@oldstudent 
I added below an incomplete list of Maple strengths that in my opinion disproves your conclusion.

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

@USPAS2014 
Just about your comment on availability: the mini-course is linked in the Maplesoft R&D Physics webpage. - last link in the column on the right.

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

@Alejandro Jakubi 

Indeed I do have it my .mapleinit. About having this as a new command: generally speaking, when certain functionality can be used through options we will not add it as a new command. This case however is one where the combination of arguments is cumbersome enough and the functionality is used frequently, kinda justifying an exception. I forwarded now your suggestion to the people who take care of plots.

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

@rashmi 

I understand you can perform this computation in current Maple 18, but am not sure if the DifferentialGeometry package in Maple 12 could help you for that purpose, probably yes, give it a try. The starting point would be the help page ?DifferentialGeometry,OverviewOfGeneralRelativity.

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

@trace 

I will have time to give a look at your paper by Friday, hopefully before that. By the way nice worksheet the one posted by Torre using the DifferentialGeometry package for this problem.

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

@trace 

I am not sure I understood your worksheet as you intended, here are two possible interpretations. In the first one you seem to want to set the metric as your expression (2)

> (dt+4*w*(1/m^2)*(sinh((1/2)*(m*r))^2)*dphi)^2-sinh(m*r)^2*(1/m^2)*(dphi^2)-dr^2-dz^2

and then compute the value of Ricci[2,2]. If this is all what you want, and assuming that the coordinates are already set, just enter

> Setup(metric = %)

And the value of Ricci[2,2] is then given by

> Ricci[2,2]

And that is all. 

If however your question is about how to perform a transformation to then compute Ricci[2,2], suppose then you want to perform a transformation on a tensorial expression, and by that I mean you have the set of transforming equations, say tr, with the old variables on the left-hand sides and the new variables on the right-hand sides, or the other way around (new = old instead of old = new). Suppose now that you want to compute the components of a tensorial expression in the new coordinates, for example: the component [2,2] of the Ricci tensor. This is the sequence of steps:

1) transform coordinates on the spacetime metric

> TransformCoordinates(tr, g_[mu,nu]);

The output will show up in matricial form. If you prefer to see it as the square of the line element, then pass also the optional argument 'output = line_element'. Either way, if the output is according to what you expect, then:

2) set this result as the new value of the metric

> Setup(metric = %);

And that is all. You can now compute the value of any tensor directly, for example, as in

> Ricci[2,2]

The relevant thing then is the transformation tr. Reading your worksheet I have the impression you meant to use a transformation that can be represented in matricial form by

> M := Matrix([[1, 0, 0, 0], [0, sin(m*r)*(1/m), 0, 0], [0, 0, 1, 0], [0, 4*w*(sinh((1/2)*(m*r))^2)*(1/m^2), 0, 1]])

Am I understanding your worksheet correctly? Note also that the ordering of coordinates I am using is [r, phi, z, t], so t is the 4th, and hence also the 0th coordinate - this is the convention in the Physics package: all 0th components are entered as 4th components and you can refer to them using 0 or 4, as you prefer.

I noticed also that the first two lines are not actually equal: simplify((1)-(2)) does not give 0 - I am not sure whether that is consistent with what you present).

In summary: I suggest you to write the transformation explicitly and then use TransformCoordinates. Please feel free to ask again if you cross with obstacles; I'd be interested in seeing your development.

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

@trace 

Thanks for posting your worksheet mm.mw; the answer appears further below

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

@trace @escorpsy

Download mmm_(reviewed).mw

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

@Alejandro Jakubi 

Generally speaking I agree with what you say about Solve and solve, and in fact that was the original idea, many years ago, when I mentioned this project of a "unified solver" to Keith Geddes. I also agree in that the so-powerful differential elimination packages in Maple are almost invisible to the average user. About an interface for them, however, I think PDEtools;-casesplit is just great, easy and as intuitive/friendly as it could get. It would be more used if it were top-level though, the same as PDEtools:-Solve.

But that picture is incomplete: Both dsolve and pdsolve make extensive use of these two 'invisible' differential/algebraic elimination packages, and also of the differential-polynomial-forms introduced with PDEtools:-dpolyform and PDEtools:-casesplit. So, through dsolve and pdsolve everybody end using these fantastic packages and the extra functionality provided while interfaceing them, even if the packages are somehow invisible to the average user. And the differences between the solving power of dsolve & pdsolve in Maple versus DSolve in Mathematica are rather well noticed by mostly everybody.

Likewise, the solve command has incorporated a few years ago a lot of elimination functionality from the RegularChains package, to some point a sort of algebraic equivalent of the DifferentialAlgebra package that also handles inequalities. And so through solve mostly everybody now take advantage of RegularChains, also an almost invisible package. By the way there is no facilitating interface for RegularChains. For some time I've been thinking of extending PDEtools:-casesplit to also (optionally) interface RegularChains for handling non-differential systems.

