ecterrab

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These are answers submitted by ecterrab

Unable to reproduce, this is what I get...

Answered: ecterrab 14370

September 04 2024

2 5

Using directly Maple TTY to avoid any interference: no error message.

I suggest you to revise you don't have anything interfering, and if you still see this error message you posted, then input 'tracelast;' and post its output here so that I could see how this message you get is produced. If entering tracelast results in nothing, executing the odetest line again, reproducing the error, then tracelast again.

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

Fixed within the Maplesoft Physics Updat...

Answered: ecterrab 14370

August 28 2024

2 1

To install the fix (works only for Maple 2024), as usual, open Maple and input Physics:-Version(latest)

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

Adjusted in the Maplesoft Physics Update...

Answered: ecterrab 14370

August 22 2024

3 0

Hi @nm
Note first that you show how to compute an implicit solution, for which dsolve([ode, IC], implicit) works fine and returns, exactly, the solution you show as my_new_sol.

The problem was related to computing an explicit solution, a process involving computing a series that happened to interrupt with that error you saw. That problem is solved and the adjustment is distributed within the latest Maplesoft Physics Updates for Maple 2024. (install via Physics:-version(latest))

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

Issue in solve (not dsolve)...

Answered: ecterrab 14370

August 19 2024

2 0

While solving the ode + IC, the following call to solve is what hangs

ee := ((y(x)-1)*(y(x)+1))^(1/2)/(y(x)-1)^(1/2)/(y(x)+1)^(1/2)*ln(y(x)+(y(x)^2-1)^(1/2))-1/2*(y(x)^2-1)^(1/2)*(-ln(-y(x)^2+1)+2*ln((y(x)^2*x+((x^2-1)*(y(x)^2-1))^(1/2)*(y(x)^2-1)^(1/2)-x)/(y(x)^2-1)^(1/2)))/(y(x)-1)^(1/2)/(y(x)+1)^(1/2)-1/2*(y0^2-1)^(1/2)*(2*ln(y0+(y0^2-1)^(1/2))+ln(-y0^2+1)-2*ln((y0^2*x0+((x0^2-1)*(y0^2-1))^(1/2)*(y0^2-1)^(1/2)-x0)/(y0^2-1)^(1/2)))/(y0-1)^(1/2)/(y0+1)^(1/2)

((y(x)-1)*(y(x)+1))^(1/2)*ln(y(x)+(y(x)^2-1)^(1/2))/((y(x)-1)^(1/2)*(y(x)+1)^(1/2))-(1/2)*(y(x)^2-1)^(1/2)*(-ln(-y(x)^2+1)+2*ln((y(x)^2*x+((x^2-1)*(y(x)^2-1))^(1/2)*(y(x)^2-1)^(1/2)-x)/(y(x)^2-1)^(1/2)))/((y(x)-1)^(1/2)*(y(x)+1)^(1/2))-(1/2)*(y0^2-1)^(1/2)*(2*ln(y0+(y0^2-1)^(1/2))+ln(-y0^2+1)-2*ln((y0^2*x0+((x0^2-1)*(y0^2-1))^(1/2)*(y0^2-1)^(1/2)-x0)/(y0^2-1)^(1/2)))/((y0-1)^(1/2)*(y0+1)^(1/2))

(1)

>	solve(ee, y(x)); # hangs

Download issue_in_solve.mw

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

This is a problem in solve, not PDEtools...

Answered: ecterrab 14370

August 15 2024

3 0

This is a problem in solve.

restart;

>	eqs := [-2arctanh((_C1-1)^(1/2)/_C1^(1/2)) = _C2, 0 = _C1^(3/2)tanh(1/2_C2)sech(1/2*_C2)^2]; unknowns := {_C1,_C2};

[-2*arctanh((_C1-1)^(1/2)/_C1^(1/2)) = _C2, 0 = _C1^(3/2)*tanh((1/2)*_C2)*sech((1/2)*_C2)^2]

${_C1, _C2}$

(1)

Good:

>	solve(eqs, unknowns)

${_C1 = 1, _C2 = 0}$

(2)

Simpler:

>	tanh(_C2/2): % = expand(%)

(3)

>	simplify((lhs-rhs)(%))

(4)

>	EQS := subs(%%, eqs)

[-2*arctanh((_C1-1)^(1/2)/_C1^(1/2)) = _C2, 0 = _C1^(3/2)*(cosh(_C2)/sinh(_C2)-1/sinh(_C2))*sech((1/2)*_C2)^2]

(5)

Not good:

>	solve(EQS, unknowns)

Warning, solutions may have been lost

The step that seems to be failing within solve: normalize functions in terms of independent exponentials

>	convert(EQS, exp)

[-2*arctanh((_C1-1)^(1/2)/_C1^(1/2)) = _C2, 0 = 4*_C1^(3/2)*(exp(_C2)-1)/((exp(_C2)+1)*(exp((1/2)*_C2)+exp(-(1/2)*_C2))^2)]

(6)

>	solve(%, unknowns)

${_C1 = 1, _C2 = 0}$

(7)

PS: I added the workaround to solve's limitation mentioned, so that now PDEtools:-Solve returns a result. The adjustment is present in the latest Maplesoft Physics Updates for Maple 2024; as usual to install input Physics:-Version(latest).

