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It works fine for me...

ecterrab 14540
September 06 2022
0 0

@lemelinm

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

Hi Mario, could you please upload a work...

ecterrab 14540
September 05 2022
0 0

Hi Mario, could you please upload a worksheet with the problem formulated? You can use the Green arrow for that purpose.

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

a worksheet with the problem?...

ecterrab 14540
September 02 2022
0 0

Hi, please use the Green arrow to upload a Maple worksheet with the input for your problem and some text making your question or expected solution clear; thanks.

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

Using Physics...

ecterrab 14540
August 24 2022
0 0

I hope it helps you in performing your calculations easier.

Download Using_Physics_instead.mw

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

@justauser Physics has Fundiff, to ...

ecterrab 14540
August 17 2022
0 0

@justauser 
Physics has Fundiff, to compute Lagrange equations in general by variating the Action as an integral of the Lagrangian density. It is true, however, that none of these three packages have a command to tackle the inverse problem, that is, to find the Lagrangian given Lagrange's equations (am I right in understanding that is what you want?)

At first sight, I would think that a direct approach, formulating the problem with L as an unknown, then tackling the system of equations with PDEtools commands, would be the way to go, instead of computing the Helmholtz conditions for the existence of the Lagrangian to then try to compute the Lagrangian itself. If Helmholtz conditions are not satisfied, the PDEtools commands will automatically return nothing.

Could you please write, within a Maple worksheet, who your 'vg' is and what solution (Lagrangian) you expect and upload the worksheet? (For that, use the green arrow you see when posting something here.) If you do not know what you expect, could you please add in that worksheet an example where you give vg and the solution you expect? With that, I can take a look

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

Hi @Carl Love I am saying it i...

ecterrab 14540
August 07 2022
0 0

Hi @Carl Love 
I am saying it is a different problem, but not just that: I fixed both problems in two different places. One is: simplifying a RootOf returned by solve may, rarely but can happen, result in a RootOf("...") where "..." has no _Z around. The other is: due to a bug in solve in 1999, there was a line of code removing solutions x = x for single equation problem. That bug got fixed in 2000, but that line in the internal differential equation (solve) routine remained in place. Then you posted what, for me, is a valid example of solution x = x, not related to RootOf(...) that has no _Z. So: two problems that required two fixes. That explains in detail why I wrote "these are two different and unrelated problems."

Then your view on this, that "something", solved with respect to x, could not have x when "something" is simplified, yes, in that sense, there is similitude; your message, however, didn't say that. Instead: "here is another one where solve and PDEtools:-Solve return different, and I (Carl) prefer the output by solve". 

Maplesoft Physics Updates v.1276...

ecterrab 14540
July 27 2022
0 0

@Rouben Rostamian, @nm 

I deleted my previous comment, "uses" Physics:-Vectors is not necessary in Rouben's program, but anyway I preferred to look at this closely. The issue is now resolved in the Maplesoft Physics Updates v.1276.

The details are:

1) Vectors:-`-`  (not a Vectors visibly exported command) is an internal routine coded as a hidden export. I would expect not to be binded by uses, but it is. Maybe my expectation is due to my lack of familiarity with uses, a useful and straightforward idea.

2) Vectors:-`-` was coded (~30 years ago) as a binary operator. Used as an internal routine, that is all that was needed. But the same way as `+`, the `-` operation is well defined with 0, 1 or N > 1 operands.

3) Within Rouben's program, "1 - p.q" is an operation (`-`) with two operands, not just one, but when "uses" is there, Vectors:-`-` receives only one operand, resulting, due to 2), in the error interruption observed. You see that by entering debug(Vectors:-`-` ) before entering my_module:-my_proc(u_,v_). If, in Rouben's program, instead of `-` you use `+`, then debug(Vector:-`+`), you see that two operands are passed to Vectors:-`+`, not just one. (Unlike Vectors:-`-`, Vectors:-`+` is a Vectors visibly exported command, expected to be binded by uses.) So up to what I can see there is an issue in "uses" here: it passes two operands to `+`, but only one to `-`.

