@Jamie128 @Hullzie16 @Rakshak
I misread the original question; sorry for that; I answered again in the same frame where I presented my first answer - see further above.

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

Without disputing preferences, there is alias:-Show, which handles arbitrarily large algebraic expressions that may (or not) contain subexpressions involving aliased things. For example

There is more; see the output of exports(alias).

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

@Hullzie16 
I hope you don't take me wrong, but what you are doing is not the proper way of comparing things. I see no bug in Define, nor in series, nor do I see two results; always one and only one:

Download Also_NO_SeriesBug.mw

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

@Hullzie16 

The way you wrote the worksheet, I cannot follow by eye that the steps in sector 1 are equivalent to those in sector 2. But there is a simpler way to compare assuring you are performing the same steps, as shown in this image (mw attached at the end), where you see the two results, where (9) is after Defining the tensor P, and (10) is after taking a=1, b=1 without defining anything, results in the same thing: (9) - (10) = 0.

UPDATE May/1: independent of "no bug that I could see here", in the Maplesoft Physics Updates for Maple 2023, v.1436 or newer, there is a new option for the Define command: computetensordependency = false (default value is true), which skips the most time-consuming verification step. Using this option, the Define(ition) of this tensorial expression of length > 50,000 is performed in ~100 seconds in a two-year-old Mac (M1 chip).

Download NO_BugMaple_(reviewed).mw

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

@sursumCorda 
That work/comparison is indeed useful. I also see that the number of DEs that Maple can solve and Mathematica cannot is much-much larger than the opposite, both for ODEs and PDEs. I do not work following anything in particular, but yes, this comparison is a good guideline for improvements; to increase the existing Maple advantage in this area.

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

@Diasaur 

Download noncommutative_prefixes_and_quantum_operators.mw

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

@dharr and @Pascal4QM,
Good catch. There was indeed one place, in one internal routine, where the old dot operator `.`, blind regarding Not(commutative) objects, was in use. This is now fixed,

The fix distributed to everyone using Maple 2023 within the Maplesoft Physics Updates v.1431 and newer.

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

@Thomas Richard 
Yes, if you include those two equations within that list, you have simplify also not introducing csch and sech.

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

@Thiago_Rangel7 
Yes, 1/cot(z) -> tan(z) looks more convenient; it is a different thing though; also working that way since ages, not new in Maple 2023. I am currently not working in simplify but will take a look, maybe it is something easy to address.

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

Hi
This post is now updated with the finished version of this material that appears in Maple 2023 as the help page Courseware-Support, Mechanics. The material - 70 solved problems on a computer algebra worksheet covering most of the syllabus of Mechanics courses - runs also in Maple 2022.2 after installing the last Maplesoft Physics Updates for Maple 2022, that is, v.1409, uploaded today. 

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

@prldb 

You asked a very similar (basically the same) question in 2020. The answer today would be the same, only emphasizing that the ordering of the NP null vectors that conform a tetrad depends on the signature, basically on where you put the timelike component in the metric. For the signature used in Maple, which is also the one used in Stephanie's book, the ordering is n, m, mb and l also in the book. And yes the convention used by Maple for the NP null vectors is the existing one (standard) - also used in Stephanie's book. All this is shown in that answer of 2020, also explained in the link I mentioned in the previous reply "What to take care of when entering a tetrad".

Now, the tetrad is defined up to an arbitrary rotation in 4 dimensions, from which you can mix everything and write a new tetrad with the imaginary unit anywhere you want. That doesn't change the fact that n, m, mb, l form a tetrad when you put them in the correct order according to the signature, and doesn't change the definitions of these NP vectors that you can see, for example with n_, via

Try TensorArray([%]) with some simplfier and you see all their defining equations are satisfied, regardless of where you put the imaginary unit.

Regarding the optimized tetrads shown in Stephanie's book: Maple is not using those tetrads - as said in a previous reply above, those book's tetrads are coded, but won't be used until they are reviewed. Instead, Maple computes a tetrad from scratch as explained in ?Tetrads:-IsTetrad, and as said there you could also enter any other different tetrad that you may prefer and the system will work with it the same way. Or if you prefer to perform transformations to optimize your tetrad in any particular way, see ?Tetrads:-TransformTetrad.

Regarding Maple's tetrad, which is constructed from the NP vectors, the standard ones, in the right ordering according to the signature used in Stephanie's book, or regarding any other tetrad, you can always test whether it is or not a correct tetrad using Tetrad:-IsTetrad,  or checking whether it satisfies the tetrad equations, basically input TensorArray(e_[definition]) and you see, by eye, if it is or not a correct tetrad, regardless of where you put complex components. Here is for the metric you are mentioning:

In summary: the tetrad computed by Maple is correct, the NP null vectors are correct too, they are the standard ones, and the conventions for how to order the NP vectors to form a tetrad are those shown in the book too, as shown step-by-step in the answer I gave to you during 2020. The only thing that is not a match to Stephanie's book is that Maple computes a tetrad that is not as the one shown there, where the case is that tetrads are defined up to an arbitrary 4D rotation.

I'm sorry but also need to move to other topics.

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

@sand15 

Physics significantly enlarges the computational domain, i.e. what can be part of the input and output. That can be felt as "a world within Maple", I suppose.

Regarding your question about the Updates, quoting from Maplesoft Physics Updates, the answer is

"... You can take advantage of this ongoing work by downloading the research version of Physics as it is updated with improvements, fixes, and the very latest new developments. ..".

This ongoing work happens on a daily basis, so you have new versions of the Maplesoft Physics Updates - on average - say 365 times per year. You see why it cannot be "in sync" with Maple's version numbers (typically only two per year).

If by 'such a special package' you refer to 'updating the package on the web around the clock', I prefer to work in close contact with people who use the package(s) and to accomplish that - in practice - requires quick feedback that people can use right away (thus, the Maplesoft Physics Updates).  As a result, Physics is what it is and the project just moves forward at its own rhythm.

Finally, note that the Maplesoft Physics Updates include not just Physics, but also Typesetting, Differential Equations, Mathematical Functions, some others, and frequently also bug fixes of things that interfere with the computational functionality in any of those areas. 

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

@mmcdara 

Your approach is also interesting in that it shows that changing the value of the summation index is - conceptually and so for the computer - a change of variables. By the way, IntegrationTools:-Change performs changes of variables calling PDEtools:-dchange.

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

@C_R 
It is unfortunate for me that you didn't bet. :)

To the side, if we disregard the word we use, and instead focus on the concept, sums and integrals have summation and integration variables and in both cases (also in sums) what we are doing is (conceptually) "a change of variables".

Incidentally, the very first, tinny routine from which dchange evolved later on into a full-change-of-variables command is this one for "changing variables in Sums", in connection with a paper I was co-authoring years ago in plasma physics.

Anyway, I can understand your comment today. I will try to remember to add an example of this in ?dchange.

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

@prldb 

Could you please also take a look at this other question: "What to take care of when entering a tetrad" - I think it is related to your question now.

Independently, yes the documentation needs to be clear about these things, I will take care of that.

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