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guru kido

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@guru kido  No worries anyway I re...

guru kido 60
May 09 2019
0 0

@guru kido 

No worries anyway I realised I can download your code so thanks!

@tomleslie  The metrics are the sa...

guru kido 60
May 09 2019
0 0

@tomleslie 

The metrics are the same, first code i simply inputted an arbitrary metric manually.

Still not getting it...

guru kido 60
May 09 2019
0 0


 

> 

restart
> 

with(Physics); Physics-Version(); Physics:-`*`(interface(rtablesize = 10), interface(version))

Physics-Version()

  

10*`Standard Worksheet Interface, Maple 2017.3, Windows 10, September 27 2017 Build ID 1265877`

 (1)
> 

Setup(mathematicalnotation = true)

[mathematicalnotation = true]

 (2)
> 

Setup(dimension = 3, coordinates = (X = [x1, x2, x3]), metric = arbitrary)

`* Partial match of 'coordinates' against keyword 'coordinatesystems'`

  

`The dimension and signature of the tensor space are set to: [3, - - +] `

  

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

  

`Systems of spacetime Coordinates are: `*{X = (x1, x2, x3)}

  

[coordinatesystems = {X}, dimension = 3, metric = {(1, 1) = _F1(X), (1, 2) = _F2(X), (1, 3) = _F3(X), (2, 2) = _F4(X), (2, 3) = _F5(X), (3, 3) = _F6(X)}]

 (3)
> 

g_[]

Physics:-g_[mu, nu] = Matrix(%id = 18446746167580001694)

 (4)
> 

Weyl[nonzero]

`[Length of output exceeds limit of 1000000]`

 (5)
   

Hi,
I don't think thats the case it has worked for me before, but anyhow I have used your code and do not get the same result? Thanks,
 

Download Weyltensor0in3d.mw

@tomleslie 

  ...

guru kido 60
May 05 2019
0 0


 

> 

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): g0:=A*v0+b; H0:=Matrix(4,4,{(1,1)=1,(2,2)=1,(3,3)=1,(4,4)=1},fill=0); tot:=0: while (Norm(g0) > 1e-6) do; d0:=-H0*g0 alpha0 := -(Transpose(g0).d0)/(Transpose(d0).A.d0); v1 := v0 +alpha0*d0; g1:=A*v1+b; p0:=v1-v0; q0:=g1-g0; H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0); v0:=v1; H0:=H1; g0:=g1; tot:=tot+1 od: v0, g(v0);"

Error, unterminated loop

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): g0:=A*v0+b; H0:=Matrix(4,4,{(1,1)=1,(2,2)=1,(3,3)=1,(4,4)=1},fill=0); tot:=0: while (Norm(g0) > 1e-6) do; d0:=-H0*g0 alpha0 := -(Transpose(g0).d0)/(Transpose(d0).A.d0); v1 := v0 +alpha0*d0; g1:=A*v1+b; p0:=v1-v0; q0:=g1-g0; H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0); v0:=v1; H0:=H1; g0:=g1; tot:=tot+1 od: v0, g(v0);"

  
> 

``

Ah I have corrected these issues (H0 is supposed to be 4x4). CODE:
 

restart;  

with(LinearAlgebra):  

A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0):  

b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}):  

f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x):  

v0 := Matrix([[0.], [1.], [0.], [0.]]):  

g0:=A*v0+b;  

H0:=Matrix(4,4,{(1,1)=1,(2,2)=1,(3,3)=1,(4,4)=1},fill=0);  

tot:=0:  

while   (Norm(g0) > 1e-6) do;  

d0:=-H0*g0    

alpha0 := -(Transpose(g0).d0)/(Transpose(d0).A.d0);  

v1 := v0 +alpha0*d0;  

g1:=A*v1+b;      

p0:=v1-v0;  

q0:=g1-g0;  

H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0);    

v0:=v1;  

H0:=H1;  

g0:=g1;  

tot:=tot+1  

od:  

v0, g(v0);

 

@tomleslie 

@vv  The post deleted itselft multi...

guru kido 60
May 05 2019
0 0

@vv 
The post deleted itselft multiple times so I keep reposting it

@guru kido  restart;  with(Li...

guru kido 60
May 05 2019
0 0

@guru kido 
restart;

 with(LinearAlgebra):

