ecterrab

missing details?...

Answered: ecterrab 13431

May 06 2015

0 0

Preben is right, without further details all I can say is that this works correctly, as shown in this image

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

yes...

Answered: ecterrab 13431

April 26 2015

0 0

@Markiyan Hirnyk

Indeed, when writing the answer I copied and pasted from trace's worksheet and didn't notice his typo, sin^2*theta instead of sin(theta)^2.. I included this correction now in the answer.

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

Answers to your two questions...

Answered: ecterrab 13431

April 17 2015

0 0

@Inacio

Your first question is: "Is there a way to remove the test function Phi in a different way?"

Yes, you can use coeff, as shown in this image:

Your second question: "Can Maple rewrite the expression above like F[mu nu]Phi? Because I have define F[mu, nu] like d_[rho](A[nu]) - d_[nu](A[rho])."

Yes, you can use simplify taking equations into account (check the help page ?simplify/siderels), as shown in this other image; note I first match the indices, and alternatively you could also use SubstituteTensor (check its help page):

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

Operatorial algebraic expressions - this...

Answered: ecterrab 13431

April 15 2015

0 0

Hi Inacio,

This functionality required a bit more of time. What you were expecting is actually a full implementation of "operatorial algebraic expressions", that is, algebraic expressions where you multiply differential operators with the meaning of applying them to what come to their right.

In Computer Algebra Systems, however, there is a strong distinction between multiplication of algebraic objects and application of differential operators: the latter are implemented as function application, (a function name applied to arguments) not multiplication. Also, when computing by hand, we multiply from the right of a differential operator with the meaning of "it gets applied to all what comes to its right", while with computer algebra "function application" there is no way you can do that. To implement such a thing in computer algebra, it is necessary to do a careful mimicry of the paper and pencil computation, i.e. one where you tell when is that the product becomes application of the differential operator.

In brief, implementing the whole thing required changes in various places, starting with the produt operator, making all differential operators non-commutative objects under multiplication, etc. The result was worth: a very nice full implementation of operatorial expressions, something really interesting to have in place, mainly for computations of Physics - we compute that way all the time.

Going to your example, good for illustration purposes

>

Note the Physics version, from today, you need to update your Physics library from the Maplesoft R&D Physics webpage to have everything working as shown below:

>

(1)

So, what follows is your worksheet with few changes to show how to use the new functionality (arriving at the results you were expecting for a Commutator involving sums of differential operators and functions of the coordinates)

>

First, note that d_[mu] operators commute between themselves, but not with functions of the coordinates

>

(2)

>

(3)

>

(4)

Now the commutator of a d_ operator with a function of the coordinates

>

(5)

Regarding the expansion of this commutator, I revised the experimental design of the previous reply for consistency: the product of an operator times a function is now expressed as a product (i.e. multiplication) instead of function application.

>

(6)

So in the above, you see , not , which would be the operator applied to . This permits multiplying from the right and allowing to have the operator applied (a posteriori) to the product that gets formed to its right:

>

(7)

>

(8)

That is, in the above, you see , not , which would be wrong (with applied only to )

Now we still want the function application of differential operators, not just their multiplication. This function aplication is now implemented using a new Physics:-Library command, called . In this example, the multiplications of differential operators you see in get transformed into their application that is, with the differential operators applied to all what comes to their right:

>

(9)

Likewise,

>

(10)

The dot operator also knows about these operatorial equations

>

(11)

Commutator already knows about operatorial equations, this is your example

>

(12)

Note first that the commutator of differential operators is now handled

>

(13)

Their multiplication and application too:

>

(14)

>

(15)

>

(16)

>

(17)

The above is the result you were expecting. Likewise,

>

(18)

But a commutator where there are non-differential operands, cannot be "applied", so it remains as a product

>

(19)

>

(20)

>

(21)

>

(22)

So regarding the expected result you mentioned:

Φ

= Φ + Φ + Φ + Φ

= Φ +

=

You are correct and that is indeed what the code returns now (see above, equation (17)) and in this way this most general concept of operatorial expressions that involve products of differential operators and functions that can then be transformed into function application at any desired moment is now available, which is really great, and thanks for bringing the subject to focus.

Download Identidades_de_Bianchi_(reviewed_II).mw

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

Heun issues: all fixed for Maple 17 or M...

Answered: ecterrab 13431

April 09 2015

0 0

@Carl Love

I don't know to what several bugs you are referring but just to clarify: there was one bug in the numerical evaluation of HeunC fixed in Maple 18 or Maple 17 (can't remember now) and another in the numerical evaluation of HeunG fixed in Maple 16. In addition to that, for Maple 2015, a new algorithm for the numerical evaluation of the same function is available. So no I'd not blame the numerical evaluation of Heun functions, unless you have some information of wrong results that I am missing, of course.

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

TimeMachine at work! :)...

