Maple Questions and Posts

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Hi there!

Sorry to ask, but I don't know how I can solve it in a smart way. I want to take the covariant derivative of a specific vector, whose components are specifically defined, which can be constant or some functions of the coordinates. But, as soon as I define that specific vector, the covariant derivative fails to compute, saying there are "too many levels of recursion". Let me show what I mean.

I have these two attempts to get this:

 

First attempt

 

restart

with(Physics)

Setup(mathematicalnotation = true)

Coordinates(X = spherical)

{X}

(1.1)

Parameters(k)

{k}

(1.2)

Setup(metric = {(1, 1) = a(t)^2/(-k*r^2+1), (2, 2) = (a(t)*r)^2, (3, 3) = (a(t)*r*sin(theta))^2, (4, 4) = -1})

[metric = {(1, 1) = a(t)^2/(-k*r^2+1), (2, 2) = a(t)^2*r^2, (3, 3) = a(t)^2*r^2*sin(theta)^2, (4, 4) = -1}]

(1.3)

Define(b[mu])

{Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(1.4)

NULL

Define(Db[nu, mu] = D_[nu](b[mu](X)))NULL

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(1.5)

Define(b[mu] = [0, 0, 0, beta], redo)

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(1.6)

Db[]

Db[mu, nu] = Matrix(%id = 18446746372997772822)

(1.7)

beta := proc (X) options operator, arrow; b0 end proc``

proc (X) options operator, arrow; b0 end proc

(1.8)

Define(b[mu] = [0, 0, 0, beta], redo)

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(1.9)

Db[]

Error, (in index/PhysicsTensor) too many levels of recursion

 

````

Second attempt

 

restart

with(Physics)

Setup(mathematicalnotation = true)

Coordinates(X = spherical)

{X}

(2.1)

Parameters(beta, k)

{beta, k}

(2.2)

Setup(metric = {(1, 1) = a(t)^2/(-k*r^2+1), (2, 2) = (a(t)*r)^2, (3, 3) = (a(t)*r*sin(theta))^2, (4, 4) = -1})

[metric = {(1, 1) = a(t)^2/(-k*r^2+1), (2, 2) = a(t)^2*r^2, (3, 3) = a(t)^2*r^2*sin(theta)^2, (4, 4) = -1}]

(2.3)

Define(b[mu])

{Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(2.4)

``

Define(Db[nu, mu] = D_[nu](b[mu](X)))``

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(2.5)

Define(b[mu] = [0, 0, 0, beta], redo)

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(2.6)

Db[]

Error, (in index/PhysicsTensor) too many levels of recursion

 

````

``

On my first attempt, expression (1.7) is almost fine, but β is supposed to be constant (and a function of "t" in the future), so I set it as a constant function. But then it causes that problem. Therefore, on my second attempt, I tried to set β as a parameter since the beginning, but it was no good either.


If I were to take the covariant derivative as "D_[nu](b[mu])" (without the "(X)"), it would not have that specific problem, but would not be a good solution, since the covariant derivative would discard the partial derivative of b[mu] and, if it would have some dependence on the coordinates, it would give a wrong result.


The only way it seems to work is by defining the tensor "Db[mu,nu]" explicitly as "d_[mu](b[nu](X)) - Christoffel[~alpha, mu, nu]*b[alpha]". (It seems that the problem comes from that "(X)" next to "b[alpha]" in the connection term.) But that would be an awful way to circumvent the problem. Isn't there any better way to get this?


Can someone help me with this, please?


Thanks for any help!

 

Download Cov._derivative_of_a_specific_vector.mw

 

In Maple 2019, I'm using the "applyrule" function to write a simple routine for contracting vector indices, following the Einstein summation convention (repeated indices are summed).

