Maple 2021 Questions and Posts

These are Posts and Questions associated with the product, Maple 2021

restart;
Proc := proc(t) local t4, l3, R, r, eq, sol; _EnvHorizontalName := 'x'; _EnvVerticalName := 'y'; t4 := thickness = 4; l3 := linestyle = dot; R := 9; r := 1/2*R; geometry:-point(OO, 0, 0); geometry:-circle(Cir, [OO, R]); geometry:-point(K, R*cos(t), R*sin(t)); geometry:-point(Omega, r*cos(t), r*sin(t)); geometry:-circle(cir, [Omega, r]); eq := geometry:-Equation(cir); geometry:-line(XXp, y = 0); geometry:-line(YYp, x = 0); geometry:-line(L1, y = x); geometry:-line(L2, y = -x); geometry:-projection(M1, K, XXp); geometry:-coordinates(M1); geometry:-point(K2, geometry:-coordinates(M1)[1] - 2*R, 0); geometry:-coordinates(K2); geometry:-segment(sT, K2, M1); geometry:-point(N1, 0, R*sin(t)); subs(y = x, eq); sol := solve(%, x); geometry:-point(N2, sol[2], sol[2]); subs(y = -x, eq); sol := solve(%, x); geometry:-point(N3, sol[2], -sol[2]); plots:-display(geometry:-draw([Cir(color = blue, t4), cir(color = grey, t4), sT(color = black, t4), XXp(color = black, l3), YYp(color = black, l3), L1(color = black, l3), L2(color = black, l3), N1(color = blue, symbol = solidcircle, symbolsize = 15), N2(color = blue, symbol = solidcircle, symbolsize = 15), N3(color = blue, symbol = solidcircle, symbolsize = 15), M1(color = blue, symbol = solidcircle, symbolsize = 15)]), axes = none, view = [-30 .. 10, -10 .. 10], size = [800, 800]); end proc;
plots:-animate(Proc, [t], t = 0 .. 2*Pi, frames = 200);
NULL;
I am trying to program  this drawing, how to improve this code ? Thank you.

restart;
with(geometry);
with(plots);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
cb := color = blue;
t3 := thickness = 3;
l3 := linestyle = 3;
xA := 3;
yA := 0;
xB := -3;
yB := 0;
c := 6;
point(A, xA, yA);
point(B, xB, yB);
R := 5;
alpha := arctan(3/4);
evalf(%*180/Pi);
                          36.86989765

xH := (xA + xB)/2;
yH := (yA + yB)/2;
point(H, xH, yH);
xO1 := 0;
yO1 := 4;
point(O1, xO1, yO1);
xO2 := 0;
yO2 := -4;
point(O2, xO2, yO2);
segment(sAB, A, B);
segment(sHO, H, O1);
segment(sAO, A, O1);
segment(sBO, B, O1);
                              sAB

alpha1 := arctan((yO1 - yA)/(xO1 - xA));
beta := Pi + arctan((yO1 - yB)/(xO1 - xB));
AR1 := plottools[arc]([xO1, yO1], R, alpha1 .. beta, l3);
AR2 := plottools[arc]([xO2, yO2], R, -beta .. -alpha1, l3);
N := 80;
dt := (beta - alpha1)/(N - 1);
dr := draw({O1, O2, sAB, sHO, sAO(cb), sBO(cb)});
tex := textplot([[xA, yA - 0.5, "A"], [xB, yB - 0.5, "B"], [xO1, yO1 + 0.5, "O1"], [xO2, yO2 - 0.5, "O2"], [xH, yH - 0.5, "H"]]);
M1 := seq(plottools[disk]([R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], 0.2, color = orange), t = 0 .. N);
M2 := seq(plottools[disk]([R*cos(dt*t + Pi + alpha1) + xO2, R*sin(dt*t + Pi + alpha1) + yO2], 0.2, color = orange), t = 0 .. N);
P1 := seq(plottools[polygon]([[R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N);
P2 := seq(plottools[polygon]([[R*cos(dt*t + Pi + alpha1) + xO1, R*sin(dt*t + Pi + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N);
for t to N do
    E[t] := display(dr, tex, AR1, AR2, M1[t], P1[t]);
    F[t] := display(dr, tex, AR1, AR2, M2[t], P2[t]);
end do;
display([seq(E[t], t = 1 .. N), seq(F[t], t = 1 .. N)], insequence = true, axes = none, scaling = constrained, view = [-10 .. 10, -10 .. 10]);

NULL;
angle that does not turn on the lower arc. Thank you.

point(A, xA, yA);
point(B, xB, yB);
point(C, xC, yC):
L3 := linestyle = 3
triangle(Tr, [A, B, C])
line(AP, [A, P]);
line(BP, [B, P]);
line(CP, [C, P]);
dr := draw([Tr(t3), AP(cbl, L3), BP(cbl, L3), CP(cbl, L3)]),
textplot([[coordinates(A)[], "A"], [coordinates(B)[], "B"], [coordinates(C)[], "C"]], align = {above, right});
display({dr, ellip}, scaling = constrained, axes = none, view = [-1 .. 14, -1 .. 11]);
Why doesn’t Maple show me the expected effect ? Thank you.

