863600880868583111111118588811111111
111111118688111111111001111111111116880850888608086855358611111111100381111111
111111160831111111110880111111111118080883885568063880505511111111118088111111
111111588811111111110668811111111180806800386888336868380511108011111006811111
111111111088600008888688861111111108888088058008068608083888386111111108301111
111116088088368860808880860311111885308508868888580808088088681111111118008111
111111388068066883685808808331111808088883060606800883665806811111111116800111
111581108058668300008500368880158086883888883888033038660608111111111111088811
111838110833680088080888568608808808555608388853680880658501111111111111108011
118008111186885080806603868808888008000008838085003008868011111111111111186801
110881111110686850800888888886883863508088688508088886800111111111111111118881
183081111111665080050688886656806600886800600858086008831111111111111111118881
186581111111868888655008680368006880363850808888880088811111111111111111110831
168881111118880838688806888806880885088808085888808086111111111111111111118831
188011111008888800380808588808068083868005888800368806111111111111111111118081
185311111111380883883650808658388860008086088088000868866808811111111111118881
168511111111111180088888686580088855665668308888880588888508880800888111118001
188081111111111111508888083688033588663803303686860808866088856886811111115061
180801111111111111006880868608688080668888380580080880880668850088611111110801
188301111111111110000608808088360888888308685380808868388008006088111111116851
118001111111111188080580686868000800008680805008830088080808868008011111105001
116800111111118888803380800830868365880080868666808680088685660038801111180881
111808111111100888880808808660883885083083688883808008888888386880005011168511
111688811111111188858888088808008608880856000805800838080080886088388801188811
111138031111111111111110006500656686688085088088088850860088888530008888811111
111106001111111111111111110606880688086888880306088008088806568000808508611111
111118000111111111111111111133888000508586680858883868000008801111111111111111
111111860311111111111111111108088888588688088036081111860803011111111863311111
111111188881111111111111111100881111160386085000611111111888811111108833111111
111111118888811111111111111608811111111188680866311111111111811111888861111111
111111111688031111111111118808111111111111188860111111111111111118868811111111
111111111118850811111111115861111111111111111888111111111111111080861111111111
111111111111880881111111108051111111111111111136111111111111188608811111111111
111111111111116830581111008011111111111111111118111111111116880601111111111111
111111111111111183508811088111111111111111111111111111111088880111111111111111
111111111111111111600010301111111111111111111111111111688685811111111111111111
111111111111111111111110811801111111111111111111158808806881111111111111111111
111111111111111111111181110888886886338888850880683580011111111111111111111111
111111111111111111111111111008000856888888600886680111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111

This is a 3900 digit prime number. It took me about 400 seconds of computation to find using Maple.

It turns out be be really easy to do because prime numbers are realy quite common.  If you have a piece of ascii art where all the characters are numerals, you could just call on it and get a prime number that is still ascii art with a couple digits in the corner messed up (for a number this size, I expect fewer than 10 of the least significant digits would be altered).  You may notice, however, that my Maple Prime has beautiful corners!  This is possible because I found the prime in a slightly different way.

To get the ascii art in Maple, I started out by using to import ( )  and process the original image.  First then and to get a nice 78 pixel wide image.  Then to make it a pure 1-bit black or white image.

Then, from the image, I create a new Array of the decimal digits of the ascii art and my prime number.  For each of the black pixels I randomly use one of the digits or and for the white pixels (the background) I use 's.  Now I convert the Array to a large integer and test if it is prime using (it probably isn't) so, I just randomly change one of the black pixels to a different digit (there are 4 other choices) and call again. For the Maple Prime I had to do this about 1000 times before I landed on a prime number. That was surprisingly fast to me! It is a great object lesson in how dense the prime numbers really are.

So that you can join the fun without having to replicate my work, here is a small interactive Maple document that you can use to find prime numbers that draw ascii art of your source images. It has a tool that lets you preview both the pixelated image and the initial ascii art before you launch the search for the prime version.

Prime_from_Picture.mw

With this application you will learn the beginning of the study of the vectors. Graphing it in a vector space from the plane to the space. You can calculate its fundamental characteristics as triangle laws, projections and strength. App made entirely in Maple for engineering students so they can develop their exercises and save time. It is recommended to first use the native syntax then the embedded components. In Spanish.

Vector_space_with_projections_and_forces_UPDATED.mw

Movie #01

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

Movie #02

https://www.youtube.com/watch?v=sIxBm_GN_h0

Movie #03

https://www.youtube.com/watch?v=LOZNaPN5TG8

Lenin Araujo Castillo

Ambassador of Maple

This is a toy example illustrating three possible ways to define a group.

An icosahedron:

with(GroupTheory): with(Student[LinearAlgebra]): with(geom3d):

icosahedron(ii): vv := vertices(ii):

PLOT3D(POLYGONS(op(evalf(faces(ii))), TRANSPARENCY(.75)),
  op(zip(TEXT, 1.1*evalf(vv), [`$`(1..12)])), AXESSTYLE(NONE));

The group of rotations is generated by the rotation around the diagonal (1,4) and the rotation around the line joining the midpoints of the edges (1,2) and (3,4), by the angles 2*Pi/5 and Pi respectively.

Define the group by how the two rotations permute the 6 main diagonals of the icosahedron:

gr := PermutationGroup({[[2, 5, 3, 4, 6]], [[1, 2], [5, 6]]});

IdentifySmallGroup(gr);
                             60, 5

Or define the group by the relations between the two rotations:

gr2 := FPGroup([a, b], [[a$5], [b$2], [a, b, a, b, a, b]]);

IdentifySmallGroup(gr2);
                             60, 5

Finally, define a group with elements that are rotation matrices:

m1 := RotationMatrix(2*Pi/5, Vector(op(4, vv) - op(1, vv))):
m2 := RotationMatrix(Pi, Vector(add(op(3..4, vv) - op(1..2, vv)))):
m1, m2 := op(evala(convert([m1, m2], radical))):

gr3 := CustomGroup([m1, m2], `.` = evala@`.`, `/` = evala@rcurry(`^`, -1), `=` = Equal);

IdentifySmallGroup(gr3);
                             60, 5
AreIsomorphic(SmallGroup(60, 5), AlternatingGroup(5));
                              true

One question is how to find out that the group is actually A5 without looking up the group (60, 5) elsewhere.

Also, it doesn't matter for this example, but adding `1`=IdentityMatrix(3) to the CustomGroup definition gives an error.

 

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?

 

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

Video

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

Lenin Araujo C

Ambassador of Maple

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

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 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;

Digits := 15

(1)

 

Enter the first-order ODEs into EqODEs list.

EqODEs:=[diff(y1(x),x)=y2(x),diff(y2(x),x)=epsilon*sinh(epsilon*y1(x))];

EqODEs := [diff(y1(x), x) = y2(x), diff(y2(x), x) = epsilon*sinh(epsilon*y1(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)]);

bc1 := [y1(x)]

(3)

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

bc2 := [y1(x)-1.]

(4)

 

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

Range:=[0.,1.];

Range := [0., 1.]

(5)

 

List any known parameters in the list.

pars:=[epsilon=2];

pars := [epsilon = 2]

(6)

 

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

unknownpars:=[];

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;

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;

atol := 0.1e-5

rtol := 0.100000000000000e-7

(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

5

(10)

sol[5]; # Maximum local RMSE

0.604570329905172e-8

(11)

 

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

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

NN := 11

(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');

node := 2

odevars := [y1(x), y2(x)]

(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

 

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?

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

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