Item 2) is corrected by removing the (unnecessary) restriction to "always expect 2 operands"; with that, the problem noticed disappears entirely. Item 3) is now tracked as an issue in uses, to be reviewed by the kernel group.

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

 

Vectors:-Identify...

ecterrab 14540
July 26 2022
0 0

Hi @Rouben Rostamian  

The Maple default type 'scalar', as several commands previous to Physics, knows nothing about this extended domain that includes vectors, tensors, abstract matrices (non-commutative symbols) etc. Instead of that, please check the help page for Vectors:-Identify. That is what you need to distinguish a scalar from a vector.

To the side: Physics and its subpackages are self-complete, including, in addition to the package's commands, the Physics:-Library (nops(with(Physics:-Library)) -> 172) and the library of Physics types (nops(with(Physics:-Library:-PhysicsTypes)) -> 114). Those are the internal routines and types with which the Physics package and its subpackages are written.

Of course, there are Maple commands that exist before Physics, and you can use all of them when writing code for the extended domain implemented with Physics, but depending on the (pre-Physics) command or type, you need to first check that they work as you intend. In part for that reason, I made the whole Physics:-Library and its types available and documented at user level - they allow you to code everything you'd need to compute with the extended domain.

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

 

  Hi Rouben, Having a ...

ecterrab 14540
July 26 2022
0 0

 

Hi Rouben,

Having a unique zero is an intentional design, not a mistake, not a bug nor something of the like.

 

First, note that the error message you show is due to the way you programmed your procedure f enforcing a type checking of its argument that only accepts a vector (what you'd call a non-zero vector). For example, if instead of that you accept input of type algebraic and use Vectors:-Norm, you have the f you seem to have tried to program

 

with(Physics:-Vectors)

 

f := proc (q::algebraic) options operator, arrow; Norm(q)^2 end proc

proc (q::algebraic) options operator, arrow; Physics:-Vectors:-Norm(q)^2 end proc

 (1)

f(a_)

Physics:-Vectors:-Norm(a_)^2

 (2)

f(k*a_)

Physics:-Vectors:-Norm(k*a_)^2

 (3)

expand(%)

k^2*Physics:-Vectors:-Norm(a_)^2

 (4)

f(0*a_)

0

 (5)

All the product and sum operations between objects of type algebraic are well defined with 0

0+a_

a_

 (6)

0.a_

0

 (7)

`&x`(0, a_)

0

 (8)

In this context, distinguishing between the algebraic 0 you see in (7) and the algebraic 0 you see in (8) is of no use, not for the purpose of performing algebraic computations (computing with objects of type algebraic). Such a distinction would then be artificial; an unnecessary complication.

 

Regarding your comment about other packages, VectorCalculus and LinearAlgebra compute with matrix representations of vectors, and there is indeed a computational distinction between the number 0 and a matrix all of whose components are equal to 0; so these two packages need to distinguish between the number 0 and a "0 vector" (represented as a column matrix of zeros). One of the main disadvantage of the matrix representation of vectors is that you cannot compute with the basic object A_, a non-projected vector, as you do when computing with Physics:-Vectors.

 

By the way, one of the key design differences between Physics and other packages is that, in Physics, instead of restricting the representations of mathematical objects to scalars or matrices, the domains of the operations were extended to handle the objects we use when computing with paper and pencil. That includes abstract vectors, tensors, Einstein's sum rule for repeated indices, anti-commutative and non-commutative objects, spinors, etc. All these objects, not present in the standard domain of computer algebra systems before Physics, are implemented with type algebraic. The number of advantages of having all these objects of type algebraic (when that is possible) is too large to list here, but for me the main one is that it allows for the closest match between what we do when computing with paper and pencil and what we can now do on a computer algebra worksheet.

 

If regardless of the above you still want to have a type that includes 0 as a vector you can use Or(0, PhysicsVectors). Then your procedure f will work as you planned, and in addition you can assign a name to this combo, as in TPV := Or(0, PhysicsVectors).

