Have you seen this Mapleprimes post: "The Zassenhaus formula and the algebra of the Pauli matrices"? It has the kind of approximation using expansion similar to what you are mentioning.

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

@Rouben Rostamian  

I don't want to enter a debate about this, but I don't think the interpretation of the wiki page is as you are saying.

Here is another page by people I respect where I read the same, perhaps more clearly said: "The use of dyadics is nearly archaic since tensors perform the same function but are notationally simpler."

This other one is from math.stackexchange: "Dyadics and polyadics are a fantastically archaic way, due to Gibbs of thermodynamics fame, to denote and work with linear transformations and tensors in general."

The word dyadic does not appear in the book on Statistical Mechanics by Feynman, nor in the ones about Fluid Mechanics and Electordynamics of Continuous Media by Landau and Lifchitz. I mentioned that about the so-complete Landau's Course of Theoretical Physics. All these textbooks work with tensors, something implemented very well in Maple - see ?Physics,Tensors.  

All in all, yes, there are books as you say that still use dyadic notation but I (personally) don't think they represent modernity as you seem to say. Yes, I realize too that you feel different about this; there is nothing wrong, diversity of opinions.

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

@Rouben Rostamian  

I can see I read your post too fast, missed what you were talking about - Nabla is certainly not the answer to your question, which indeed is more like a request for a feature, not a question. Sorry for that.

By the way, tensor products exist in Physics, but using Dirac's notation, see the help page ?Updates,Maple2019,Physics.  That notation can be mapped one to one into 3D vector notation (equivalent to the dyadic notation you are mentioning) . I will think about your request. I note however that I never heard of people in need of dyadic vector notation. People, e.g. working with continuum mechanics as you say, frequently use tensors. Regardless of opinions, it is a fact that the whole collection of books on Theoretical Physics by Landau & Lifshitz, which use tensors all around, does not use dyadic notation. 

Also, from the Wikipedia help page: "Dyadic notation was first established by Josiah Willard Gibbs in 1884. The notation and terminology are relatively obsolete today." If I were to point out a hole in Physics[Vectors], the main development priority, I'd mention Vector Integral calculus.

Anyway, as you probably know, Physics is a project I care about a lot but that takes time, working alone on it, and with so many areas into which extend. For Maple 2021, particle physics, strengthening all around, and more documentation examples, were the focus. I need to come back to this less advanced but in some sense with more frequently used functionality like Integral Vector Calculus.

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

@mthkvv 

I returned to this today, and have a few comments. I fail to see the problem formulated as in the book. Instead of 96.5 you start from some other expression,

where, in addition, you missed the red dot (multiplication) you see inserted in this image above. The way you entered, %g_[determinant])(...) was a function, not a product, and that generated that error message about the indices (so not an index issue, the error message is correct, but an input issue in your worksheet).

In my previous reply, I suggested a step-by-step based on entering 96.5, that is 

So, define your t[i,k], your h[i,k,l] as in the book, then isolate t[i,k] etc. I am not sure the steps I outlined in the previous reply will take you to 96.8, I didn't try, but if you formulate this correctly I can help you in doing that derivation in the computer.

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

@nm 

The issues you mention now are those I mentioned, to be resolved in a next Updates (v.977).

About breqn, I respectfully disagree. I find it nowadays an excellent package. By the way, the use of breqn passed almost 3 months of intensive testing in the beta forum, and passed that test with flying colours. As an example of that, see this Mapleprimes post.

Best

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

@Sradharam 

The way to go with computer algebra systems is to type a couple of words in their help system. Try that. Type Lie algebras in Maple's help. You will see two packages popping up, not one. DifferentialGeometry, the other package, is the state-of-the-art. That said, your statement of the problem is also unclear. What do you mean, exactly, by "Lie subalgebra for finding optimal solutions"? You can help people help you by clicking that green arrow and uploading a worksheet with your problem formulated, up to what you know how to do it, and then text with your question.

So please check the help system, and if what you need is not found in the help pages (or you are lost in the amount of them), please post a worksheet with your problem; people here will try to help you.

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

@subzero 

Also, in 3-D the Weyl tensor has all its components = 0. BTW, in the input that results in (12), it should be Ricci, not Ricch.

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

@subzero 

Setup(dimension = 3). Check the help page of Physics:-Setup

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

@subzero 

Download Gamma_R_Ri_W.mw

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

@subzero 

Your question is too specific to answer without seeing details. First of all, do you have Maple? If so, open please the help page ?Physics:-Christoffel. There is a defining formula there. The starting point is for you to tell, in formulas, on a Maple worksheet, exactly what (apparently different?) definition you want to use. Without that I can't help you more than the generic comments in the previous two replies, or saying my opinion, that Maple has more thorough and flexible tools for working with general relativity than any other platform (as in: for GR, if not in Maple, then nowhere).

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

UPDATE Mar/21: This is now fixed and the fix distributed for everybody using Maple 2021 within the Maplesoft Physics Updates v.934 and newer.

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

UPDATE Mar/21: This is now fixed and the fix distributed for everybody using Maple 2021 within the Maplesoft Physics Updates v.934 and newer.

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

@subzero 

You can't do it the way you say, the same way you can note redefine the exponential function and expect that everything else uses your new definition. Christoffel has a unique definition. That definition is used in several places. If you post your problem or an example that illustrates what you need more specifically, then one could imagine a solution.

Meantime, what you can do is ignore entirely that all these tensors exist predefined according to textbook definitions (Christoffel, Ricci and Riemann) and define yours, say C, R, Ri, where your C I imagine is in terms of the metric g and its derivatives, R is some contraction of Ri and Ri is a function of C, g and its derivatives. To define a W traceless tensor related to Ri is also trivial. All that can be done defining tensors as explained in the Physics,Tensors help page, the sections on how to define tensors. Actually, it wouldn't take more than 5 minutes to define three or four tensors like those.

To summarize, you can always ignore the existing definition of these tensors and work with an entire set of other ones that you define with ease in a few minutes at most. It is simpler if you could show, on a Maple worksheet, more specifically, what you need and how do you intend to use it. There may be other more convenient solutions.

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

@nm 

Copy As LaTeX works pretty-well, and is a very nice feature of Maple 2021. Don't give up that. Why don't you write an email to support@maplesoft.com telling the problem? I'm sure they will do their best to help you diagnose and fix the problem, which, by the way, is not related to Physics, the Updates or anything else. I would try to uninstall Maple and then use a File Manager to wipe out every remaining for-sure. Then re-install; though I suppose you already tried that, anyway, support is the way to go, I think.

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

@nm 

It is difficult to imagine the problem; guessing here, could it be that you have another library interfering? Maybe some Maple 2020 library around in a personal directory ... or the Physics Updates installed by hand?

Take a look at the directories mentioned in the variable libname and make sure you don't have any additional libraries around. Then, Copy As LaTeX should work right away, even if you don't have the Maplesoft Physics Updates installed. And if you have it, as said below, I implemented a new option so that things get copied as output, not as input.

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