Download why_PDEtools_Solve_fail_august_14_2024_(reviewed).mw

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

Several comments....

Answered: ecterrab 14370

August 13 2024

5 6

First, according to odeadvisor's help page,

The matching of the types is checked sequentially, and odeadvisor might return more than one type; otherwise, the first matching of a pattern interrupts the process and a classification is returned.

In fact, enter DEtools:-odeadvisor(ode, [dAlembert]) and you see your ode also matches dAlembert, not just separable.

Second, you realize the scope of a command like dsolve requires more than a classification: several ODEs can be tackled using different approaches, and choosing an appropriate one for each case requires considering multiple things. Your ode is a simple example of that. To make the point, a much more common, and complicated case is when the ODE does not match any standard classification but there are different symmetries available, sometimes also different integrating factors, all these leading to different forms of the solution. Anyway, solving your example using dAlembert's formulation is just more appropriate than tackling it as a separable equation.

Third, see ?dsolve,setup and you see you can force dsolve to use methods in an ordering different than the one I coded (mind you, the choice you see on that page is nontrivial).

Fourth, this ode you brought, as several other ones you have been posting, is nonlinear in the highest derivative; several methods require isolating the highest derivative, which involves choosing branches. Typically, the choice of a branch makes the solution valid only in some regions. This "problem" is unavoidable: either you accept an ode nonlinear in its highest derivative as a valid problem or not. If yes, you need to decide on what to do with these choices of branches. It is mainly for this reason I coded the implicit option.

With the above paragraphs in mind, this is what I see for your ode

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

Formula free of singularities...

Answered: ecterrab 14370

August 09 2024

3 5

First reply: "the indicated formula has singularities". I goofed with that comment (deleted now), as @vv tells below. The formula suggested by Wolinski above is now returned by the FunctionAdvisor and by convert(tan(z), Sum) as per pic below.

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

Fixed in v.1780 of the Maplesoft Physics...

Answered: ecterrab 14370

August 06 2024

3 5

To install the fix, open Maple and input Physics:-Version(latest).
The solution returned, however, is not the simplest you show but this other one:

ode := diff(y(x), x, x) = (diff(y(x), x))^3-(diff(y(x), x))^2

(1)

(2)

(3)

To test the IC requires using limit

(4)

(5)

(6)

Download Fixed_in_v.1780.mw

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

alias...

Answered: ecterrab 14370

August 04 2024

2 0

Download representation_of_code_(reviewed).mw

PS: if you use Maple 2024, remember to input Physics:-Version(latest) to work with the current Physics version.

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

r := ..., Not r = ......

Answered: ecterrab 14370

August 01 2024

2 0

Hi
d_[1](g(r)) will return different from 0 if and only if r depends on the Cartesian coordinates (that you defined as such). When you enter r = sqrt(x^2 + y^2 + z^2), so using the `=` operator, r does not get assigned anything. Input r and you see r as output. Use the assignment operator `:=` and i you will get what you were expecting.

Regarding diff(r, x) after you loaded the Physics:-Vectors package, that package comes with a diff command that automatically use the interdependency between Cartesian (x, y, z), Cylindrical (rho, phi, z) and Spherical coordinates (r, theta, phi) without you having to assign or state anything, and that in turn is also a setting: input Setup(geometricdifferentiation), and you see its value is false after loading Physics but before loading Vectors and true after loading it. This setting only affects the output of Physics:-Vectors:-diff, not Physics:-d_. I realize this is more of a design issue; intentional though: more often than otherwise, when using d_ we do not want geometricdifferentiation to interfere with computations, and if you want, it suffices to use := instead of = as mentioned above.

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

A few corrections. What is what you were...

Answered: ecterrab 14370

July 25 2024

3 1

This is your worksheet with some minor corrections, comments, and three different ways of arriving at the same result (note you didn't say what is the result you expect)

(1)

$Setup(realobjects = {g, diff(x, x), diff(y(x), x), diff(z(x), x), f__A(X[1])})$

$[realobjects = {g, phi, r, rho, theta, x, `x'`, y, `y'`, z, `z'`, f__A(r)}]$

(2)

I see you are using primed variables, which is perfectly fine, just recalling the prime in this context does not mean derivative.

Here is the first adjustment: you use the assignment operator :=, but on the left you put functionality: in doing so, you create a function, as you see in the output of your post, but after that, you only use the primed variables as symbols, not as functions, so your definition is not used (this is not related to Physics, but to how computer algebra systems work).