 A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0):

 b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}):

 f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x):

 v0 := Matrix([[0.], [1.], [0.], [0.]]):

 g0:=A*v0+b;  

H0:=Matrix(2,2,{(1,1)=1,(2,2)=1},fill=0);  

tot:=0:

 for i from 1 by 1 while   (Norm(g(v0)) > 1e-6) do;  

d0=-H0*g0

    alpha0 := -(Transpose(g0).d0)/(Transpose(d0).A.d0);

  v1 := v0 +alpha0*d0;

 g1:=A*v1+b

     p0:=v1-v0;

 q0:=g1-g0;

  H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0);

 f1:=(1/(2))*Transpose(v1).A.v1+Transpose(b)*v1  

 v0:=v1;

 H0:=H1;  

g0:=g1;

 tot:=i

 od:

 v0, g(v0);

 

mw@tomleslie    ...

guru kido 60
May 05 2019
0 0

mw@tomleslie 

 

> 

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): g0:=A*v0+b; H0:=Matrix(2,2,{(1,1)=1,(2,2)=1},fill=0); tot:=0: for i from 1 by 1 while (Norm(g(v0)) > 1e-6) do; d0=-H0*g0 alpha0 := -(Transpose(g0).d0)/(Transpose(d0).A.d0); v1 := v0 +alpha0*d0; g1:=A*v1+b p0:=v1-v0; q0:=g1-g0; H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0); f1:=(1/(2))*Transpose(v1).A.v1+Transpose(b)*v1 v0:=v1; H0:=H1; g0:=g1; tot:=i od: v0, g(v0);"

Error, unterminated loop

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): g0:=A*v0+b; H0:=Matrix(2,2,{(1,1)=1,(2,2)=1},fill=0); tot:=0: for i from 1 by 1 while (Norm(g(v0)) > 1e-6) do; d0=-H0*g0 alpha0 := -(Transpose(g0).d0)/(Transpose(d0).A.d0); v1 := v0 +alpha0*d0; g1:=A*v1+b p0:=v1-v0; q0:=g1-g0; H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0); f1:=(1/2)*Transpose(v1).A.v1+Transpose(b)*v1 v0:=v1; H0:=H1; g0:=g1; tot:=i od: v0, g(v0);"

  
> 

``


The big green up arrow just inserts this as a photo...

Download new_Try.mw

  ...

guru kido 60
May 05 2019
0 0


 

> 

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): x := Matrix([[x1], [x2], [x3], [x4]]): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): H0:=Matrix(2,2,{(1,1)=1,(2,2)=1},fill=0); g := x -> (A . x) + b: tot:=0: for i from 1 by 1 while (Norm(g(v0)) > 1e-6) do; p0:=v1-v0; q0:=g(v1)-g(v0); H1:=H0-(H0*q0*Transpose(q0).H0)/(Transpose(q0).H0.q0)+(p0*Transpose(p0))/(Transpose(p0).q0); alpha0 := -(Transpose(H0*g(v0))g(v0))/(Transpose(H0*g(v0)).A.H0.g(v0)); v1 := v0 - alpha0*g(v0)*H0; v0:=v1; H0:=H1; g(v0):=g(v1); tot:=i od: v0, g(v0);"

_rtable[18446746302871797382]

  

Error, (in LinearAlgebra:-Transpose) matrix dimensions don't match: 4 x 1 vs 4 x 4

  

Matrix(%id = 18446746302871797262), Matrix(%id = 18446746302871785710)

 (1)
> 

``
> 

``


Updated

 

@guru kido 

@tomleslie  Hi,    I have...

guru kido 60
May 05 2019
0 0

@tomleslie 
Hi,
   I have given you the code in my question. And they are all dots to represent matrix multiplication

  ...

guru kido 60
May 03 2019
0 0


 

> 

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): x := Matrix([[x1], [x2], [x3], [x4]]): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): g := x -> (A . x) + b: p0=v1-v0 q0=g(v1)-g(v0) tot:=0: for i from 1 by 1 while (Norm(g(v0)) > 1e-6) do; H1:=H0-(H0*q0*Transpose(q0)*H0)/(Transpose(q0)*H0*q0)+(p0*Transpose(p0))/(Transpose(p0)*q0) alpha0 := -(Transpose(H0*g(v0))*g(v0))/(Transpose(H0*g(v0))); v1 := v0 - alpha0*g(v0)*H(0); tot:=i od: v0, g(v0);"