Input:

applyrule(
  vec(a::symbol, i::symbol) * vec(b::symbol, i::symbol) = dotproduct(a,b),
  vec(a,k) * vec(b,k)
);

Output:

dotproduct(a, b)

So far it works as expected. Now I add an "uncontracted" vector, and the code seems to become unpredictable. First, let me show a case in which the code still works:

Input:

applyrule(
  vec(a::symbol, i::symbol) * vec(b::symbol, i::symbol) = dotproduct(a,b),
  vec(u,k) * vec(u,i) * vec(c,k)
);

Output:

dotproduct(c, u)*vec(u, i)

As expected, contraction has been done on the repeated index "k", but vec(u,i) was left alone. However, the above code fails if I simply change the variable name "u" to "a":

Input:

applyrule(
  vec(a::symbol, i::symbol) * vec(b::symbol, i::symbol) = dotproduct(a,b),
  vec(a,k) * vec(a,i) * vec(c,k)
);

Output:

vec(a, k)*vec(a, i)*vec(c, k)

Apparently, the rule was not applied because I used "a:symbol" in the transformation rule. But I thought "a::symbol" should be considered a wildcard that represents any symbol, including "a" itself?

Depending on the exact input, this bug sometimes shows up, sometimes not, but the above examples are verified to be reproducible. A simple workaround is always avoiding the use of the same variable, but this seems to be hack. Is there a proper solution?

How can add a PNG within a text section of a Maple Worksheet without having the image appear with a black background?

Does Maple have an alpha channel or transparency setting that must be toggled for this to work?

Maple 2019
macOS 10.14.6

I am added pictures from the clipboard via drag+drop.

(Yes, I know that I can use JPEG without this problem and that is NOT my question).

--
JJW

Hello Everyone, 

I am new to Maple and I have designed an algorithm to compute multivariate differential dimension Grobner bases. Now I want to add this Maple as a package but I do not how to do that.

Can someone please help with this.

Thank you so much for your help.

My code can be found at https://github.com/Gsparsh/Grobner-bases

Hi everyone, I have a problem in the code solving coupled partial differential equations. I could not find out the solution. Please help me out with this. Find the code in the attachment.

Dear maple users,
Greetings.
I am solving an ode problem with an analytical solution.
programming running properly, but my plot not exact with the already existing article plot. 
how to get the exact plot.

Thanking you.

Code:JVB.mw
 

restart

N := 3;

3

 

1

(1)

dsolve(diff(f(x), `$`(x, 3)));

f(x) = (1/2)*_C1*x^2+_C2*x+_C3

(2)

Rf := 2*(diff(f[m-1](x), x, x, x))-(2*mh*mh)*(diff(f[m-1](x), x))+sum(f[m-1-n](x)*(diff(f[n](x), x)), n = 0 .. m-1)-bet*(sum(sum(2*f[m-1-n](x)*(diff(f[n-t](x), x))*(diff(f[t](x), x, x))+f[m-1-n](x)*f[n-t](x)*(diff(f[t](x), x, x, x))+x*(diff(f[m-1-n](x), x))*(diff(f[n-t](x), x))*(diff(f[t](x), x, x)), t = 0 .. n), n = 0 .. m-1));

2*(diff(diff(diff(f[m-1](x), x), x), x))-2*(diff(f[m-1](x), x))+sum(f[m-1-n](x)*(diff(f[n](x), x)), n = 0 .. m-1)-.2*(sum(sum(2*f[m-1-n](x)*(diff(f[n-t](x), x))*(diff(diff(f[t](x), x), x))+f[m-1-n](x)*f[n-t](x)*(diff(diff(diff(f[t](x), x), x), x))+x*(diff(f[m-1-n](x), x))*(diff(f[n-t](x), x))*(diff(diff(f[t](x), x), x)), t = 0 .. n), n = 0 .. m-1))

(3)

dsolve(diff(f[m](x), x, x, x)-CHI[m]*(diff(f[m-1](x), x, x, x)) = h*H*Rf, f[m](x));

f[m](x) = Int(Int(Int(CHI[m]*(diff(diff(diff(f[m-1](x), x), x), x))+2*h*(diff(diff(diff(f[m-1](x), x), x), x))-2*h*(diff(f[m-1](x), x))+h*(sum(f[m-1-n](x)*(diff(f[n](x), x)), n = 0 .. m-1))-(1/5)*h*(sum(sum(2*f[m-1-n](x)*(diff(f[n-t](x), x))*(diff(diff(f[t](x), x), x))+f[m-1-n](x)*f[n-t](x)*(diff(diff(diff(f[t](x), x), x), x))+x*(diff(f[m-1-n](x), x))*(diff(f[n-t](x), x))*(diff(diff(f[t](x), x), x)), t = 0 .. n), n = 0 .. m-1)), x), x)+_C1*x, x)+_C2*x+_C3