I am then removing the functionality on the lhs so that the primed variables have the value you indicated

(3)

(4)

(5)

${Physics:-Dgamma[mu], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(6)

You see how the value is now taken into account:

(7)

Next you assigned to a sum, I imagine to want automatic simplification ... it is not how it works: if you want the following simplification you need to invoke it. I am changing that := by = and show

(8)

(9)

Next you use , not wrong but perhaps more clear: you could have used its value

(10)

${A[mu, `~a`], Physics:-Dgamma[mu], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(11)

A[mu, `~a`] = (1-f__A(r))*Physics:-LeviCivita[4, a, j, mu]*R[j]/(g*r)

(12)

Here again you see the value of the primed variables, not themselves

A[mu, a] = Matrix(%id = 36893488158540457972)

(13)

${A[mu, `~a`], Physics:-Dgamma[mu], F[mu, nu, a], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(14)

(15)

Again, the output involves the value of the primed variables

F[mu, nu, a] = _rtable[36893488153612237268]

(16)

Now the Sum over all the repeated indices

(1/2)*(g^2*r^2*(sin(theta)^2*cos(phi)^2+1)*(diff(f__A(r), r))^2+2*g^2*r*(-1+f__A(r))*(sin(theta)*sin(phi)*cos(theta)*cos(phi)*r-sin(theta)^2*cos(phi)^2-1)*(diff(f__A(r), r))-(-1+f__A(r))^2*(1+f__A(r)^2-2*f__A(r)+g^2*(r^2*cos(theta)^2-sin(theta)^2)*cos(phi)^2+2*sin(theta)*sin(phi)*cos(theta)*cos(phi)*g^2*r+(-2*r^2-1)*g^2))/(g^4*r^4)

(17)

Is this a scalar? Yes, in the sense that there are no tensors around. Is this the result you were expecting? I don't know, it would help if you type in the result you were expecting.

Also, you see that (17) above depends on and . You can determine the conditions on such that depends only on r by differentiating

(1/2)*(cos(phi)*((diff(f__A(r), r))^2*r^2-2*r*(-1+f__A(r))*(diff(f__A(r), r))+(-1+f__A(r))^2*(r^2+1))*sin(2*theta)+2*cos(2*theta)*r*sin(phi)*(-1+f__A(r))*(1-f__A(r)+r*(diff(f__A(r), r))))*cos(phi)/(r^4*g^2)

(18)

-(sin(theta)*sin(phi)*r*(diff(f__A(r), r))-(-1+f__A(r))*(r*cos(phi)*cos(theta)+sin(theta)*sin(phi)))*(sin(theta)*cos(phi)*(diff(f__A(r), r))*r+(-1+f__A(r))*(cos(theta)*r*sin(phi)-sin(theta)*cos(phi)))/(r^4*g^2)

(19)

Any that cancels both equations result in depending only on r.

Alternatively, you can compute the same but keeping the primed variables around till the last step, as follows

(20)

$Setup(realobjects = {g, diff(x, x), diff(y(x), x), diff(z(x), x), f__A(X[1])})$

$[realobjects = {g, phi, r, rho, theta, x, `x'`, y, `y'`, z, `z'`, f__A(r)}]$

(21)

So do not assign the primed variables. Instead, create a set of substitution equations (i.e. use =, not :=) to be used only when you want

(22)

(23)

(24)

(25)

(26)

Proceed now using the primed variables themselves instead of their expression in terms of spherical coordinates

${Physics:-Dgamma[mu], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(27)

You see now the definition in terms of the primed variables

(28)

which all depend on the spherical coordinates

(29)

Next, the following is an identity, I will assign it here to some name, eq, in order to use it for simplification purposes later

(30)

(31)

(32)

Define now A

${A[mu, a], Physics:-Dgamma[mu], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(33)

A[mu, a] = (1-f__A(r))*Physics:-LeviCivita[4, a, j, mu]*R[j]/(g*r)

(34)

Here again you see the primed variables themselves, not their value

A[mu, a] = Matrix(%id = 36893488158429633340)

(35)

All primed ones are functions of the spherical coordinates:

A[mu, a] = Matrix(%id = 36893488158429633340)

(36)

Define now your F

${A[mu, a], Physics:-Dgamma[mu], F[mu, nu, a], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(37)

(38)

Again, the output involves the primed variables, and their derivatives, denoted with an index

F[mu, nu, a] = _rtable[36893488152118081044]

(39)

Now the Sum over all the repeated indices

(40)

Simplify the above taking into account eq

(41)