Download unique_zero.mw

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

 

Same procedure solves the problem...

ecterrab 14540
July 15 2022
0 0

@bikramphy 

You have now added information, the metric, OK. Below is basically the same worksheet, with small changes, taking into account the form of the metric.

 

with(Physics); with(Tetrads)

_______________________________________________________

 (1)

Simplify steps setting the signature as close as possible to the one you want, and set coordinates as you now indicated they are

Setup(signature = "+++-", coordinates = (X = [Q, q, u, t]))

[coordinatesystems = {X}, signature = `+ + + -`]

 (2)

 

Input the metric that you are now indicating

ds2 := dQ*dq+dQ*dq+du*dt+du*dt-Typesetting[delayDotProduct](2*Q/m.(diff(V(q, t), q)), dt, true)*dt

2*dQ*dq+2*du*dt-2*Q*(diff(V(q, t), q))*dt^2/m

 (3)

Set compact display to make things more readable 

CompactDisplay(2*dQ*dq+2*du*dt-2*Q*(diff(V(q, t), q))*dt^2/m)

V(q, t)*`will now be displayed as`*V

 (4)

Setup(metric = ds2)

[metric = {(1, 2) = 1, (3, 4) = 1, (4, 4) = -2*Q*(diff(V(q, t), q))/m}]

 (5)

The tetrad computed by default is orthonormal

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 36893488153642513460)

 (6)

"IsTetrad(?)"

true

 (7)

From

e_[definition]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

 (8)

You see the form of the tetrad metric eta[a, b] that indicates the signature, is "(+++-)"

e_[a, mu]*e_[b, mu]; % = TensorArray(%, simplifier = `@`(simplify, evala))

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Matrix(%id = 36893488153619389964)

 (9)

You need to find a matrix A such that when redefining the tetrad as A * e_ = E, and substituting into the above, you get "E[a,mu] E[b]^(mu)=([[[1,0,0,0],[0,1,0,0],[0,0,-1,0],[0,0,0,-1]]])", where in the right-hand side you see the signature "(++--)".


I'd search for a matrix of this form, adding one element out of the diagonal, mostly for illustration purposes (you see in the result that was actually not necessary)

A := subs(`~`[`=`]([a[1, 2], a[1, 3], a[1, 4], a[2, 4], a[3, 4]], 0), Matrix(4, symbol = a, shape = symmetric))

Matrix(%id = 36893488151942187836)

 (10)

E[a, mu] = A.rhs(e_[])

E[a, mu] = Matrix(%id = 36893488151939865164)

 (11)

Define this as a tensor, to use the tensor functionality to build a set of equations for the `a__i,j`

"Define(?)"

{Physics:-D_[mu], Physics:-Dgamma[mu], E[a, mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-Tetrads:-e_[a, mu], Physics:-Tetrads:-eta_[a, b], Physics:-g_[mu, nu], Physics:-gamma_[i, j], Physics:-Tetrads:-gamma_[a, b, c], Physics:-Tetrads:-l_[mu], Physics:-Tetrads:-lambda_[a, b, c], Physics:-Tetrads:-m_[mu], Physics:-Tetrads:-mb_[mu], Physics:-Tetrads:-n_[mu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (12)

Introduce the expected form of the tetrad metric to build the set of equations to be solved for the `a__i,j`

Eta[a, b] = Matrix(4, 4, [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]])

Eta[a, b] = Matrix(%id = 36893488153673893876)

 (13)

"Define(?)"