(4)

f[0](x) := 1-exp(x);

1-exp(x)

(5)

for m to N do CHI[m] := `if`(m > 1, 1, 0); f[m](x) := int(int(int(2*CHI[m]*(diff(f[m-1](x), x, x, x))-(2*h*H*mh*mh)*(diff(f[m-1](x), x))+h*H*(sum(f[m-1-n](x)*(diff(f[n](x), x)), n = 0 .. m-1)), x)-h*H*(sum(sum(2*f[m-1-n](x)*(diff(f[n-t](x), x))*(diff(f[t](x), x, x))+f[m-1-n](x)*f[n-t](x)*(diff(f[t](x), x, x, x))+x*(diff(f[m-1-n](x), x))*(diff(f[n-t](x), x))*(diff(f[t](x), x, x)), t = 0 .. n), n = 0 .. m-1))*bet, x)+_C1*x, x)+_C2*x+_C3; s1 := evalf(subs(x = 0, f[m](x))) = 0; s2 := evalf(subs(x = 0, diff(f[m](x), x))) = 0; s3 := evalf(subs(x = 1, f[m](x))) = 0; s := {s1, s2, s3}; f[m](x) := simplify(subs(solve(s, {_C1, _C2, _C3}), f[m](x))) end do:

f(x) := sum(f[l](x), l = 0 .. N);

1-0.7644444444e-1*exp(5.*x)*h^2*x-0.1333333333e-1*x^2*exp(5.*x)*h^2-2.675700596*exp(2.*x)*h^2*x-0.5876096022e-1*exp(6.*x)*h^3*x-0.9282030175e-2*x^2*exp(6.*x)*h^3+.9962792493*exp(3.*x)*h^3*x+.1647896790*exp(5.*x)*h^3*x+0.2066962962e-1*x^2*exp(5.*x)*h^3+3.357118680*exp(2.*x)*h^3*x-.3264340965*exp(4.*x)*h^3*x+0.3999999998e-1*exp(2.*x)*ln(exp(x))*h^2+58.61348006*h^3+1.023148148*h^2*x^3+0.1364197531e-1*ln(exp(x))*h^3*x^3-0.8954734530e-1*exp(2.*x)*h^3*x^4-.1353159884*x^3*exp(4.*x)*h^3+.7542645986*exp(3.*x)*h^3*x^2-0.2830138323e-1*x^3*h^3*exp(3.*x)-0.6455420536e-1*exp(x)*h^3*ln(exp(x))*x+0.4775858416e-1*exp(x)*h^3*ln(exp(x))*x^2+0.8888888887e-3*exp(x)*h^3*ln(exp(x))^2+8.400000000*h*exp(x)-exp(x)-0.6666666666e-1*h*ln(exp(x))+.1416666666*exp(4.*x)*h^2*x-.4790123458*exp(3.*x)*h^2*x+.1333333333*exp(3.*x)*h*x+.3791666665*exp(4.*x)*h^2-1.340020575*exp(3.*x)*h^2+.3111111109*exp(3.*x)*h+5.570191338*h^2*exp(2.*x)-.4500000000*h*exp(2.*x)-0.9874869443e-1*exp(6.*x)*h^3+.4125877323*exp(3.*x)*h^3-4.984787877*h^3*exp(2.*x)-.8010958741*exp(4.*x)*h^3+.3215641638*exp(5.*x)*h^3-5.930474628*h^2*x+36.04284024*exp(x)*h^3*x+8.324321524*x^2*h^2-.5362260993*h^3*x^3-6.207072379*exp(x)*x^2*h^3+1.664189246*exp(x)*h^3*x^3-8.237962963*h+.1200000000*exp(x)*h^2*ln(exp(x))+0.2222222222e-1*exp(3*x)*h*x+24.00299428*h^3*x-2.098561083*x^2*h^3-53.48457977*h^3*exp(x)+0.9949705035e-2*ln(exp(x))*h^3*x^4-0.7308641971e-2*ln(exp(x))*exp(4.*x)*h^3+0.8984910834e-2*ln(exp(x))*exp(3.*x)*h^3-0.3741666666e-1*ln(exp(x))*h^3*exp(2.*x)-.1188740741*exp(5.*x)*h^2-12.53662834*x^2*h+25.90916526*h^2*exp(x)-30.39962862*h^2-0.7499999999e-1*h*exp(2*x)+0.5185185185e-1*exp(3*x)*h+5.372840718*exp(x)*x^2*h^2-25.09181716*exp(x)*h^2*x+0.8976305409e-1*h^3*x^5+0.2158026099e-1*exp(7.*x)*h^3+0.8606919260e-1*h^3*x^4+0.5079365079e-3*x^3*exp(7.*x)*h^3-.3215468487*x^2*exp(4.*x)*h^3+0.1762236380e-1*exp(7.*x)*h^3*x+0.5048727639e-2*exp(7.*x)*x^2*h^3-3.116709690*exp(2.*x)*x^2*h^3+.1066289908*exp(2.*x)*h^3*x^3-8.527777777*h*x-0.2814814814e-2*ln(exp(x))*exp(4.*x)*h^3*x-0.1053497943e-2*ln(exp(x))*exp(3.*x)*h^3*x+0.4848332783e-1*h^3*x^6+.7462278773*h^2*x^4+.5519508187*exp(x)*h^3*x^4+0.9367631194e-1*exp(x)*h^3*ln(exp(x))+3.581893812*exp(2.*x)*x^2*h^2