$simplify(-(1/2)*(-f__A(r)^2*(diff(`z'`(r, theta, phi), theta))^2*g^2*r^4-f__A(r)^2*(diff(`x'`(r, theta, phi), phi))^2*g^2*r^4-f__A(r)^2*(diff(`y'`(r, theta, phi), r))^2*g^2*r^4-f__A(r)^2*(diff(`x'`(r, theta, phi), theta))^2*g^2*r^4-f__A(r)^2*(diff(`y'`(r, theta, phi), phi))^2*g^2*r^4-f__A(r)^2*(diff(`z'`(r, theta, phi), r))^2*g^2*r^4-`y'`(r, theta, phi)^2*(diff(f__A(r), r))^2*g^2*r^4-`z'`(r, theta, phi)^2*(diff(f__A(r), r))^2*g^2*r^4-2*f__A(r)^2*(diff(`x'`(r, theta, phi), r))^2*g^2*r^4-2*f__A(r)^2*(diff(`y'`(r, theta, phi), theta))^2*g^2*r^4-2*f__A(r)^2*(diff(`z'`(r, theta, phi), phi))^2*g^2*r^4-2*(diff(f__A(r), r))^2*`x'`(r, theta, phi)^2*g^2*r^4+2*f__A(r)*(diff(`z'`(r, theta, phi), theta))^2*g^2*r^4+2*f__A(r)*(diff(`x'`(r, theta, phi), phi))^2*g^2*r^4+2*f__A(r)*(diff(`y'`(r, theta, phi), r))^2*g^2*r^4+4*f__A(r)*(diff(`x'`(r, theta, phi), r))^2*g^2*r^4+4*f__A(r)*(diff(`y'`(r, theta, phi), theta))^2*g^2*r^4+2*f__A(r)*(diff(`x'`(r, theta, phi), theta))^2*g^2*r^4+2*f__A(r)*(diff(`y'`(r, theta, phi), phi))^2*g^2*r^4+4*f__A(r)*(diff(`z'`(r, theta, phi), phi))^2*g^2*r^4+2*f__A(r)*(diff(`z'`(r, theta, phi), r))^2*g^2*r^4-4*f__A(r)^2*`y'`(r, theta, phi)^2*g^2*r^2-4*f__A(r)^2*`z'`(r, theta, phi)^2*g^2*r^2-8*f__A(r)^2*`x'`(r, theta, phi)^2*g^2*r^2-4*`y'`(r, theta, phi)^2*(diff(f__A(r), r))*g^2*r^3-4*`z'`(r, theta, phi)^2*(diff(f__A(r), r))*g^2*r^3-8*(diff(f__A(r), r))*`x'`(r, theta, phi)^2*g^2*r^3-2*(diff(`x'`(r, theta, phi), r))*(diff(`y'`(r, theta, phi), theta))*g^2*r^4-2*(diff(`x'`(r, theta, phi), r))*(diff(`z'`(r, theta, phi), phi))*g^2*r^4-2*(diff(`y'`(r, theta, phi), theta))*(diff(`z'`(r, theta, phi), phi))*g^2*r^4+8*f__A(r)*`y'`(r, theta, phi)^2*g^2*r^2+8*f__A(r)*`z'`(r, theta, phi)^2*g^2*r^2+16*f__A(r)*`x'`(r, theta, phi)^2*g^2*r^2+4*`y'`(r, theta, phi)*(diff(`y'`(r, theta, phi), r))*g^2*r^3+4*`z'`(r, theta, phi)*(diff(`z'`(r, theta, phi), r))*g^2*r^3+8*`x'`(r, theta, phi)*(diff(`x'`(r, theta, phi), r))*g^2*r^3+4*`x'`(r, theta, phi)*(diff(`y'`(r, theta, phi), theta))*g^2*r^3+4*`x'`(r, theta, phi)*(diff(`z'`(r, theta, phi), phi))*g^2*r^3+f__A(r)^4*`y'`(r, theta, phi)^4+f__A(r)^4*`z'`(r, theta, phi)^4+f__A(r)^4*`x'`(r, theta, phi)^4-4*f__A(r)^3*`y'`(r, theta, phi)^4-4*f__A(r)^3*`z'`(r, theta, phi)^4-4*f__A(r)^3*`x'`(r, theta, phi)^4+6*f__A(r)^2*`y'`(r, theta, phi)^4+6*f__A(r)^2*`z'`(r, theta, phi)^4+6*f__A(r)^2*`x'`(r, theta, phi)^4-4*f__A(r)*`y'`(r, theta, phi)^4-4*f__A(r)*`z'`(r, theta, phi)^4-4*f__A(r)*`x'`(r, theta, phi)^4+2*`y'`(r, theta, phi)^2*`z'`(r, theta, phi)^2+2*`y'`(r, theta, phi)^2*`x'`(r, theta, phi)^2+2*`z'`(r, theta, phi)^2*`x'`(r, theta, phi)^2-(diff(`z'`(r, theta, phi), theta))^2*g^2*r^4-(diff(`x'`(r, theta, phi), phi))^2*g^2*r^4-(diff(`y'`(r, theta, phi), r))^2*g^2*r^4-(diff(`x'`(r, theta, phi), theta))^2*g^2*r^4-(diff(`y'`(r, theta, phi), phi))^2*g^2*r^4-(diff(`z'`(r, theta, phi), r))^2*g^2*r^4+2*f__A(r)^4*`y'`(r, theta, phi)^2*`z'`(r, theta, phi)^2+2*f__A(r)^4*`y'`(r, theta, phi)^2*`x'`(r, theta, phi)^2+2*f__A(r)^4*`z'`(r, theta, phi)^2*`x'`(r, theta, phi)^2-2*(diff(`x'`(r, theta, phi), r))^2*g^2*r^4-2*(diff(`y'`(r, theta, phi), theta))^2*g^2*r^4-2*(diff(`z'`(r, theta, phi), phi))^2*g^2*r^4-8*f__A(r)^3*`y'`(r, theta, phi)^2*`z'`(r, theta, phi)^2-8*f__A(r)^3*`y'`(r, theta, phi)^2*`x'`(r, theta, phi)^2-8*f__A(r)^3*`z'`(r, theta, phi)^2*`x'`(r, theta, phi)^2+12*f__A(r)^2*`y'`(r, theta, phi)^2*`z'`(r, theta, phi)^2+12*f__A(r)^2*`y'`(r, theta, phi)^2*`x'`(r, theta, phi)^2+12*f__A(r)^2*`z'`(r, theta, phi)^2*`x'`(r, theta, phi)^2-4*`y'`(r, theta, phi)^2*g^2*r^2-4*`z'`(r, theta, phi)^2*g^2*r^2-8*`x'`(r, theta, phi)^2*g^2*r^2-8*f__A(r)*`y'`(r, theta, phi)^2*`z'`(r, theta, phi)^2-8*f__A(r)*`y'`(r, theta, phi)^2*`x'`(r, theta, phi)^2-8*f__A(r)*`z'`(r, theta, phi)^2*`x'`(r, theta, phi)^2-2*f__A(r)^2*(diff(`x'`(r, theta, phi), r))*(diff(`y'`(r, theta, phi), theta))*g^2*r^4-2*f__A(r)^2*(diff(`x'`(r, theta, phi), r))*(diff(`z'`(r, theta, phi), phi))*g^2*r^4-2*f__A(r)^2*(diff(`y'`(r, theta, phi), theta))*(diff(`z'`(r, theta, phi), phi))*g^2*r^4+4*f__A(r)^2*`y'`(r, theta, phi)*(diff(`y'`(r, theta, phi), r))*g^2*r^3+4*f__A(r)^2*`z'`(r, theta, phi)*(diff(`z'`(r, theta, phi), r))*g^2*r^3+8*f__A(r)^2*`x'`(r, theta, phi)*(diff(`x'`(r, theta, phi), r))*g^2*r^3+4*f__A(r)^2*`x'`(r, theta, phi)*(diff(`y'`(r, theta, phi), theta))*g^2*r^3+4*f__A(r)^2*`x'`(r, theta, phi)*(diff(`z'`(r, theta, phi), phi))*g^2*r^3+4*f__A(r)*`y'`(r, theta, phi)^2*(diff(f__A(r), r))*g^2*r^3+4*f__A(r)*`z'`(r, theta, phi)^2*(diff(f__A(r), r))*g^2*r^3+8*f__A(r)*(diff(f__A(r), r))*`x'`(r, theta, phi)^2*g^2*r^3+4*f__A(r)*(diff(`x'`(r, theta, phi), r))*(diff(`y'`(r, theta, phi), theta))*g^2*r^4+4*f__A(r)*(diff(`x'`(r, theta, phi), r))*(diff(`z'`(r, theta, phi), phi))*g^2*r^4+4*f__A(r)*(diff(`y'`(r, theta, phi), theta))*(diff(`z'`(r, theta, phi), phi))*g^2*r^4+2*`y'`(r, theta, phi)*(diff(`y'`(r, theta, phi), r))*(diff(f__A(r), r))*g^2*r^4+2*`z'`(r, theta, phi)*(diff(f__A(r), r))*(diff(`z'`(r, theta, phi), r))*g^2*r^4+4*(diff(f__A(r), r))*`x'`(r, theta, phi)*(diff(`x'`(r, theta, phi), r))*g^2*r^4+2*(diff(f__A(r), r))*`x'`(r, theta, phi)*(diff(`y'`(r, theta, phi), theta))*g^2*r^4+2*(diff(f__A(r), r))*`x'`(r, theta, phi)*(diff(`z'`(r, theta, phi), phi))*g^2*r^4-8*f__A(r)*`y'`(r, theta, phi)*(diff(`y'`(r, theta, phi), r))*g^2*r^3-8*f__A(r)*`z'`(r, theta, phi)*(diff(`z'`(r, theta, phi), r))*g^2*r^3-16*f__A(r)*`x'`(r, theta, phi)*(diff(`x'`(r, theta, phi), r))*g^2*r^3-8*f__A(r)*`x'`(r, theta, phi)*(diff(`y'`(r, theta, phi), theta))*g^2*r^3-8*f__A(r)*`x'`(r, theta, phi)*(diff(`z'`(r, theta, phi), phi))*g^2*r^3+`y'`(r, theta, phi)^4+`z'`(r, theta, phi)^4+`x'`(r, theta, phi)^4-2*f__A(r)*`y'`(r, theta, phi)*(diff(`y'`(r, theta, phi), r))*(diff(f__A(r), r))*g^2*r^4-2*f__A(r)*`z'`(r, theta, phi)*(diff(f__A(r), r))*(diff(`z'`(r, theta, phi), r))*g^2*r^4-4*f__A(r)*(diff(f__A(r), r))*`x'`(r, theta, phi)*(diff(`x'`(r, theta, phi), r))*g^2*r^4-2*f__A(r)*(diff(f__A(r), r))*`x'`(r, theta, phi)*(diff(`y'`(r, theta, phi), theta))*g^2*r^4-2*f__A(r)*(diff(f__A(r), r))*`x'`(r, theta, phi)*(diff(`z'`(r, theta, phi), phi))*g^2*r^4)/(r^8*g^4), {eq})$

