@macrobbair 

All OK with you loading a metric or tetrad by hand, but It is not clear what you are doing or how to help you just from text (for example, I don't understand what you meant by something 'going wrong with the indices' or 'red comment'). Note as well that you can always edit a previous post, for example, to make it more clear, just click More -> Edit and proceed.)

Could you please upload a worksheet with the formulation and related text comments? For that purpose you can use the green arrow you see when you write a comment / question here in Mapleprimes.

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

@denitsastaicova

Download Tetrads_for_the_most_general_case.mw

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

@Carl Love 

Somehow I didn't get notified by Mapleprimes and only saw this one by chance now. Dmitry, the definition of normal form for an ODE depends largely on the person speaking - see also the help page ?convert/ODE. For hyper-normal forms I only found a paper with 40 pages.

Could you please upload a worksheet with what you expect and one single (simple) example?

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

@lastgunslinger 

I revised your worksheet. You say "The problem arises in the above computation, namely, the inner product taken between f[~mu,nu]*h[mu,beta,alpha],the inner product should be taken on the first index of each of the tensors, yet the results do not correspond to reults already computed. The antisymmetry is gone and the corresponding values do not coincide."

But neither you show what are these results "already computed" nor what antisymmetry are you talking about (none of the tensors you show is antisymmetric). From your worksheet, I have no way to understand these two statements.

I also see that you define tensors using TensorArray but without the freeindices keyword, and with some indices contravariant. The freeindices tells what is the ordering of the free indices - this is key when there are more than two indices, as in your case. Defining tensors with some indices contravariant, although possible, it is problem to problems since you don't know, from start, what are the components of the all covariant tensor (will depend on the metric, you understand).

Attached is a reviewed version of your worksheet, removing some lines that are not necessary and using some other tools to investigate the symmetries of the resulting tensors. I suggest you

1. Use the keyword freeindices of TensorArray to see exactly what is the ordering of the indices in the rhs of your definition (I show how to do that in the reviewed worksheet below).
2. Make your indices be all covariant by multiplying by the metric, then taking TensorArray to see, exactly, what are the covariant components of the tensors you are defining.
3. Use Library:-GetTensorSymmetryProperties to see the symmetry properties of the resulting tensors, and/or do some verification in any other way regarding these symmetry properties, before constructing the inner product you expect to be antisymmetric.

Then it would be possible to understand whether the last result should or should not be antisymmetric.

Download Maple_inner_tensor_product_(reviewed).mw

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

@lastgunslinger 

I see a worksheet with a rather long computation, but not using current Maple Physics but the old tensor. That old code worked fine but it is very old, not as capable as Physics, and not what I was asking for. Sorry for that, I do not work using that code since too many years.

In order to help you further please attach a worksheet with your computation using the current Maple Physics. Please also have in mind: the idea is to help you, that requires you to rewrite the worksheet (expressing the computation using Physics). Where you have a doubt, just present please the worksheet up to what you understand and the question.

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

@lastgunslinger 

Would you mind please putting all your computation into a worksheet? Use the green arrow for that purpose. And in the worksheet I will be able to see the input (without having to do copy and paste from your post above), the output, and you can place your comments in text regions (check the help to see how you do that, or take any of the worksheets I attached in the previous replies as template).

From there I think I can give you some help more concretely.

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

@lastgunslinger 

There are several ways of accomplishing the same thing, perhaps the simplest is to input

> Define(k2[sigma, ~mu, nu] = k[~mu, nu, sigma]);

Likewise you can use

> Define(k[sigma, ~mu, nu] = rhs(k[~mu, nu, sigma, array]));

or

> Define(k[sigma, ~mu, nu] = TensorArray(k[~mu, nu, sigma]));

By the way: to define a tensor (lhs) in terms of other tensors (rhs) you do not need to express the rhs in array form. You could use, directly, the symbolic expression. If so, the only thing you need to be consistent is with having the same free indices and character (covariant or contravariant) on both sides of the defining equation. For example, you could define 

> Define( j[sigma, ~mu, nu] = (9) )

where (9) is the symbolic tensorial expression e[nu, ~alpha] * f[~sigma, beta] * Christoffel[~nu, sigma, mu], but then the free indices, and their covariant/contravariant character, on the lhs and rhs, are not the same. If you make them be the same it will work, or if you really intended something different, using the array form (via TensorArray, check its option to indicate the ordering of the free indices) also works.

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

Hi
Replacing your line 
t:= textplot3d([-1.1,1.1,1,v_])
by
 t := textplot3d([-1.1, 1.1, 1, Typesetting:-Typeset(v_)])
also works.

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

Download How_to_split_symmetries_into_cases_(III).mw

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

Download Maple_inner_tensor_product_answers.mw

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

@Rouben Rostamian  

Enjoyable reading as usual, only one comment: convert(%, set) is not necessary, dsolve accepts differential equations also directly in matricial form.

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

@otaviosama 

You realize you sent a document with 23 pages. Man, it shouldn't be that way. Think about. Could you please indicate one page where I can see the formulas you would like help proving?

@mwahab 

Having m and/or k as exponents in powers, or as coefficients of non-linear terms, makes no difference: they are just parameters, things that don't depend on the independent or dependent variables of the problem. You can try replacing pde (equation (1) in the post) by pde := U[t] + U[]^m*U[x] + U[]^k = 0 and run the same computation. Of course, the problem is more complicated, and so are the infinitesimals. You can then indicate u as the dependent variable, or the list [u, k, m] as I did in answer to the recent question in Mapleprimes, or replace the functionality (x, t) by (x, t, k, m)as I did in the post above. It all works (Maple 2019, latest Physics Updates), and what you do depends on what you want to achieve.

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

@otaviosama 

My impression is that you are not typing the right equations. For instance: your algebra rules involve J[y], but your expected result (righ-hand side of the equation you added in red in the original post and after the post) has no J[y]. (Also: when changing the worksheet, do not rewrite the original, but attach a new one so that we can follow and find things right away.)

Likewise, although I understood your now CommuRules are not the ones for the Pauli sigma matrices, from your title, if you are computing with angular momentum operators, their algebra rules and identities are all shown in this Wikipedia page and they all check OK when entered and simplified on a Maple worksheet.

Would you mind revise your input and, if necessary, please create an image from the book (also what book) you are taking these formulas and paste the image here? That may help to identify a typo or what is going on. From what I've seen so far, Maple's result is just correct, and not what you expect.

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

@mwahab 

I looked again at your worksheet. This is not a Differential Geometry problem, but one of algebraic manipulation of simple form. You can do what you want, entirely, using PDEtools only, and that will avoid the confusion caused by the occurrence of diff(u(x,y,z),y,y) together with u[y,y] in the same expression representing the same thing. The PDEtools commands you want: diff_table (makes your input tremendously simpler), ToJet / FromJet (see the notation options you have to match the one you show in your question) and of course, dcoeffs (use it the way I indicated in my first answer). And then you won't need to write a program or anything of that sort in order to "get the coefficients of u(x,y,z) and/or v(x,y,z)", including of course the coefficients of their derivatives (that is why it is called 'd'coeffs).

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