 

 

NULL


 

Download JVB.mw

 

Analytical solution approach:

 

 

 

 

How to solve this DE by using  Differential transformation method?

diff(f(x),x$3)+1/2 * f(x) *diff(f(x),x$2)=0;

with boundary conditions

  f(0)=1 ; f '(0)= lamda * f ''(0)   and   f ' (x) -> 1  as x -> infinity
where lamda is some constant...

Explore the values of km_digit(n,m) using km_ for all m,0 < or equal to m and less than or equal to 8. 

Look at the output until you can make a conjecture that concerns the pattern obtained.

Hint use km_list(m,6,20) when m is not a multiple of 4 and km_list(m,6,50) when m is a multiple of 4

I am studying the motion of a beam coupled to piezoelectric strips. This continuous system is modelled by two DE:

YI*diff(w(x,t), x$4)-N[0]*cos(2*omega*t)*diff(w(x,t), x$2)+c*diff(w(x,t), t)+`&rho;A`*diff(w(x,t), t$2)+C[em,m]*v(t) = 0;

And:

C[p]*diff(v(t), t)+1/R[l]*v(t) = C[em,e]*(D[1,2](w)(0,t)-D[1,2](w)(ell,t));
 

where "w(x,t)" stands for the beam's vibration and "v(t)" means the electric voltage, which is constant throught the beam. I would like to numerically solve both DE simultaneosly, but maple will not let me do it. I would like to know why. I am getting the following error:

Error, (in pdsolve/numeric/process_PDEs) number of dependent variables and number of PDE must be the same
 

I suppose it is because "w(x,t)" depends on "x" and "t", while "v(t)" depends solely on time, but I am not sure. Could someone help me out? Here is my current code:

restart:
with(PDEtools):
declare(w(x,t), v(t)):

YI*diff(w(x,t), x$4)-N[0]*cos(2*omega*t)*diff(w(x,t), x$2)+c*diff(w(x,t), t)+`&rho;A`*diff(w(x,t), t$2)+C[em,m]*v(t) = 0;
pde1:= subs([YI = 1e4, N[0] = 5e3, c = 300, omega = 3.2233993, C[em,m] = 1], %):
ibc1:= w(0,t) = 0, D[1,1](w)(0,t) = 0, w(ell,t) = 0, D[1,1](w)(ell,t) = 0, D[2](w)(x,0) = 0, w(x,0) = sin(Pi*x/ell):

C[p]*diff(v(t), t)+1/R[l]*v(t) = C[em,e]*(D[1,2](w)(0,t)-D[1,2](w)(ell,t));
pde2:= subs([C[p] = 10, R[l] = 1000, C[em,e] = 1, ell = 5], %):
ibc2:= v(0) = 0:

pdsolve({pde1, pde2}, {ibc1, ibc2}, numeric);

Thanks.