(42)

Express now the primed variables as functions of the spherical coordinates

(43)

So your now is given by

(1/2)*(g^2*r^2*(sin(theta)^2*cos(phi)^2+1)*(diff(f__A(r), r))^2+2*g^2*r*(-1+f__A(r))*(r*sin(theta)*cos(phi)*cos(theta)*sin(phi)-sin(theta)^2*cos(phi)^2-1)*(diff(f__A(r), r))-(1+f__A(r)^2-2*f__A(r)+g^2*(r^2*cos(theta)^2-sin(theta)^2)*cos(phi)^2+2*g^2*r*sin(theta)*cos(phi)*cos(theta)*sin(phi)+(-2*r^2-1)*g^2)*(-1+f__A(r))^2)/(r^4*g^4)

(44)

This is the same result obtained following the previous approach, where

(45)

(46)

A last verification, also resulting in the same: compute with everything inert, to revise by eye each step. This is the Lagrangian

(47)

You see that the su2 indices, last index in F, run from 1 to 3 only

(48)

Compute the value of each of these inert F

${%F[1, 1, 1] = 0, %F[1, 1, 2] = 0, %F[1, 1, 3] = 0, %F[1, 2, 1] = (((diff(f__A(r), r))*`z'`(r, theta, phi)*r-(1-f__A(r))*(diff(`z'`(r, theta, phi), r))*r+2*(1-f__A(r))*`z'`(r, theta, phi))*g*r+(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[1, 2, 2] = ((-1+f__A(r))*(diff(`z'`(r, theta, phi), theta))*r^2*g+(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[1, 2, 3] = ((1-f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g+(2*(-1+f__A(r))*`x'`(r, theta, phi)-(diff(f__A(r), r))*`x'`(r, theta, phi)*r-(-1+f__A(r))*(diff(`x'`(r, theta, phi), r))*r)*g*r-(-1+f__A(r))*`z'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[1, 3, 1] = ((2*(-1+f__A(r))*`y'`(r, theta, phi)-(diff(f__A(r), r))*`y'`(r, theta, phi)*r-(-1+f__A(r))*(diff(`y'`(r, theta, phi), r))*r)*g*r-(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[1, 3, 2] = ((-1+f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g+((diff(f__A(r), r))*`x'`(r, theta, phi)*r-(1-f__A(r))*(diff(`x'`(r, theta, phi), r))*r+2*(1-f__A(r))*`x'`(r, theta, phi))*g*r+(1-f__A(r))*`y'`(r, theta, phi)^2*(-1+f__A(r)))/(r^4*g^2), %F[1, 3, 3] = ((1-f__A(r))*(diff(`y'`(r, theta, phi), phi))*r^2*g-(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[1, 4, 1] = 0, %F[1, 4, 2] = 0, %F[1, 4, 3] = 0, %F[2, 1, 1] = ((-(diff(f__A(r), r))*`z'`(r, theta, phi)*r+(1-f__A(r))*(diff(`z'`(r, theta, phi), r))*r-2*(1-f__A(r))*`z'`(r, theta, phi))*g*r-(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[2, 1, 2] = (-(-1+f__A(r))*(diff(`z'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[2, 1, 3] = ((-2*(-1+f__A(r))*`x'`(r, theta, phi)+(diff(f__A(r), r))*`x'`(r, theta, phi)*r+(-1+f__A(r))*(diff(`x'`(r, theta, phi), r))*r)*g*r-(1-f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g+(-1+f__A(r))*`z'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[2, 2, 1] = 0, %F[2, 2, 2] = 0, %F[2, 2, 3] = 0, %F[2, 3, 1] = ((1-f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g-(-1+f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))*`x'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[2, 3, 2] = (-(1-f__A(r))*(diff(`x'`(r, theta, phi), theta))*r^2*g+(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[2, 3, 3] = ((-1+f__A(r))*(diff(`x'`(r, theta, phi), phi))*r^2*g+(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[2, 4, 1] = 0, %F[2, 4, 2] = 0, %F[2, 4, 3] = 0, %F[3, 1, 1] = ((-2*(-1+f__A(r))*`y'`(r, theta, phi)+(diff(f__A(r), r))*`y'`(r, theta, phi)*r+(-1+f__A(r))*(diff(`y'`(r, theta, phi), r))*r)*g*r+(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[3, 1, 2] = ((-(diff(f__A(r), r))*`x'`(r, theta, phi)*r+(1-f__A(r))*(diff(`x'`(r, theta, phi), r))*r-2*(1-f__A(r))*`x'`(r, theta, phi))*g*r-(-1+f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g-(1-f__A(r))*`y'`(r, theta, phi)^2*(-1+f__A(r)))/(r^4*g^2), %F[3, 1, 3] = (-(1-f__A(r))*(diff(`y'`(r, theta, phi), phi))*r^2*g+(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[3, 2, 1] = ((-1+f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g-(1-f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g+(-1+f__A(r))*`x'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[3, 2, 2] = ((1-f__A(r))*(diff(`x'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[3, 2, 3] = (-(-1+f__A(r))*(diff(`x'`(r, theta, phi), phi))*r^2*g-(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[3, 3, 1] = 0, %F[3, 3, 2] = 0, %F[3, 3, 3] = 0, %F[3, 4, 1] = 0, %F[3, 4, 2] = 0, %F[3, 4, 3] = 0, %F[4, 1, 1] = 0, %F[4, 1, 2] = 0, %F[4, 1, 3] = 0, %F[4, 2, 1] = 0, %F[4, 2, 2] = 0, %F[4, 2, 3] = 0, %F[4, 3, 1] = 0, %F[4, 3, 2] = 0, %F[4, 3, 3] = 0, %F[4, 4, 1] = 0, %F[4, 4, 2] = 0, %F[4, 4, 3] = 0}$