Answered: ecterrab 13431

April 01 2015

0 0

@EugeneKalentev

The Mapleprimes site manager answered me saying that the worksheet and reply "Easier using Physics", that you are asking, is lost in unrecoverable way. Fortunately, however, I've found an old TimeMachine backup of my older laptop actually containing this worksheet! I am amazed at how well this simple Macintosh TimeMachine backup system works. So I restored the worksheet and the answer, including a link to the worksheet, all in "Easier using Phyiscs".

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

It is implemented - update your Physics...

Answered: ecterrab 13431

April 01 2015

0 0

Regarding commutators involving the differential operators of the Physics package (both d_[mu] and D_[mu]), the Commutator and AntiCommutator commands now work with them, including the case of vectorial operators. You need to update your Physics library from the Maplesoft R&D Physics webpage.

>

The version of Physics is from today:

>

(1)

Commutator of Nabla with a scalar function

>

(2)

>

(3)

>

(4)

Commutator of Nabla with a vector function

>

(5)

>

(6)

Define A as a tensor

>

${A, Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}$

(7)

>

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x1, x2, x3, x4)}$

(8)

Compact display for A

>

(9)

This commutator is from your worksheet

>

(10)

>

(11)

So the order of the operands is respected: d_[mu] is now considered a non-commutative object with regards to multiplication. As usual you can query:

>

(12)

>

(13)

Now, the title of your worksheet is 'identidades de Bianchi, so I guess you are talking of a curved spacetime? Set the spacetime to something curved then, for instance,

>

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}$

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = r/(-r+2*m), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 4) = (r-2*m)/r}, storage = triangular[upper], shape = [symmetric]))$

(14)

Check the commutation of the covariant derivatives

>

(15)

Hence,

>

(16)

>

(17)

Apply this expanded commutator to

>

(18)

>

(19)

>

(20)

Likewise,

>

(21)

>

(22)

So now it is applying that you get the result

>

(23)

although you could have also multiplied, then applied

>

(24)

>

(25)

Get for instance the component 1,2,2 of this tensorial expression

>

$T := Array(1..4, 1..4, 1..4, {(1, 1, 1) = 0, (1, 1, 2) = 0, (1, 1, 3) = 0, (1, 1, 4) = 0, (1, 2, 1) = A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 2, 2) = A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 2, 3) = A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 2, 4) = A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 3, 1) = A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 3, 2) = A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 3, 3) = A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 3, 4) = A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 4, 1) = -2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 4, 2) = -2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 4, 3) = -2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (1, 4, 4) = -2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (2, 1, 1) = -A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (2, 1, 2) = -A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (2, 1, 3) = -A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (2, 1, 4) = -A[2](r, theta, phi, t)*m/(r^2*(-r+2*m)), (2, 2, 1) = 0, (2, 2, 2) = 0, (2, 2, 3) = 0, (2, 2, 4) = 0, (2, 3, 1) = 0, (2, 3, 2) = 0, (2, 3, 3) = 0, (2, 3, 4) = 0, (2, 4, 1) = 0, (2, 4, 2) = 0, (2, 4, 3) = 0, (2, 4, 4) = 0, (3, 1, 1) = -A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (3, 1, 2) = -A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (3, 1, 3) = -A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (3, 1, 4) = -A[3](r, theta, phi, t)*m/(r^2*(-r+2*m)), (3, 2, 1) = 0, (3, 2, 2) = 0, (3, 2, 3) = 0, (3, 2, 4) = 0, (3, 3, 1) = 0, (3, 3, 2) = 0, (3, 3, 3) = 0, (3, 3, 4) = 0, (3, 4, 1) = 0, (3, 4, 2) = 0, (3, 4, 3) = 0, (3, 4, 4) = 0, (4, 1, 1) = 2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (4, 1, 2) = 2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (4, 1, 3) = 2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (4, 1, 4) = 2*A[4](r, theta, phi, t)*m/(r^2*(-r+2*m)), (4, 2, 1) = 0, (4, 2, 2) = 0, (4, 2, 3) = 0, (4, 2, 4) = 0, (4, 3, 1) = 0, (4, 3, 2) = 0, (4, 3, 3) = 0, (4, 3, 4) = 0, (4, 4, 1) = 0, (4, 4, 2) = 0, (4, 4, 3) = 0, (4, 4, 4) = 0})$

(26)

>

(27)

>

Download Commutator_and_d.mw

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

updating your Physics...

Answered: ecterrab 13431

March 31 2015

0 0

@mpre

I know it may sound as advertisement but, if you could trust me ... it is not that. What it is: Physics in Maple 2015 is already a big step forward with regards to Physics in Maple 18, and it has been this way at every release since Maple 15. So if this package is of relevance for your work in any particular way, I strongly recommend you to update your copy of Maple.

Not less important: since Maple 18 (August of 2013), Maplesoft releases an update of the Physics package on the web every week (sometimes every day of a week). What this means:

you do not need to wait one year (for a new release) to see a bug fix or feature request in place
you can participate direclty in the development of the package by presenting your concrete real-work problem and when possible the package gets its functionality extended to be of use in your real case scenario.
you have access to the latest developments in the package 1 release advanced.