Greetings! I need a maple code of logarithm of residuals for graphical comparison of iterative methods. Can anyone help me out by sharing a code comparing log of residuals of 2 or 3 iterative methods? 

I am trying to decompose an isprime into sum of 2 squares.
Can you tell me why yhse procedure are not goog.
                       

Sumof2Squares:= proc(p::And(prime, satisfies(p-> irem(p,4)=1)))
local x, y:= 1;
   x:= mods(Roots(x^2+y^2), p)[2,1];
   while x^2+y^2 > p do
      (x,y):= FermatDescent(x,y,p)
   end do;
   (x,y)
end proc:

FermatDescent:= proc(x::posint, y::posint, p::posint)
local 
   m:= (x^2+y^2)/p,
   a:= mods(x,m),  
   b:= mods(y,m);

   (abs((a*x+b*y)/m), abs((a*y-b*x)/m))
end proc:
   
trace(FermatDescent);

Sumof2Squares(1973);
Thank you.

Dear Maple experts,

I am running a Maple code with several lines/ curves in a plot.

For two of the curves, I use style=pointline. But in the legend the symbols on the lines are not shown. Can you advise how we can make them appear?

I have attached the file. Thank you so much! 

Download LegendPlot.mw

I am currently unable to type in maple 2021, there is no cursor, nothing shows up when I type etc. I am also unable to save files. I went into document mode, nothing. Some of the dialogs don't show up. Where does maple save documents and how do I change that setting? That might be my problem. Furthermore, there are no tabs visable, like I can't access any documents I open.

I have some large systems of linear equations.  The solutions are probability generating functions.  I can get solutions in a few minutes for systems of up to n= 200 eqns or so, but Maple just cycles indefinitely if I try to solve much larger systems.  I really only need to perform Gaussian Elimination, as I only need to solve for one of the n solutions.  The matrices are sparse, there are only 3 non-zero entries per row.  I tried to get help from the manuals but I get the impression that sparse solutions are only available for numeric computations.   Doesn't Maple allow for sparse symbolic solutions?  If so, how to do it?

How can I solve Einstein’s equation and calculus of the value of the K constant in Einstein's equation and the value of the tensor stress energy that fits in this equation?

   

 

QTBend.docxSqBend.mw

After a long hiatus I have come back to the issue of null tetrads in the physics package in light of the updates to  Maple in 2021. I have uploaded a document file to illustrate. See below. My first question concerns the labelling of elements of a null tetrad. After calling the metric 27,37 from Stephani et al, and using Setup to specify a null tetrad, Maple's choice is such that the elements labelled m and bar m are not complex. Rather, both these elements are in fact real, while the elements lablled l and n are complex, with one being the negative complex conjugate of the other. While these are just labels, they don't agree with the usual conventions for the Newman-Penrose formalism, which is disorienting. What convention is Maple using to label the elments of a null tetrad?

Next, I try to specify the null tetrad used by Stephani et al., first by converting it into covariant form (which I did by hand rather than in Maple). In Maple's default null tetrad, the order in which Maple listed the elements of the null tetrad is n, m, bar m, l (as rows in the matrix display for e_[ ]), so I followed that convention (in the conventions of Stephani et al., the first and fourh element should have scalar product -1, the second and third scalar product 1, and all other scalar products zero, which is the case). After entering the matrix and using Setup to specify the null tetrad by the matrix, I get an error message saying that the components of the metric with respect to my tetrad are not just 0, 1, and -1. Yet,  executing eta_[ ]  does not confirm this warning; nor does a computation by hand.

Finally, IsTetrad asserts the tetrad is not null, contrary to the fact that it is a null tetrad.

Since I have followed the conventions implicit in Maple's default null tetrad for this metric, I am puzzled as to what has gone wrong.SKMHH27_37_2021_New.mw

On the other hand, taking into account how Maple 2019 orders the coordinates in Stephani et al 27.37 and labels the null vectors in a null ttetrad, if I translate accordingly what I have in the 2021 Maple file, Maple 2019 confirms Stephani et al.'s null tetrad is indeed a null tetrad, as one would expect. See the following file.SKMHH27_37_2019_Var.mw.

Hi MaplePrimes, 

I am interested in obtaining some gravitational field equations from an action using the FunDiff command. I have been able to write what I think is a pretty short and quick worksheet(with an arbitrary metric) and I am easily able to obtain the Einstein Field Equations. However, when I introduce some new more complicated terms into the action and apply the Simplify command maple does not appear to be able to evaluate and I end up halting the computation. When I specify a metric Maple, for example Schwarzschild, Maple will easily be able to Simplify my expression but it will use metric components during the process. Where what I am interested in is just the Tensor expression with respective indices. 

I was wondering if anyone had any thoughts on how I could resolve this. 

I have attached the worksheet that I am working with. I look forward to your thoughts/comments.

Thank you.  

ActionFieldEquations.mw

So I have a problem when copying multiple output from Maple to Word. I am using a simple example here. 

When I copy each part of the output as an image SEPERATELY, it comes out much cleaner than if I highlight everything together and then copy. Is there a reason this is the case and is there a way to fix this problem?

Here is the Maple Code:

with(plottools);
with(plots);
ttt1 := textplot([0, 1, "3  +  5  =  __", color = "black", font = ["Arial", "bold", 120]]);
display(ttt1, size = [1000, 200], axes = none);
ttt1 := textplot([0, 1, "1  +  3  =  __", color = "black", font = ["Arial", "bold", 120]]);
display(ttt1, size = [1000, 200], axes = none);

I also uploaded the Maple Worksheet

Here is the output:

There are TWO imges in the output in this example. I tried copying both together and pasting them, and then I tried copying them individually. The output is very different in quality.

The one copied alot is much cleaner once copied and pasted in Word. Does anyone why this is happening? I don't want to have to copy images individally for obvoius reasons. I know this example is only two images but what if there was alot more, etc.

Thank you

NULL

with(plottools); with(plots); ttt1 := textplot([0, 1, "3  +  5  =  __", color = "black", font = ["Arial", "bold", 120]]); display(ttt1, size = [1000, 200], axes = none); ttt1 := textplot([0, 1, "1  +  3  =  __", color = "black", font = ["Arial", "bold", 120]]); display(ttt1, size = [1000, 200], axes = none)

 

 

NULL

Download Copy_Images.mw

Hi Everyone,

I intend to find the invariants of a 2nd-order Tensor using the representation of the rotational group as an Infinitesimal generator. However, maple does not seem to find the correct invariants $\tr \mathbf{a}$, $\tr \mathbf{a}^2$, $\tr \mathbf{a}^3$. Maple only returns an empty set. I can't seem to figure out why that is the case.

Here is my code

NULL

restart

with(PDETools)

assume(r >= 0)

interface(showassumed = 0)

X_z2 := (-a__21-a__12)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__11))+(a__11-a__22)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__12))-a__23*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__13))+(a__11-a__22)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__21))+(a__12+a__21)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__22))+a__13*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__23))-a__32*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__31))+a__31*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__32)) = 0

(-a__21-a__12)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__11))+(a__11-a__22)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__12))-a__23*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__13))+(a__11-a__22)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__21))+(a__12+a__21)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__22))+a__13*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__23))-a__32*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__31))+a__31*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__32)) = 0

(1)

X_y2 := (-a__31-a__13)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__11))-a__32*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__12))+(a__11-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__13))-a__23*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__21))+a__21*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__23))+(a__11-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__31))+a__12*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__32))+(a__13+a__31)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__33)) = 0

(-a__31-a__13)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__11))-a__32*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__12))+(a__11-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__13))-a__23*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__21))+a__21*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__23))+(a__11-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__31))+a__12*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__32))+(a__13+a__31)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__33)) = 0

(2)

X_x2 := -a__13*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__12))+a__12*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__13))-a__31*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__21))+(-a__32-a__23)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__22))+(a__22-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__23))+a__21*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__31))+(a__22-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__32))+(a__23+a__32)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__33)) = 0

-a__13*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__12))+a__12*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__13))-a__31*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__21))+(-a__32-a__23)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__22))+(a__22-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__23))+a__21*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__31))+(a__22-a__33)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__32))+(a__23+a__32)*(diff(f_1(x, y, z, a__11, a__12, a__13, a__21, a__22, a__23, a__31, a__32, a__33), a__33)) = 0

(3)

sys := [X_z2, X_y2, X_x2]

sol := pdsolve(sys)

(4)

NULL

Download SO(3)_spherical_2.mw

Any input would be appreciated!

Hello;

Hope you are fine. I need to learn how can I call my code in the other file of maple, define new value for A and can get results. For this purpose, I just take an simple example as

questionmaple.mw

restart; A := 5; B := 10; AAA[0] := 10; for i from 0 while i <= A do AAA[i+1] := B*i*AAA[0] end do

5

 

10

 

10

 

0

 

100

 

200

 

300

 

400

 

500

(1)
 

``

Download questionmaple.mw

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