(49)

Now for the contravariant ones

${%F[`~1`, `~1`, 1] = 0, %F[`~1`, `~1`, 2] = 0, %F[`~1`, `~1`, 3] = 0, %F[`~1`, `~2`, 1] = ((2*(-1+f__A(r))*`z'`(r, theta, phi)-(diff(f__A(r), r))*`z'`(r, theta, phi)*r-(-1+f__A(r))*(diff(`z'`(r, theta, phi), r))*r)*g*r+(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~1`, `~2`, 2] = ((1-f__A(r))*(diff(`z'`(r, theta, phi), theta))*r^2*g+(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[`~1`, `~2`, 3] = ((-1+f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g+((diff(f__A(r), r))*`x'`(r, theta, phi)*r-(1-f__A(r))*(diff(`x'`(r, theta, phi), r))*r+2*(1-f__A(r))*`x'`(r, theta, phi))*g*r-(-1+f__A(r))*`z'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[`~1`, `~3`, 1] = (((diff(f__A(r), r))*`y'`(r, theta, phi)*r-(1-f__A(r))*(diff(`y'`(r, theta, phi), r))*r+2*(1-f__A(r))*`y'`(r, theta, phi))*g*r-(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~1`, `~3`, 2] = ((1-f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g+(2*(-1+f__A(r))*`x'`(r, theta, phi)-(diff(f__A(r), r))*`x'`(r, theta, phi)*r-(-1+f__A(r))*(diff(`x'`(r, theta, phi), r))*r)*g*r+(1-f__A(r))*`y'`(r, theta, phi)^2*(-1+f__A(r)))/(r^4*g^2), %F[`~1`, `~3`, 3] = ((-1+f__A(r))*(diff(`y'`(r, theta, phi), phi))*r^2*g-(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[`~1`, `~4`, 1] = 0, %F[`~1`, `~4`, 2] = 0, %F[`~1`, `~4`, 3] = 0, %F[`~2`, `~1`, 1] = ((-2*(-1+f__A(r))*`z'`(r, theta, phi)+(diff(f__A(r), r))*`z'`(r, theta, phi)*r+(-1+f__A(r))*(diff(`z'`(r, theta, phi), r))*r)*g*r-(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~2`, `~1`, 2] = (-(1-f__A(r))*(diff(`z'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[`~2`, `~1`, 3] = ((-(diff(f__A(r), r))*`x'`(r, theta, phi)*r+(1-f__A(r))*(diff(`x'`(r, theta, phi), r))*r-2*(1-f__A(r))*`x'`(r, theta, phi))*g*r-(-1+f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g+(-1+f__A(r))*`z'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[`~2`, `~2`, 1] = 0, %F[`~2`, `~2`, 2] = 0, %F[`~2`, `~2`, 3] = 0, %F[`~2`, `~3`, 1] = ((-1+f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g-(1-f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))*`x'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[`~2`, `~3`, 2] = (-(-1+f__A(r))*(diff(`x'`(r, theta, phi), theta))*r^2*g+(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~2`, `~3`, 3] = ((1-f__A(r))*(diff(`x'`(r, theta, phi), phi))*r^2*g+(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~2`, `~4`, 1] = 0, %F[`~2`, `~4`, 2] = 0, %F[`~2`, `~4`, 3] = 0, %F[`~3`, `~1`, 1] = ((-(diff(f__A(r), r))*`y'`(r, theta, phi)*r+(1-f__A(r))*(diff(`y'`(r, theta, phi), r))*r-2*(1-f__A(r))*`y'`(r, theta, phi))*g*r+(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~3`, `~1`, 2] = ((-2*(-1+f__A(r))*`x'`(r, theta, phi)+(diff(f__A(r), r))*`x'`(r, theta, phi)*r+(-1+f__A(r))*(diff(`x'`(r, theta, phi), r))*r)*g*r-(1-f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g-(1-f__A(r))*`y'`(r, theta, phi)^2*(-1+f__A(r)))/(r^4*g^2), %F[`~3`, `~1`, 3] = (-(-1+f__A(r))*(diff(`y'`(r, theta, phi), phi))*r^2*g+(-1+f__A(r))^2*`y'`(r, theta, phi)*`z'`(r, theta, phi))/(r^4*g^2), %F[`~3`, `~2`, 1] = ((1-f__A(r))*(diff(`y'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))*(diff(`z'`(r, theta, phi), phi))*r^2*g+(-1+f__A(r))*`x'`(r, theta, phi)^2*(1-f__A(r)))/(r^4*g^2), %F[`~3`, `~2`, 2] = ((-1+f__A(r))*(diff(`x'`(r, theta, phi), theta))*r^2*g-(-1+f__A(r))^2*`y'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~3`, `~2`, 3] = (-(1-f__A(r))*(diff(`x'`(r, theta, phi), phi))*r^2*g-(-1+f__A(r))^2*`z'`(r, theta, phi)*`x'`(r, theta, phi))/(r^4*g^2), %F[`~3`, `~3`, 1] = 0, %F[`~3`, `~3`, 2] = 0, %F[`~3`, `~3`, 3] = 0, %F[`~3`, `~4`, 1] = 0, %F[`~3`, `~4`, 2] = 0, %F[`~3`, `~4`, 3] = 0, %F[`~4`, `~1`, 1] = 0, %F[`~4`, `~1`, 2] = 0, %F[`~4`, `~1`, 3] = 0, %F[`~4`, `~2`, 1] = 0, %F[`~4`, `~2`, 2] = 0, %F[`~4`, `~2`, 3] = 0, %F[`~4`, `~3`, 1] = 0, %F[`~4`, `~3`, 2] = 0, %F[`~4`, `~3`, 3] = 0, %F[`~4`, `~4`, 1] = 0, %F[`~4`, `~4`, 2] = 0, %F[`~4`, `~4`, 3] = 0}$