{Physics:-D_[mu], Physics:-Dgamma[mu], E[a, mu], Eta[a, b], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-Tetrads:-e_[a, mu], Physics:-Tetrads:-eta_[a, b], Physics:-g_[mu, nu], Physics:-gamma_[i, j], Physics:-Tetrads:-gamma_[a, b, c], Physics:-Tetrads:-l_[mu], Physics:-Tetrads:-lambda_[a, b, c], Physics:-Tetrads:-m_[mu], Physics:-Tetrads:-mb_[mu], Physics:-Tetrads:-n_[mu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (14)

This is the identity you use to adjust the `a__i,j`

E[a, mu]*E[b, `~mu`] = Eta[a, b]

E[a, mu]*E[b, `~mu`] = Eta[a, b]

 (15)

TensorArray(E[a, mu]*E[b, `~mu`] = Eta[a, b], simplifier = simplify)

Matrix(%id = 36893488153624419132)

 (16)

"convert(?,setofequations)"

{0 = 0, a[1, 1]^2 = 1, a[2, 3]*(a[2, 2]+a[3, 3]) = 0, -a[4, 4]^2 = -1, a[2, 2]^2+a[2, 3]^2 = 1, a[2, 3]^2+a[3, 3]^2 = -1}

 (17)

solve({0 = 0, a[1, 1]^2 = 1, a[2, 3]*(a[2, 2]+a[3, 3]) = 0, -a[4, 4]^2 = -1, a[2, 2]^2+a[2, 3]^2 = 1, a[2, 3]^2+a[3, 3]^2 = -1})

{a[1, 1] = 1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = 1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}, {a[1, 1] = 1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = 1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}, {a[1, 1] = -1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = -1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}, {a[1, 1] = -1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = -1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}

 (18)

Any of these solutions serve your purpose, take the first one, remove that RootOf

sol := convert(({a[1, 1] = 1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = 1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}, {a[1, 1] = 1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = 1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}, {a[1, 1] = -1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = -1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1}, {a[1, 1] = -1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = 1}, {a[1, 1] = -1, a[2, 2] = -1, a[2, 3] = 0, a[3, 3] = RootOf(_Z^2+1), a[4, 4] = -1})[1], radical)

{a[1, 1] = 1, a[2, 2] = 1, a[2, 3] = 0, a[3, 3] = I, a[4, 4] = 1}

 (19)

The tetrad that results in the tetrad metric "(++--)"is then given by

subs(sol, E[])

E[a, mu] = Matrix(%id = 36893488153642495860)

 (20)

Set now this matrix as the tetrad

"Setup(tetrad = rhs(?))"

[tetrad = {(1, 1) = ((1/2)*I)*2^(1/2), (1, 2) = -((1/2)*I)*2^(1/2), (2, 3) = (1/2)*2^(1/2)/((Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2), (2, 4) = -(1/2)*2^(1/2)*(Q*(diff(V(q, t), q))-(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2))/(((Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2)*m), (3, 3) = ((1/2)*I)*2^(1/2)/(-(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2), (3, 4) = -((1/2)*I)*2^(1/2)*(Q*(diff(V(q, t), q))+(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2))/((-(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2)*m), (4, 1) = -((1/2)*I)*2^(1/2), (4, 2) = -((1/2)*I)*2^(1/2)}]

 (21)

And that is all.

 

Verify now:

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 36893488151918118900)

 (22)

This is the tetrad metric you were looking for, associated to a signature of the form "(++--)"

eta_[]

Physics:-Tetrads:-eta_[a, b] = Matrix(%id = 36893488151947672860)

 (23)

This is the tetrad definition

e_[definition]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

 (24)

This definition is satisfied (by construction)

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

Matrix(%id = 36893488153642493340)

 (25)

This is the other way of the definition of the tetrad, relating it to the spacetime metric

e_[a, mu]*e_[a, nu] = g_[mu, nu]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[`~a`, nu] = Physics:-g_[mu, nu]

 (26)

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[`~a`, nu] = Physics[g_][mu, nu])

Matrix(%id = 36893488151942188916)

 (27)

The equation components are all identities by eye, also by computer (useful when the expressions are difficult to read by eye).

"map(evalb,?)"

Matrix(%id = 36893488151918155164)

 (28)

Summarizing: the procedure is the same one shown in the first reply, just that instead of solving the system by eye as I did from (6) to (7) of the previous workshet-reply (something correct in that simple case of Schwarzschild solution) you solve it using the computer as done above from (15) to (18), also arriving at a simple solution.

