Hi
After you Setup(dimension = 3) you can use Coordinates(X = [r, theta, phi]) or Setup(coordinates = (X = [r, theta, phi])). You can also set both things in one go via Setup(dimension = 3, coordinates = (X = [r, theta, phi])).

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

Download TransformTetrad.mw

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

Hi
There is a command, DEtools[MatrixDE] that I imagine that does what you are looking for.
Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi
First of all: I recommend you to update your Physics library with the one distributed in the Maplesoft R&D Physics webpage. With the update, that exists for Maple 2016 and the two previous releases, Dagger(d_[mu]) returns d_[mu].

Next, before answering your question, some comments on the input you show.

You set spaceindices to be uppercaselatin, then by default spacetimeindices are represented by greek lowercase letters (you can always check all settings rapidly by entering > Setup(); so without arguments), and then you define mu, mu5 and Pi as tensors. But mu and mu5 are indices ... you can but it is better not to use them also as tensor names. Also Pi is the constant 3.14.... it is not a good idea to use it as a tensor name.

Then you defined M but you use M_ These two things are not the same.

You also seem to have a problem in the keyboard: I imagine that in the input you show you intended multiplication but you used "power *", that displays like a product operator but as a superscript (exponent) thus triggering "Hermitian conjugation" instead of multiplication. Concretely, nowwhere in your input I see the 'Dagger' command being used but for these multiplication operators appearing as exponents.

There must also be something wrong in that the output of the two input lines you show look the same, while the input liens are not the same.

To avoid this kind of confusion it is better if you post a worksheet instead of copy and paste expressions.

Now on your question: to indicate that some objects are real (and therefore, if A is real, Dagger(A) = A), you can use Setup, as in Setup(realobjects = A), or Setup(realobjects = {A, B, C}) and so on. To set more specific assumptions, you can use the Physics:-Assume command, for example as in Assume(A > 0).

Finally, it is helpful if you tag your questions with 'physics' - somehow I was unable to fix this myself (I was in previous versions of Mapleprimes).

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

Download SimplificationOfDgamma.mw

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

Hi
Just do it as with paper and pencil:

1. You specify in your mind what indices you will use for spacetime and for space, e.g. respectively A and a, so uppercaselatin and lowercaselatin; tell the computer about that: Setup(spacetimeindices=uppercaselatin, spaceindices=lowercaselatin);

And that is all. You can now work using these two different kind of indices at the same time with exactly the meaning you attribute to them when computing with paper and pencil. Simplify, TensorArray, Library:-TensorComponents; all will perform the operations distinguising between 4 and 3 dimensions.

Two CAVEATS.

1. If you work with a Minkowski spacetime, the default signature is ---+; in this case the 3D metric is Euclidean while the 3D components of the 4D spacetime metric, g_[a,b] all have a `-` sign in front. In this case, for 3D tensor operations, instead of g_[a,b] you can use the Physics:-gamma_[a,b] command, which is the true 3D metric. Alternatively, you can change the signature (Setup(signature = `-+++`)) and now the components of g_[a,b] and gamma_[a,b] are the same. And of course if you work with an Euclidean 4D spacetime, g_[a,b] = gamma_[a,b].
2. If you work with an arbitrary curved spacetime, you need to use gamma_[a,b] as the 3D metric because the 3D components of g_, that is g_[a,b] are not a metric, ie. A[a] * A[~a] is different from g_[a,b] A[~a] A[~b].

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

I see the error message, about a Physics setting (geometricdifferentiation), but cannot reproduce the problem in current Maple. What version of Maple are you using? In any case I suggest you to update your Physics library with the one distributed in the Maplesoft R&D Physics webpage -- there are udpates also for Maple 2015 and the former Maple18.

Independent of solving the problem by updating your library, I think a workaround in your machine without changing anything would be to load Physics[diff] before using it; for instance: at the top of your worksheet, enter > with(Physics, diff);

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

Hi

Yes, there is a package; see the help page for ?PDEtools, the section on symmetry methods.

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

Hi nm
Check the help page, it is explained there, what a general, particular and singular solution means, what dsolve returns (the general and essentially singular solutions) and how to obtain the other ones if you prefer (using the optional argument singsol, or DEtools[particularsol]. As a very brief summary, a) there is no point in returning particular solutions, there are just infinitely many; b) singular solutions are returned only when they cannot be obtained by specializing the integration constants in any way (including here infinite values for them and limiting processes). The singular solutions returned are thus called essential. For a more technical description of singular solutions (essential versus non-essential ones) see ?diffalg,essential_components.