Error, unterminated loop

"restart; with(LinearAlgebra): A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0): b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}): x := Matrix([[x1], [x2], [x3], [x4]]): f := x -> (((1/2*Transpose(x)) . A) . x) + ((Transpose(b)) . x): v0 := Matrix([[0.], [1.], [0.], [0.]]): g := x -> (A . x) + b: p0=v1-v0 q0=g(v1)-g(v0) tot:=0: for i from 1 by 1 while (Norm(g(v0)) > 1e-6) do; H1:=H0-(H0*q0*Transpose(q0)*H0)/(Transpose(q0)*H0*q0)+(p0*Transpose(p0))/(Transpose(p0)*q0) alpha0 := -(Transpose(H0*g(v0))*g(v0))/(Transpose(H0*g(v0))); v1 := v0 - alpha0*g(v0)*H(0); tot:=i od: v0, g(v0);"

  
> 

``


 

Download op_cwk.mw

@guru kido 

@guru kido    How do I do sim...

guru kido 60
May 03 2019
0 0

@guru kido 
 

How do I do similar iterations with the formulae shown? I have tried to do this but I am stuck at the first hurdle, i do not know how to programme p(k) and q(k). Thanks in advance your help is much appreciated!


 

> 

restart; with(LinearAlgebra); A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0); b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}); x := Matrix([[x1], [x2], [x3], [x4]]); f := proc (x) options operator, arrow; (1/2)*Transpose(x).A.x+Transpose(b).x end proc; v0 := Matrix([[0.], [1.], [0.], [0.]]); g := proc (x) options operator, arrow; A.x+b end proc; tot := 0; for i while Norm(g(v0)) > 0.1e-5 do alpha0 := solve(diff(f(v0-g(v0)*alpha)[1, 1], alpha) = 0); v0 := v0-alpha0*g(v0); tot := i end do; v0, g(v0)

Matrix(%id = 18446745664466056126), Matrix(%id = 18446745664466057086)

 (1)
> 

tot

7151

 (2)
> 

``


 

Download optimisation_courseworkdone.mw

  ...

guru kido 60
May 03 2019
0 0


 

> 

restart; with(LinearAlgebra); A := Matrix(4, 4, {(1, 1) = 1, (1, 3) = -1, (1, 4) = 3, (2, 2) = 2, (2, 3) = 1, (3, 1) = -1, (3, 2) = 1, (3, 3) = 6, (3, 4) = -1, (4, 1) = 3, (4, 3) = -1, (4, 4) = 10}, fill = 0); b := Matrix(4, 1, {(1, 1) = 0, (2, 1) = -2, (3, 1) = -1, (4, 1) = -1}); x := Matrix([[x1], [x2], [x3], [x4]]); f := proc (x) options operator, arrow; (1/2)*Transpose(x).A.x+Transpose(b).x end proc; v0 := Matrix([[0.], [1.], [0.], [0.]]); g := proc (x) options operator, arrow; A.x+b end proc; tot := 0; for i while Norm(g(v0)) > 0.1e-5 do alpha0 := solve(diff(f(v0-g(v0)*alpha)[1, 1], alpha) = 0); v0 := v0-alpha0*g(v0); tot := i end do; v0, g(v0)

Matrix(%id = 18446745664466056126), Matrix(%id = 18446745664466057086)

 (1)
> 

tot

7151

 (2)
> 

``


scrap that

Download optimisation_courseworkdone.mw

@guru kido 

@vv Hi, How can I see how many iter...

guru kido 60
May 03 2019
0 0

@vv Hi,
How can I see how many iterations this has gone through? 

Hi,   I was aware of the misor...

guru kido 60
March 25 2019
0 0

Hi,
   I was aware of the misordering I was just ignoring it but I have corrected now. Now this all makes sense and thank you for your help, but I have made changes and I am still not getting zero. Do you think I need to input more equations so that the computer can simplify more? I am stuck. Thanks in advance. How do I find help pages also? ALSO, the 'half flat metrics in 4D' is not a command, it is simply a note for myself. Thanks.