Taking all into account, I don't see the picture as grim as you seem to present it. I do believe mainly that a step forward would be to make top level both the the concept of a unified solver (PDEtools:-Solve) and this general 'split into cases' command (PDEtools:-casesplit). Beyond what they unify or interface, both commands also bring new functionality that is highly regarded and just not available elsewhere in Maple or other computer algebra systems.

By the way I don't see how could I help you but it is unfortunate that you stopped posting.

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

@trace 

Look, you need to show what you wrote so that one could give an opinion, for instance whether "R_0,rho,alpha,beta R_0^rho,alhpa,beta" or "R_1,rho,alpha,1 R^rho,alpha , Riemann[1,rho,alpha,1].Ricci[rho,alpha]" are correct. Also, correct with regard to what?

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

@Alejandro Jakubi 

Regarding the post in this tread, you ask for an explicit form of the function call using PDEtools:-Solve or PDEtools:-casesplit: just replace solve by PDEtools:-Solve or PDEtools:-casesplit.

Now if you give a look at the corresponding help pages you will also see that you can do more than with solve: you can indicate, precisely the solving ordering, which is the most relevant thing when solving nonlinear systems of equations. Moreover: you can split your solving variables into subsets (with 1 or more variables each) and specify not only the ordering of the subsets, and the ordering of the variables within a subset, but, more important, specify to use an elimination or an orderly ranking (for the terminology explained in simple terms see the help page for PDEtools:-casesplit).

For example, PDEtools:-Solve({…system…}, [m, {x1, x2, x3, x4, x5}]); will first triangularize the system in order to obtain one subsystem that expresses m in terms of {x1, x2, x3, x4, x5}, then another subsystem that only involves {x1, x2, x3, x4, x5}; solve the latter for {x1, x2, x3, x4, x5}, then substitute into the former for m. If the solving variables are passed as [{x1, x2, x3, x4, x5}, m], the solving process will run the other way around. In this example the equations however are simple and so switching the order between m and {x1, x2, x3, x4, x5} makes no difference, but try passing the xn as a list and change the order in different ways and you will see the effect in the output. And this is all I have to say about this post.

About other issues like help pages, unified commands, your opinion that this unified command shall be solve, etc. I think these are other matters, you may want to post about them. And indeed I think there is always room for improvements, wether in the documentation or the commands themselves, and I agree that the visibility of the actual solving power could be bigger. For example, PDEtools:-Solve is in fact a unified solving command (it combines solve, dsolve, pdsolve, fsolve, incluing the options series and numeric of dsolve and pdsolve), that can also do more things than some of these solving commands it combines, but few people know about PDEtools:-Solve.

For DEtools[rifsimp] and DifferentialAlgebra I think the issue is different: these packages are rather powerful, but a bit more technical in what they do and so their presentation in help pages is complicated for the average Maple user. The presentation of their techniques in the help page for PDEtools:-casesplit aimed at facilitating their use, also through a simpler interface. Anyway all these are different topics.

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

@toaskfrommaple 

Say the maximum differential order with regard to a function within your equations is N, for example: in your single equation, for R, N = 2. Then, generally speaking, the conserved currents J can depend at most on derivatives of R of order N-1. In your example that is: 1. This is so because the conserved currents satisfy Nabla J = 0 "modulo the equations themselves", and Nabla is a differentiation operator of order 1. For more details see the help page for ConservedCurrents. This is the same as with Lie symmetry generators, and in fact these two problems are interrelated. Another way of viewing this is to recall that conserved currents are the PDE equivalent of 'first integrals" in the ODE case, also of order N-1.

Now, regarding "distinguishing the independent variables from the dependent ones", it is natural that the dependency of the conserved currents includes the dependent variables, and that can be up to derivatives of order N-1, as said. So what is exactly your question?

Note anyway that you can set the dependency (Q in the previous reply) to be what you want, and so, for example, compute conserved currents that depend only on the independent variables of the PDE system - this is in general a simpler problem.

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

@Markiyan Hirnyk 

... I'm not sure I understand your question. The interpretation of the output of

> solve({…same system…}, {m, x1, x2, x3, x4, x5});

i.e. the output you show, is what I already wrote: it contains all the solution cases that bluehotel was expecting, not receiving, and asking what could be wrong, the answer is: missing m in the set containing the solving variables. This output also contains the same solution cases shown below as 'the better output obtained in Mathematica', which is not better or worse but just the same.

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

The conversion of this function call FromMma is fixed. The fix, together with other developments in the DE and Mathematical Functions area, can be downloaded on the Maplesoft R&D Differential Equations and Mathematical Functions webpage.

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

@Markiyan Hirnyk 
It seems we still have this problem with posting images of formulas. I corrected this typing the input, without using an image. All this development about Wirtinger calculus is advertised in the help page ?updates,Maple18,Physics under "New Enhanced Modes in Physics Setup".

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