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

In eval(g(x),x=1), first g(x) is computed, resulting in the integrand f(2x), then eval is applied, finding "no x" in the integrand because this is an definite integral: 'x' is just a dummy here.

In eval(g(z),z=1), computing g(z) results in the integrand f(x+z), then eval at z = 1 is applied, finding z in the integrand (in this case z is not the dummy integration variable of a definite integral), so that it becomes f(x+1).

All looks right and enough. By the way what were you expecting?

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

Hi

To obtain the triangularization you mention, just remove 'integrabilityconditions = false' from the input, and DeterminingPDE will do that triangularization for you automatically. Now, if you prefer to do it manually, you can 'casesplit(DetSys)' and will get the same triangularization, which in turn can be performed using the RIF (Reduced Involutive Form) or DifferentialAlgebra packages. By default we use RIF (it is faster) but you can request to casesplit to do the task using DifferentialAlgebra by passing the optional argument diffalg. All this is explained in the help pages for PDEtools:-Determining and PDEtools:-casesplit .

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

The error message tells you only that dsolve cannot handle derivatives like

> diff(1/(ln(X)-Psi(1-f(X))-Psi(f(X))), [X$n-k[1]-k[2]]); # input this in the worksheet to see what it is

while solving w.r.t f(X). Indeed, this derivative is of order n-k[1] -k[2] but, mainly, the derivand is of the form 1/(ln(X)-Psi(1-f(X))-Psi(f(X))), which makes it be beyond what dsolve can handle for solving purposes. The error message actually collects all such derivatives that cannot be handled and were found in the input you sent to dsolve.

Now, none of these strange derivative constructions were introduced by dsolve. They were in the input you sent to the command. I may be missing something here but - besides the fact that these derivatives you sent to dsolve are rare constructions - the error message is actually simple to understand.

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

Hi Mehdi

I am not sure I understand your question correctly. Are you taking of D_[mu](d_[nu](psi(X)))? I.e. psi(X) is a scalar function, then d_[nu](Psi(X)) is what you call the "ordinary derivative" (... unlike the ordinary diff, d_[nu] is a rather special differential operator, whose effect, when applied to a scalar function, is actually a tensor, also in a curved spacetime). Then you want to take the covariant derivative of that - so just proceed, the covariant differentiation operator is D_, and if you enter all in one, that would be D_[mu](d_[nu](psi(X))). If so, check please the help page for ?Physics:-D_.

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

Hi
for x[k] use X[k], where X is the (main) coordinates system (i.e the one used by d_ and D_, the differentiation operators in the Physics package). For R[ijkl] you can, for instance, define a tensor R that actually has the same components of the Riemann tensor, then you go and define the metric using the expression you gave (REMARK: you are presenting a truncated power series ... that is not possible, for the purpose of symbolic computing with a metric, you really need to give an exact expression). For the defintions use the Define command. Hope this is clear enough, or else it would help if you could give more details, within a Maple worksheet (instead of the image of a formula).

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

Hi Chris

Look closer to your input lines: first you define k[mu] such that it depends on X through k0(X), k1(X) and k2(X), and epsilon. That is all OK. Then you define OpticalDeformationMatrix[mu,nu] - not in terms of k[mu] but in terms of k[mu](X) - that is k0(X)(X), k1(X)(X), k2(X)(X), sin(theta)(X), and epsilon(X). Give a look at the output of

> indets(OpticalDeformationMatrix[], unknown);

and you will see those undesired objects around. It is important to have in mind that once you defined a tensor with its dependency, you do not need to indicate its dependency on top of the one used in the definition. Remove then this additional dependency of k[mu] in your second definition input line. I.e. use:

> Define(NullGDE[nu]=(k[~mu]*D_[mu](k[nu])),OpticalDeformationMatrix[mu,nu]=(D_[mu](k[nu])));

Check it out now:

> selectremove(type, indets(OpticalDeformationMatrix[], Or(unknown, symbol)), function);

            {Phi(X), a(t), k0(X), k1(X), k2(X), k3(X), psi(X)}, {epsilon, phi, r, t, theta}

So: as expected.

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