 

> 

with(Physics)
> 

Setup(mathematicalnotation = true)
> 

u22 := diff(u(x1, x2, x3, x4), x2, x2)

Physics:-diff(Physics:-diff(u(X), x2), x2)

 (1)
> 

u12 := diff(u(x1, x2, x3, x4), x1, x2)

Physics:-diff(Physics:-diff(u(X), x1), x2)

 (2)
> 

u11 := diff(u(x1, x2, x3, x4), x1, x1)

Physics:-diff(Physics:-diff(u(X), x1), x1)

 (3)
> 

u24 := diff(u(x1, x2, x3, x4), x2, x4)

Physics:-diff(Physics:-diff(u(X), x2), x4)

 (4)
> 

u13 := diff(u(x1, x2, x3, x4), x1, x3)

Physics:-diff(Physics:-diff(u(X), x1), x3)

 (5)
> 

Setup(Coordinatesystem = (X = [x1, x2, x3, x4]), metric = -dx3^2*u22+2*dx3*dx4*u12-dx4^2*u11+dx1*dx3+dx2*dx4)

`* Partial match of 'Coordinatesystem' against keyword 'coordinatesystems'`

  

`Systems of spacetime Coordinates are: `*{X = (x1, x2, x3, x4)}

  

[coordinatesystems = {X}, metric = {(1, 3) = 1/2, (2, 4) = 1/2, (3, 3) = -(diff(diff(u(X), x2), x2)), (3, 4) = Physics:-diff(diff(u(X), x1), x2), (4, 4) = -(diff(diff(u(X), x1), x1))}]

 (6)
> 

Setup(spacetimeindices = lowercaselatin)
> 

g_[]

Physics:-g_[a, b] = Matrix(%id = 18446747324625331134)

 (7)
> 

Define(W, quiet)
> 

"W[~i,j,k,l]=1/(2)sqrt('g_[determinant]')g_[~i,~a]g_[~b,~c]LeviCivita[a,j,b,d]Weyl[~d,c,k,l]"

W[`~i`, j, k, l] = (1/2)*Physics:-g_[determinant]^(1/2)*Physics:-g_[`~a`, `~i`]*Physics:-g_[`~b`, `~c`]*Physics:-LeviCivita[a, b, d, j]*Physics:-Weyl[`~d`, c, k, l]

 (8)
> 

Define(W[`~i`, j, k, l] = (1/2)*Physics[g_][determinant]^(1/2)*Physics[g_][`~a`, `~i`]*Physics[g_][`~b`, `~c`]*Physics[LeviCivita][a, b, d, j]*Physics[Weyl][`~d`, c, k, l])

`Defined objects with tensor properties`

  

{Physics:-D_[a], Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-Ricci[a, b], Physics:-Riemann[a, b, c, d], W[`~i`, j, k, l], Physics:-Weyl[a, b, c, d], Physics:-d_[a], Physics:-g_[a, b], Physics:-Christoffel[a, b, c], Physics:-Einstein[a, b], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c, d], Physics:-SpaceTimeVector[a](X)}

 (9)
> 

W[definition]

W[`~e`, f, g, h] = (1/32)*16^(1/2)*Physics:-g_[`~a`, `~e`]*Physics:-g_[`~b`, `~c`]*Physics:-LeviCivita[a, b, d, f]*Physics:-Weyl[`~d`, c, g, h]

 (10)
> 

u11*u22-u12^2+u13+u24 = 0

diff(diff(u(X), x1), x3)+diff(diff(u(X), x2), x4)+(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))-(diff(diff(u(X), x1), x2))^2 = 0

 (11)
> 

u113 := diff(-u11*u22+u12^2-u24, x1)

-(diff(diff(diff(u(X), x1), x2), x4))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x1), x2))

 (12)
> 

u123 := diff(-u11*u22+u12^2-u24, x2)

-(diff(diff(diff(u(X), x2), x2), x4))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2))

 (13)
> 

u133 := diff(-u11*u22+u12^2-u24, x3)

-(diff(diff(diff(u(X), x2), x3), x4))-(diff(diff(diff(u(X), x1), x1), x3))*(diff(diff(u(X), x2), x2))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x3))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x3))

 (14)
> 

u134 := diff(-u11*u22+u12^2-u24, x4)

-(diff(diff(diff(u(X), x2), x4), x4))-(diff(diff(diff(u(X), x1), x1), x4))*(diff(diff(u(X), x2), x2))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x4))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x4))

 (15)
> 

u234 := diff(u(x1, x2, x3, x4), x2, x3, x4)

Physics:-diff(Physics:-diff(Physics:-diff(u(X), x2), x3), x4)