And that happens for a whole year. You realize the disadvantage of staying with Maple 13. Besides, it is very difficult to help you with regards to the use of Maple 13 - the package is just not the same.

If you have the opportunity to update your copy, please feel free to post what you describe as your more complicated problem, and from there we move ahead, I may be able to help you, or adjust the package as required.

Best

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

...hmm...

Answered: ecterrab 13431

March 29 2015

0 0

@Michael_Watson

I recently answered in Mapleprimes a question you made about a simliar issue: when using labels, the output was comming as 0 = 0 (wow with the coincidence!). As said in that reply, there is a bug in the interface (GUI). There is a workaround to this issue. In order to have the workaround installed, you need to update the Physics library that comes with the Maple 2015 installation, as usual by downloading it from the Maplesoft R&D Physics webpage. Could you please do that and let me know whether this problem with using GUI labels gets "resolved"? Thanks.

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

missing info...

Answered: ecterrab 13431

March 28 2015

0 0

Hi

When executing your worksheet, I do not get 0=0 in equation (19) as you show. Could you please post a separate worksheet with the output you expect from algsubs ((9), (18))? That will help understanding what is the problem you are experiencing, and from there we move ahead.

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

posting a worksheet?...

Answered: ecterrab 13431

March 28 2015

0 0

@patient

Would you mind posting a worksheet with the input equations? That would help avoiding mistakes while reading the image you posted to retype the system. Thanks.

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

Set the metric to 'Euclidean'...

Commented: ecterrab 13431

March 21 2015

@alljoyland

Give a look at the documentation (help pages for g_ and Setup): you can set the metric to what is appropriate for your problem. For example, if you set it to 'Euclidean', all the powerful tools you see working in curved spacetimes (general relativity) become available for tackling problems for instance in elasticity (what you are asking).

Independent of that, you can work with space indices, independently of working with spacetime indices. Recalling, the space part of a Minkowski metric is actually Euclidean (Minkowski is the default type of spacetime that is set automatically when you load Physics, i.e. a flat non-curved space, not for general relativity). This use of space indices is illustrated in the page ?Physics,examples.

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

Yes ......

Answered: ecterrab 13431

March 08 2015

0 0

@trace

When indexing tensors, say in a 4D spacetime, 1,2,3,4 refer to the positions of the components of the tensor in the corresponding tensor array. For example, if the tensor has 2 indices, the array is a matrix, so - say - the component [2, 3] is the component you see in the 2nd row, third column. Yes there is of course a relatioinship between these positions and the order you indcate for the coordinates as a list, so 1=t, 2=r, 3=theta, 4=phi, this is clear, for instance, in the spacetime metric (see the reviewed worksheet). And yes: the number itself means covariant, so 1 means 1st position, covariant, and ~1 means contravariant 1st position, so g_[~1, ~1] is the 1st line, 1st column, of the all-contravariant g_, and Christoffel[~1, 2, 2] means 1st index contravariant, position 1 in the array, and 2nd and 3rd index both covariant, position 2 in the array.

Try

> Christoffel[~1, alpha, beta, array];

or

> Christoffel[~alpha, 2, 2, array];

Instead of 'array' you can also use the keyword matrix. All these things and some more are explained in the documentation.

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

tracked and workaround...

Answered: ecterrab 13431

March 04 2015

0 0

Hi

This GUI issue has been tracked - and a solution circumvecting the problem is available - for that you need to update your Physics library from the usual place, the Maplesoft R&D Physics webpage.

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

different ics?...

Answered: ecterrab 13431

February 27 2015

0 0

@Codge

Wouldn't your reply suggest that the ics you are using are not convenient for producing sections with p1 > 0?

Independent of that, have you explored the option 'scene'? Note that you can use it as in 'scene = [p1 = 0..3, q2 = -2..2, p2 = -2..2]' or the like, ie you can specify the ranges for each of the three variables, together with the option '3' to produce a 3D section that you could then rotate with the mouse to have different 2D projection views

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

E-Mail Address:
Password:
Remember Me:	Automatically sign in on future visits

E-Mail Address:
Password:
Remember Me:	Automatically sign in on future visits

Ask a Question

Create a Post

13431 Reputation

24 Badges

MaplePrimes Activity

These are replies submitted by ecterrab

missing details?...

yes...

Answers to your two questions...

Operatorial algebraic expressions - this...

Heun issues: all fixed for Maple 17 or M...

TimeMachine at work! :)...

It is implemented - update your Physics...

updating your Physics...

...hmm...

missing info...

posting a worksheet?...

Set the metric to 'Euclidean'...

Yes ......

tracked and workaround...

different ics?...

Save this setting as your default sorting preference?

Ask a Question

Create a Post

Generating PDF…

Save this setting as your default sorting preference?
Note: You can change your preference any time in your account settings
Don't show this again