I gave a look at your worksheet, Deniscr. What I see: this is not a math issue but a typesetting one. I use 2D Math input, but type things as in 1D math input. Sometimes, in addition, I right-click and convert to 2D Math input to have nice typeset math already in the input. This is then a picture of what I see, the derivative is computed right away (first input) instead of the error message (second input, that I kept there for comparison).

Once the problem is understood, it remains to know what did you type to get this input that results in invalid derivative?

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

Hi

The practical dynamics is one where you post the formulation of the problem in a workshseet, up to what you think is the correct way to formulate it (do your best, taking advantage of the help pages) and use the green arrow to upload the worksheet here. From there, intercalating comments and input/output in that worksheet we take the question to a resolution - no doubt.

In advance to your worksheet: yes, you need to specify the functions vs and f(rs) explicitly in terms of the coordinates x, y, z, t. Otherwise, the symbols vs or rs are just some symbols not dependent on the coordinates, and therefore the metric is constant and so the Christoffel symbols are naturally equal to 0.

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

Addition_of_vectors_(reviewed).mw

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

Perpendicular_Vectors_(reviewed).mw

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

It is surprising how this problem went below the radar for so long. It is fixed. The fix, by people working with the simplifier, is available to everybody withing the Maplesoft Physics Updates v.643 or higher. Note that, as it's been the case for the last 5 years, these Updates only work with the current Maple release, Maple 2020, not retroactively with older releases.

Attached is your worksheet, reviewed, with the output after installing the Physics Updates v.643.

That said, I'd like to comment on the Title of this Question. Behind this software, there are people, proud of their work, happy to participate in this great forum and willing to help. But Titles like the one of this Question do not help, are not minimally polite and are not respectful. Please revise the communication style.

simplify_(reviewed).mw

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

Enter ?libname to understand what that is. Then please enter libname at the Maple prompt and reply here showing the output. What the message you show is telling is that

• the package is installed, typically in Physics Updates below the Maple/toolbox directory, that exists below the directory shown by kernelopts(homedir);
• that in you libname some other directory (D:\\Program Files\\Maple 2020\\lib\\maple.mla) contains the Physics package and appears in libname before the directory Physics Updates below Maple/toolbox.

Because D:\\Program Files\\Maple 2020\\lib\\maple.mla comes first in libname and contains Physics, that is the active version of Physics you are using, not the one that you installed with the Physics Updates.

That may happen for several reasons. For example if you manually set the value of libname or manually install the Physics Updates somewhere else (not that you have done any of that - just as examples).

In summary, what is the output when you input libname?

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

There is this command, PDEtools:-dcoeffs, give a look at its help page, I imagine this is what you want. You also have DifferentialAlgebra:-Tools:-Coeffs, but that is a more advanced command - not sure that is what you need.

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

@digerdiga 

Besides Pascal4QM worksheets, there is something new in Maple 2020 that you may want to take advantage: the SU(2) tensor space has dimension 3 and is Euclidean. So, in Maple 2020, instead of changing the dimension of spacetime using Setup(dimension = 3, metric = Euclidean, spacetimeindices = lowercaselatin) as you did, just use Setup(su2indices = lowercaselatin). Then set the algebra rule using KroneckerDelta, that in Maple 2020 works as a tensor for su2, su3, spinor and gauge indices (not for spacetime, space or tetrad indices for which you already have metric commands available, g_, gamma_ and Tetrads:-eta_).

Regarding your other question, on Why the sum of two terms that cancel each other are not cancelling when I called Simplify? The answer is: these expressions involve not just tensors but noncommutative objects subject to commutator rules, and in that case, when the two things are taken at the same time, up to what I know, there is no fully systematic algorithm to get all the cases. In those situations, the two commands that help are Library:-SortProducts and SubstituteTensor. You can see several examples of how that is done in the post The hidden SO(4) symmetry of the Hydrogen Atom.

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

Download The_Riemann_tensor_for_an_arbitrary_metric.mw

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

I can't tell why it hangs; I'd guess it has to do with some operations that moved into kernel extensions for performance ... but I am not sure in this case. What I can tell is that this is a problem in the handling of trig expressions that should not be happening. I am not working in the simplifier these days but will forward the example to the appropriate people.