 (16)
> 

u223 := diff(u(x1, x2, x3, x4), x2, x2, x3)

Physics:-diff(Physics:-diff(Physics:-diff(u(X), x2), x2), x3)

 (17)
> 

u133 = -u11*u223-u113*u22+2*u12*u123-u234

-(diff(diff(diff(u(X), x2), x3), x4))-(diff(diff(diff(u(X), x1), x1), x3))*(diff(diff(u(X), x2), x2))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x3))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x3)) = -(diff(diff(diff(u(X), x2), x3), x4))-(-(diff(diff(diff(u(X), x1), x2), x4))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x1), x2)))*(diff(diff(u(X), x2), x2))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x3))+2*(diff(diff(u(X), x1), x2))*(-(diff(diff(diff(u(X), x2), x2), x4))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2)))

 (18)
> 

ArrayElems(TensorArray(Weyl[`~i`, j, k, l]+W[`~i`, j, k, l], simplifier = simplify))

{(1, 1, 1, 3) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 1, 1, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (1, 1, 2, 3) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 1, 2, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 1, 3, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 1, 3, 2) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 1, 3, 4) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 1, 4, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (1, 1, 4, 2) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 1, 4, 3) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 2, 1, 3) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 2, 1, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 2, 2, 3) = -2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 2, 2, 4) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 2, 3, 1) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 2, 3, 2) = 2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 2, 3, 4) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 2, 4, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 2, 4, 2) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 2, 4, 3) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 3, 1, 3) = 4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 3, 1, 4) = -4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 3, 2, 3) = 4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 3, 2, 4) = -4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 3, 3, 1) = -4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 3, 3, 2) = -4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 3, 3, 4) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2))+4*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+8*(diff(diff(u(X), x1), x2))^2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-12*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (1, 3, 4, 1) = 4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 3, 4, 2) = 4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 3, 4, 3) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2))-4*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-8*(diff(diff(u(X), x1), x2))^2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+12*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (1, 4, 1, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 4, 1, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 4, 2, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 4, 2, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 4, 3, 1) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 4, 3, 2) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 4, 3, 4) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+4*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (1, 4, 4, 1) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (1, 4, 4, 2) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (1, 4, 4, 3) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-4*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (2, 1, 1, 3) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 1, 1, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1)), (2, 1, 2, 3) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 1, 2, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 1, 3, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 1, 3, 2) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 1, 3, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 1, 4, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1)), (2, 1, 4, 2) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 1, 4, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 2, 1, 3) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 2, 1, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 2, 2, 3) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 2, 2, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 2, 3, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 2, 3, 2) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 2, 3, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 2, 4, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 2, 4, 2) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 2, 4, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 3, 1, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 3, 1, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 3, 2, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (2, 3, 2, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 3, 3, 1) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 3, 3, 2) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (2, 3, 3, 4) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+4*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (2, 3, 4, 1) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 3, 4, 2) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 3, 4, 3) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-4*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (2, 4, 1, 3) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 4, 1, 4) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 4, 2, 3) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 4, 2, 4) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 4, 3, 1) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 4, 3, 2) = -4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (2, 4, 3, 4) = -4*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+12*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-8*(diff(diff(u(X), x1), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1)), (2, 4, 4, 1) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (2, 4, 4, 2) = 4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (2, 4, 4, 3) = 4*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-12*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+8*(diff(diff(u(X), x1), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1)), (3, 3, 1, 3) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 3, 1, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (3, 3, 2, 3) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (3, 3, 2, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 3, 3, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 3, 3, 2) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (3, 3, 3, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (3, 3, 4, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (3, 3, 4, 2) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 3, 4, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (3, 4, 1, 3) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (3, 4, 1, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1)), (3, 4, 2, 3) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 4, 2, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (3, 4, 3, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (3, 4, 3, 2) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 4, 3, 4) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (3, 4, 4, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1)), (3, 4, 4, 2) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (3, 4, 4, 3) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 3, 1, 3) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 3, 1, 4) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 3, 2, 3) = 2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (4, 3, 2, 4) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 3, 3, 1) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 3, 3, 2) = -2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (4, 3, 3, 4) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (4, 3, 4, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 3, 4, 2) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 3, 4, 3) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x2)), (4, 4, 1, 3) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 4, 1, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (4, 4, 2, 3) = -2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 4, 2, 4) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 4, 3, 1) = 2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 4, 3, 2) = 2*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 4, 3, 4) = -2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2)), (4, 4, 4, 1) = -2*(diff(diff(diff(diff(u(X), x1), x1), x1), x2)), (4, 4, 4, 2) = -2*(diff(diff(diff(diff(u(X), x1), x1), x2), x2)), (4, 4, 4, 3) = 2*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))}