Hello everyone!

I'm interesting in "zcoloring" funciton in colorscheme option.

I wrote simple programm which compares two results: spectrogram of signal drawn with "colormap" list and spectrogram which was plotted with zcoloring function. I use red, green, blue functions to construct JET-colormap: list and expressions in "zcoloring".

My result:

As I understand, when I use:

colorscheme = ["zcoloring", [z-> Red color function(z), z-> Green color function(z), z-> Blue color function(z)], colorspace = "RGB"]

Maple plots z-value with color RGB color coordinates defined from "zcoloring". For example, if "zcoloring" is

colorscheme = ["zcoloring", [z-> 5*z, z-> 3*z, z-> 2*z], colorspace = "RGB"]

and z value is 10, then 10 value will correspond [50,30,20]-RGB color.

My test program:

Spectrogram_zcoloring.mw

Spectrogram of my test signal:

list_test.txt

Hello,

a follow up question.

I am solving some overdetermined system of ODEs in cylindrical coordinates r,phi,Z. I obtain some equations of the following type:

(diff(_F1(phi, Z), phi)*r + diff(diff(s_r(phi, Z), phi), phi))/r = -s_r(phi, Z)/r

As can be seen, the differentiated functions do not depend on r, which is an independent variable. Thus, the correct solution is to separate the equation and have 

_F1(phi,Z)=_F1(Z), s_r(phi,Z)=s_r(Z).

By using dsolve, I always obtain a solution containing r.

A similar problem that does no contain derivatives is solved by solve/identity.

Is there something similar for dsolve?

EDIT: I again put here more info and file. I solve some overdetermined system of differential equations.

[diff(s_r(r, phi, Z), r) = 0, diff(s_r(r, phi, Z), phi) = -diff(s_phi(r, phi, Z), r)*r^2,
diff(s_Z(r, phi, Z), r) = -diff(s_r(r, phi, Z), Z), diff(s_phi(r, phi, Z), phi) = -s_r(r, phi, Z)/r, 
diff(s_Z(r, phi, Z), phi) = -diff(s_phi(r, phi, Z), Z)*r^2, diff(s_Z(r, phi, Z), Z) = 0, 
diff(m(r, phi, Z), r) = s_Z(r, phi, Z)*B_phi(r, phi, Z) - s_phi(r, phi, Z)*B_Z(r, phi, Z),
 diff(m(r, phi, Z), phi) = s_r(r, phi, Z)*B_Z(r, phi, Z) - s_Z(r, phi, Z)*B_r(r, phi, Z),
 diff(m(r, phi, Z), Z) = s_phi(r, phi, Z)*B_r(r, phi, Z) - s_r(r, phi, Z)*B_phi(r, phi, Z), 
s_r(r, phi, Z)*diff(W(r, phi, Z), r) + s_phi(r, phi, Z)*diff(W(r, phi, Z), phi) + 
s_Z(r, phi, Z)*diff(W(r, phi, Z), Z) = 0]

After some time, I arrive at the equation in the original question. So the independance of the other functions on r is the consequence of the other equations.

Here is the file (shortened): mwquestion2.mw

   1 1 1 2 2 3 3  

(  1 3 3 2 2 1 2 )

how to input this permutation into permgroup ?

 

This is another problem I just found in Maple 2019.2 on windows 10. professional.

I wanted to close Maple, so did  File->Exit 

 

But Maple did nothing. It did not close.  Also Alt-F4 did not close Maple. I had to click on the little X on top right corner of the open window to close Maple.  

In earlier version this used to work to close Maple.

Do others see this as well?  To reproduce, simply start Maple, and do File->Exit.

Here is a movie also

 

This may be a bug.  In Maple 2019.0

plot3d(x^2+y^2)

Now grab the graphing window and drag to adjust it's size.  It doesn't adjust it's size until you let go of the mouse button.

If maple can auto discover derived equations, 

then most variable are unknown name which may or may not be physics variable.

so, how to guess which physics data suitable for these look like meaningless variable?

or these tools only visualize the relationship of existing known equations?

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