 

NULL


 

Download arbitrary_signature_(II).mw
 

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

Hi, could you please upload a worksheet ...

ecterrab 14540
July 13 2022
0 0

Hi, could you please upload a worksheet with the metric? You can use the green arrow for that.

DiracComb...

ecterrab 14540
June 03 2022
0 0

Hi
Wikipedia mentions the DiracComb function. Anyone has an idea of how frequently used is this function? Maybe it is worth implementing it in Maple. Opinions?

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

May 27: Answer and add a worksheet ...

ecterrab 14540
May 27 2022
0 0

May 27: Answer and add a worksheet showing how to do this with Physics

First the answer to your question: change you input 

Oneform := RaiseLowerIndices(g, ON[1], [1])

by

Oneform := convert(RaiseLowerIndices(g, ON[1], [1]), DGform)

And that resolves the problem you encountered. Next is how you do the same computation using Physics

> 

NULL
> 

Delta := -2*M*R+R^2+a^2
> 

Sigma2 := (R^2+a^2)^2-Delta*a^2*sin(Theta)^2
> 

rho2 := R^2+a^2*cos(Theta)^2
> 

z := 2*M*R/rho2

  

Load Physics

> 

with(Physics)

  

Set the coordinates

> 

Coordinates(X = [T, R, Theta, Phi])

`Systems of spacetime coordinates are:`*{X = (T, R, Theta, Phi)}

  

{X}

 (1)

Set the metric, I am doing this in two steps so that you see how it is done: just remove the "&t" from around

> 

`#msup(mi("ds"),mn("2"))` := (-1+z)*dT^2+dT*dR+dT*dR+rho2*dTheta^2-z*a*sin(Theta)^2*(dPhi*dT+dPhi*dT)-a*sin(Theta)^2*(dR*dPhi+dR*dPhi)+Sigma2*sin(Theta)^2*dPhi^2/rho2

(-1+2*M*R/(R^2+a^2*cos(Theta)^2))*dT^2+2*dT*dR+(R^2+a^2*cos(Theta)^2)*dTheta^2-4*M*R*a*sin(Theta)^2*dPhi*dT/(R^2+a^2*cos(Theta)^2)-2*a*sin(Theta)^2*dR*dPhi+((R^2+a^2)^2-(-2*M*R+R^2+a^2)*a^2*sin(Theta)^2)*sin(Theta)^2*dPhi^2/(R^2+a^2*cos(Theta)^2)

 (2)

Set the metric and visually check it out (if its not right, adjust the line element above accordingly)

> 

Setup(metric = (-1+2*M*R/(R^2+a^2*cos(Theta)^2))*dT^2+2*dT*dR+(R^2+a^2*cos(Theta)^2)*dTheta^2-4*M*R*a*sin(Theta)^2*dPhi*dT/(R^2+a^2*cos(Theta)^2)-2*a*sin(Theta)^2*dR*dPhi+((R^2+a^2)^2-(-2*M*R+R^2+a^2)*a^2*sin(Theta)^2)*sin(Theta)^2*dPhi^2/(R^2+a^2*cos(Theta)^2))

_______________________________________________________

  

`Coordinates: `[T, R, Theta, Phi]*`. Signature: `(`- - - +`)

  

_______________________________________________________

  

Physics:-g_[mu, nu] = Matrix(%id = 36893488152586874996)

  

_______________________________________________________

  

`Setting `*lowercaselatin_is*` letters to represent `*space*` indices`

 (3)

To work with tetrads and everything related, there is a specialized package:

> 

with(Tetrads)

_______________________________________________________

  

`Setting `*lowercaselatin_ah*` letters to represent `*tetrad*` indices`

  

(`Defined as tetrad tensors `(`see ?Physics,tetrads`)*`, `*`𝔢`[a, mu]*` , `*eta[a, b]*` , `)*gamma[a, b, c]*` , `*lambda[a, b, c]

  

(`Defined as spacetime tensors representing the NP null vectors of the tetrad formalism `(`see ?Physics,tetrads`)*`, `*l[mu]*` , `*n[mu]*` , `)*m[mu]*` , `*conjugate(m[mu])

  

_______________________________________________________

  

[IsTetrad, NullTetrad, OrthonormalTetrad, PetrovType, SegreType, TransformTetrad, WeylScalars, e_, eta_, gamma_, l_, lambda_, m_, mb_, n_]

 (4)

For a brief description of each command, see the overview help page Tetrads

Set the signature, it follows the tetrad, by default it is of orthonormal type

> 

Setup(signature = "-+++")

[signature = `- + + +`]

 (5)
> 

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 36893488159811021868)

 (6)
> 

"IsTetrad(?)"