 (19)
> 

ArrayElems(TensorArray(Weyl[`~i`, j, k, l]-W[`~i`, j, k, l], simplifier = simplify))

{(1, 4, 3, 4) = (4*(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))+8*(diff(diff(u(X), x1), x2))^2)*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-8*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x3))-8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x3))+4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x4))+2*(diff(diff(diff(diff(u(X), x1), x1), x3), x3))-8*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x3))-8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x4))+4*(diff(diff(diff(diff(u(X), x1), x2), x3), x4))+2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x4))+2*(diff(diff(diff(diff(u(X), x2), x2), x4), x4))-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))^2+(4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))-4*(diff(diff(diff(u(X), x1), x2), x3))+2*(diff(diff(diff(u(X), x2), x2), x4)))*(diff(diff(diff(u(X), x1), x1), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))^2+(4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))+2*(diff(diff(diff(u(X), x1), x1), x3))-4*(diff(diff(diff(u(X), x1), x2), x4)))*(diff(diff(diff(u(X), x1), x2), x2))+(-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x2), x2), x2))+2*(diff(diff(diff(u(X), x2), x2), x3)))*(diff(diff(diff(u(X), x1), x1), x1))+2*(diff(diff(diff(u(X), x1), x1), x4))*(diff(diff(diff(u(X), x2), x2), x2)), (1, 4, 4, 3) = (-4*(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))-8*(diff(diff(u(X), x1), x2))^2)*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+8*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x3))+8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x3))-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x4))-2*(diff(diff(diff(diff(u(X), x1), x1), x3), x3))+8*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x3))+8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x4))-4*(diff(diff(diff(diff(u(X), x1), x2), x3), x4))-2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x4))-2*(diff(diff(diff(diff(u(X), x2), x2), x4), x4))+4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))^2+(-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))+4*(diff(diff(diff(u(X), x1), x2), x3))-2*(diff(diff(diff(u(X), x2), x2), x4)))*(diff(diff(diff(u(X), x1), x1), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))^2+(-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))-2*(diff(diff(diff(u(X), x1), x1), x3))+4*(diff(diff(diff(u(X), x1), x2), x4)))*(diff(diff(diff(u(X), x1), x2), x2))+(4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x2), x2), x2))-2*(diff(diff(diff(u(X), x2), x2), x3)))*(diff(diff(diff(u(X), x1), x1), x1))-2*(diff(diff(diff(u(X), x1), x1), x4))*(diff(diff(diff(u(X), x2), x2), x2)), (2, 3, 3, 4) = (-4*(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))-8*(diff(diff(u(X), x1), x2))^2)*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))-2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))+8*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x3))+8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x3))-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x4))-2*(diff(diff(diff(diff(u(X), x1), x1), x3), x3))+8*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x3))+8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x4))-4*(diff(diff(diff(diff(u(X), x1), x2), x3), x4))-2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x4))-2*(diff(diff(diff(diff(u(X), x2), x2), x4), x4))+4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))^2+(-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))+4*(diff(diff(diff(u(X), x1), x2), x3))-2*(diff(diff(diff(u(X), x2), x2), x4)))*(diff(diff(diff(u(X), x1), x1), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))^2+(-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))-2*(diff(diff(diff(u(X), x1), x1), x3))+4*(diff(diff(diff(u(X), x1), x2), x4)))*(diff(diff(diff(u(X), x1), x2), x2))+(4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x2), x2), x2))-2*(diff(diff(diff(u(X), x2), x2), x3)))*(diff(diff(diff(u(X), x1), x1), x1))-2*(diff(diff(diff(u(X), x1), x1), x4))*(diff(diff(diff(u(X), x2), x2), x2)), (2, 3, 4, 3) = (4*(diff(diff(u(X), x1), x1))*(diff(diff(u(X), x2), x2))+8*(diff(diff(u(X), x1), x2))^2)*(diff(diff(diff(diff(u(X), x1), x1), x2), x2))+2*(diff(diff(u(X), x2), x2))^2*(diff(diff(diff(diff(u(X), x1), x1), x1), x1))-8*(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x2))+4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x1), x3))-8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x3))+4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(diff(u(X), x1), x1), x2), x4))+2*(diff(diff(diff(diff(u(X), x1), x1), x3), x3))-8*(diff(diff(u(X), x1), x2))*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x1), x2), x2), x3))-8*(diff(diff(u(X), x1), x2))*(diff(diff(diff(diff(u(X), x1), x2), x2), x4))+4*(diff(diff(diff(diff(u(X), x1), x2), x3), x4))+2*(diff(diff(u(X), x1), x1))^2*(diff(diff(diff(diff(u(X), x2), x2), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(diff(u(X), x2), x2), x2), x4))+2*(diff(diff(diff(diff(u(X), x2), x2), x4), x4))-4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))^2+(4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2))+4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))-4*(diff(diff(diff(u(X), x1), x2), x3))+2*(diff(diff(diff(u(X), x2), x2), x4)))*(diff(diff(diff(u(X), x1), x1), x2))-4*(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))^2+(4*(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))+2*(diff(diff(diff(u(X), x1), x1), x3))-4*(diff(diff(diff(u(X), x1), x2), x4)))*(diff(diff(diff(u(X), x1), x2), x2))+(-4*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x2), x2), x2))+2*(diff(diff(diff(u(X), x2), x2), x3)))*(diff(diff(diff(u(X), x1), x1), x1))+2*(diff(diff(diff(u(X), x1), x1), x4))*(diff(diff(diff(u(X), x2), x2), x2))}