Meantime, the place where the computational flow hits this problem is when trying to solve an intermediate ODE of the characteristic strip related to your PDE. I added some 'look closer' before jumping into a generic combination of trig functions (where the problem is currently hanging, in the presence of non-constant powers of trig functions). The change is distributed for everybody as usual within the Maplesoft Physics Updates (for Maple 2020), v.627 and higher, and with that, the flow doesn't hit the problem anymore. Your timelimit then works as expected.

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

Hi nm,
The first version of the Physics Updates for Maple 2020, with number 620, is there. It includes some improvements in assuming, unapply (using intats when unapplying integrals) and improvements in Physics (displaying metric and signature now by default), things requested by Beta testers during the testing of the version of Maple released today. The Maplesoft R&D Physics webpage, however, still needs to be updated telling this version is for Maple 2020, not Maple 2019. 

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

Right before your equation (4.3), instead of subs(Maxwell_2, %) use subs(diff(Maxwell_2, t), %), where `%` represents equation (4.2). That results in what you are asking, ie "replace Curl of H with an expression in D."

Alternatively, you can formulate the same using noncommutative products of differential operators (using * as multiplication, representing application) but that is a more advanced (unnecesary, I'd say) use of the package. For details on how to do that, see the help page ?Physics,Tensors, Sec I.6.

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

Download Christoffel_symbols_of_de_Sitter_metric_research_(reviewed).mw

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

Executing your worksheet, the error says something different: "Error, (in Physics:-Define) received a definition where the number of free indices on the left-hand side (2) is not equal to the number of dimensions of the array on the right-hand side (1)"

Indeed, if I now enter

ArrayDims(rhs(%));
                                  1 .. 1, 1 .. 4

So the right-hand-side of your defining equation is not a 4x4 matrix (that is what the error message is saying).

On the other hand, your e[~mu,nu] is a 4x4 matrix so a plan would be 1) Define your e[~mu,nu] as a tensor then 2) enter f[~mu,nu] = rhs(e[definition])^(-1), then 3) Define that (say as in Define(%);) in order  to have f be the inverse tensor of e. While computing the inverse of the matrix defining e you may need to use some tricks since it is a 4x4 matrix involving huge expressions. For example, try taking the inverse of an arbitrary matrix with components thare are simple names, that is fast:

M := Matrix(4, symbol=m)^(-1)

Then substitute each of the m[i,j] in this arbitary inverse matrix by the components of your tensor e.

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

Hi

It shouldn't be necessary, but for now use Dgamma[5] instead of Dgamma[~5]. I will fix this latter today and upload a new Physics Updates with the fix for everybody.

Note also that the sign in the expected result you mention, "-4*i*epsilon[~mu,~nu,~rho,~sigma]", depends on the convention for Dgamma[5], explained in the help page, which by the way shows this result you mention with the sign you mention: during an overhaul of Dgamma that happened for Maple 2018 the help page missed an update of that sign. In summary, with the convention for Dgamma[5] used in Maple following the literature mentioned in ?Dgamma, expect + 4 i ..., not - 4 i ...

The output of Physics:-Trace comes in terms of a sum of products of metric factors g_[mu, n] with n an integer. That output is equal to the one using LeviCivita. That is something that could be improved: the output with LeviCivita is naturally simpler.

To verify the sign (is it + or -) for correctness, or for the case everything else, you can proceed as follows:

1. Substitute some values for the indices, e.g. eval(formula, [~mu = 1, ~nu = 2, ~rho = 3, ~sigma = 4]). You will get Dgamma[1]*Dgamma[2]*Dgamma[3]*Dgamma[4]*Dgamma[5] = 4*I.
2. use Library:-RewriteInMatrixForm(%) to see the actual matricial form of this equation.
3. Input % to produce the evaluation of the product of matrices and give a look: you see the expected sign (as said, for the convention used for Dgamma[5], it is +, not - ).

I will add a note to this answer later today when the correction to this ~5 issue is uploaded in the next Physics Updates.

NOTE: this issue of not handling ~5 in equal footing as 5 when this is the index of a Dirac matrix is resolved and the solution uploaded within the Physics Updates v.603 or higher.

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