`Type of tetrad: `*orthonormal

  

true

 (7)

Check its definition

> 

e_[definition]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

 (8)

Compute the components of this definition

> 

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b], simplifier = simplify)

Matrix(%id = 36893488159798381732)

 (9)

In the context of Physics you don't need to use a command to raise or lower an index, instead: just index with a covariant or contravariant index (the latter, by convention, are preceded by tilde ~). So to take one component of the tetrad, with either index covariant or contravariant, just input the tetrad with the desired indicesFor example, "`𝔢`[2]^(1) "

> 

"e_[~1,2]"

I*(2*a^2*cos(Theta)^2+2*R^2)^(1/2)/(-2*a^2*cos(Theta)^2+4*M*R-2*R^2)^(1/2)

 (10)

To compute the exterior derivative of an expression (see Physics:-ExteriorDerivative ), e.g. of  "`𝔢`[2]^(1) "

> 

"ExteriorDerivative[mu](e_[~1,2])"

I*(-(1/2)*(2*a^2*cos(Theta)^2+2*R^2)^(1/2)*(4*a^2*cos(Theta)*Physics:-d_[mu](Theta, [X])*sin(Theta)+4*M*Physics:-d_[mu](R, [X])-4*R*Physics:-d_[mu](R, [X]))/(-2*a^2*cos(Theta)^2+4*M*R-2*R^2)^(3/2)+(1/2)*(-4*a^2*cos(Theta)*Physics:-d_[mu](Theta, [X])*sin(Theta)+4*R*Physics:-d_[mu](R, [X]))/((-2*a^2*cos(Theta)^2+4*M*R-2*R^2)^(1/2)*(2*a^2*cos(Theta)^2+2*R^2)^(1/2)))

 (11)

To compute the ExteriorDerivative of a column of the tetrad, not just one component, do the same

> 

ExteriorDerivative[mu](e_[1, nu])

(1/2)*Physics:-d_[mu](Physics:-Tetrads:-e_[1, nu], [X])-(1/2)*Physics:-d_[nu](Physics:-Tetrads:-e_[1, mu], [X])

 (12)

To see the components of this expression always use TensorArray

> 

TensorArray((1/2)*Physics[d_][mu](Physics:-Tetrads:-e_[1, nu], [X])-(1/2)*Physics[d_][nu](Physics:-Tetrads:-e_[1, mu], [X]))

Matrix(%id = 36893488159805085084)

 (13)

You can also play around with these things, e.g. set a tensor whose components are given by (12)

> 

T[mu, nu] = (1/2)*Physics[d_][mu](Physics:-Tetrads:-e_[1, nu], [X])-(1/2)*Physics[d_][nu](Physics:-Tetrads:-e_[1, mu], [X])

T[mu, nu] = (1/2)*Physics:-d_[mu](Physics:-Tetrads:-e_[1, nu], [X])-(1/2)*Physics:-d_[nu](Physics:-Tetrads:-e_[1, mu], [X])

 (14)
> 

Define(T[mu, nu] = (1/2)*Physics[d_][mu](Physics:-Tetrads:-e_[1, nu], [X])-(1/2)*Physics[d_][nu](Physics:-Tetrads:-e_[1, mu], [X]))

`Defined objects with tensor properties`

  

{Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], T[mu, nu], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-Tetrads:-e_[a, mu], Physics:-Tetrads:-eta_[a, b], Physics:-g_[mu, nu], Physics:-Tetrads:-gamma_[a, b, c], Physics:-gamma_[i, j], Physics:-Tetrads:-l_[mu], Physics:-Tetrads:-lambda_[a, b, c], Physics:-Tetrads:-m_[mu], Physics:-Tetrads:-mb_[mu], Physics:-Tetrads:-n_[mu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (15)

Compute now any component of this tensor, e.g. T[`~1`, 2]

> 

"T[~1, 2]"

(4*I)*M*a^2*(-sin(Theta)^2*cos(Theta)^4*a^4-cos(2*Theta)*cos(Theta)^4*a^4+2*M*sin(Theta)^2*cos(Theta)^2*R*a^2+a^4*cos(Theta)^4+M*cos(2*Theta)*cos(Theta)^2*R*a^2-2*M*sin(Theta)^2*R^3-M*R*a^2*cos(Theta)^2+R^4*sin(Theta)^2-M*cos(2*Theta)*R^3+cos(2*Theta)*R^4+M*R^3-R^4)/((R^2+a^2*cos(Theta)^2)*(2*a^2*cos(Theta)^2+2*R^2)^(3/2)*(-2*a^2*cos(Theta)^2+4*M*R-2*R^2)^(3/2))

 (16)

 

Download WithPhysicsTetrads.mw

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

A comment and Maplesoft Physics Updates ...

ecterrab 14540
May 24 2022
0 0

In the latest Maplesoft Physics Updates (v.1235) there is a fix to "not placing the default nonnegint assumption" I mentioned a couple of replies above. So with the latest Physics Updates for Maple 2022, we now get

> D[1 $ j, 2 $ k](f)(x, y);  # computes both derivatives, x$j and y$k

                  /   k                                                 /

                  | -----                                               |

                  |  \                                      (2 j - _k1) |

pochhammer(-j, j) |   )    binomial(k, _k1) GAMMA(1 + 2 j) y            | ...

                  |  /                                                  |

                  | -----                                               |

                  |_k1 = 0                                              |

                  \                                                     \

                                     ...  

> eval(value(eval(%, [ j = 2, k = 4 ])), y = 0);

                                       6

 

PS: @Carl Love, although I see the motivation for your comment, I don't foresee adding to the diff command an option like "return a piecewise with all the cases".  The design of diff is to return a default value, and handle assumptions when possible - only that. It is similar to how the conversion network for mathematical functions works (see ?convert, to_special_function).

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

@vv I pointed out that the symbolic...

ecterrab 14540
May 23 2022
0 0

@vv 
I pointed out that the symbolic differentiation result, for symbols j and k, is correct. You objected to that. I only replied showing that, indeed, the result is correct. That closes the question.

Next you pointed out that the symbolic differentiation result is fragile. That concept is debatable. I do agree though, with a previous comment by you, that the most common situation is j, k nonnegint; so if choosing a way to express the result using GAMMA functions, do that for the most common case. The symbolic differentiation code is already doing that. But not in one branch of a subroutine, and the problem posted hits that branch. Although the result is correct, it can be better. It is something to adjust.

But then you come up with applying blindly the manipulation I used for the default output, after (you) using the opposite assumptions. With due respect, @vv, I don't see your point. You realize that if the result contains GAMMA as in this example, under direct evaluation it will either work with j negint, or for j posint, and not with both at the same time.

Add to that my opinion, that returning piecewise functions all around is not a convenient idea. Instead, I prefer presenting results with a convenient default and leave to the user to use assumptions for the other cases. Also relevant: understand that having Sums, and especially in the higher-order symbolic differentiation case, the results may require additional algebraic manipulations when evaluating the symbols involved at some values; typically, cases where correctness would require using limits or removing apparent singularities.

Returning to your example assuming j::nonegint, k::nonegint (above this reply), the manipulation after ex1 is as follows:

ex2 := simplify(eval(ex1, [j=2,k = 4, Sum = add]), GAMMA);
simplify(eval(ex2, [y=0]));
                                              6


So, the result by the symbolic differentiation code is also correct when assuming j::nonegint, k::nonegint.

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

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