 (20)
> 

``
> 

``
> 

``
> 

``
> 

``
> 

``
> 

``
> 

`in`(half*flat*metrics, 4*D)

`in`(half*flat*metrics, 4*D)

 (21)
> 

``


 

Download NEED_11.mw

  ...

guru kido 60
March 15 2019
0 0


 

> 

with(Physics)
> 

Setup(mathematicalnotation = true)
> 

Setup(Coordinatesystem = (X = [x1, x2, x3, x4]), metric = -dx3^2*u22+2*dx3*dx4*u12-dx4^2*u11+dx1*dx3+dx2*dx4)

`* Partial match of 'Coordinatesystem' against keyword 'coordinatesystems'`

  

`Systems of spacetime Coordinates are: `*{X = (x1, x2, x3, x4)}

  

[coordinatesystems = {X}, metric = {(1, 3) = 1/2, (2, 4) = 1/2, (3, 3) = -(diff(diff(u(X), x2), x2)), (3, 4) = Physics:-diff(diff(u(X), x1), x2), (4, 4) = -(diff(diff(u(X), x1), x1))}]

 (1)
> 

Setup(spacetimeindices = lowercaselatin)
> 

u22 := diff(u(x1, x2, x3, x4), x2, x2)

Physics:-diff(Physics:-diff(u(X), x2), x2)

 (2)
> 

u12 := diff(u(x1, x2, x3, x4), x1, x2)

Physics:-diff(Physics:-diff(u(X), x1), x2)

 (3)
> 

u11 := diff(u(x1, x2, x3, x4), x1, x1)

Physics:-diff(Physics:-diff(u(X), x1), x1)

 (4)
> 

g_[]

Physics:-g_[a, b] = Matrix(%id = 18446744921781828414)

 (5)
> 

Define(W, quiet)
> 

"W[~i,j,k,l]=1/(2)sqrt('g_[determinant]')g_[~i,~a]g_[~b,~c]LeviCivita[a,j,b,d]Weyl[~d,c,k,l]"

W[`~i`, j, k, l] = (1/2)*Physics:-g_[determinant]^(1/2)*Physics:-g_[`~a`, `~i`]*Physics:-g_[`~b`, `~c`]*Physics:-LeviCivita[a, b, d, j]*Physics:-Weyl[`~d`, c, k, l]

 (6)
> 

Define(6)

Error, (in Physics:-Define) expected objects with a tensor structure, from a symbol, say A, to an indexed function, say A[mu](X). Received: {6}

  
> 

W[definition]

W[definition]

 (7)
> 

"W[~1,2,3,4]"

W[`~1`, 2, 3, 4]

 (8)
> 

NULL

``

 (9)
> 

NULL
> 

``
> 

NULL
> 

NULL


 

Download NEED_11.mw

@ecterrab 

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