(50)

At this point, you have the Lagrangian in terms of inert F in (47), so that you can follow by eye that the computation is running as expected, and in (49),(50) you have the value of each inert F. So go ahead and de-inertize all F in (47)

(51)

Introduce the value of the primed variables in terms of spherical coordinates to get your

(52)

This is again equation (17), the Lagrangian you got in the first approach

(53)

Download SU(2)-field-strength-tensor_(reviewed).mw

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

OK, it is fixed and the fix distributed ...

Answered: ecterrab 14370

July 18 2024

1 1

OK, it is fixed and the fix distributed as usual within Maplesoft's Physics Updates package, so input Physics:-Version(latest) and that's it.

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

Particular solutions that are more gener...

Answered: ecterrab 14370

July 14 2024

4 1

You know, PDE solutions depend on arbitrary functions, so specializing them you have infinitely many particular solutions. The one you show is very so, it has no arbitrary functions and no arbitrary constants. The pdsolve command attempts to find general solutions, and if that is not possible, then complete solutions or else it does not attempt to find all and any possible particular solution, but for separable (simple or more general forms) or travelling wave ones.

Anyway, for more specific particular forms of solutions, e.g. polynomial ones as the one you show, there are specific commands in PDEtools, in this case PDEtools:-PolynomialSolutions.

So here are some more general - still particular - solutions, these ones depending on arbitrary constants.

(1)

(2)

This one depends on 2 arbitrary constants

${Hamil(x, y, u, v) = (1/6)*(6*x^2*y-2*y^3+3*u^2+3*v^2+3*x^2+3*y^2)*c__2+c__1}$

(3)

(4)

This other one depends on 4 arbitrary constants,

${Hamil(x, y, u, v) = (2/3)*(y+1/2)^3*c__4*x^6+(2*c__3+c__4*(-(2/3)*y^3+u^2+v^2+y^2))*(y+1/2)^2*x^4+(1/2)*(-(2/3)*y^3+u^2+v^2+y^2)^2*c__3-(2/81)*(y-3/2)^3*y^6*c__4+(1/2)*(y+1/2)*(4*(-(2/3)*y^3+u^2+v^2+y^2)*c__3+v^4*c__4+2*(-(2/3)*y^3+u^2+y^2)*c__4*v^2+u^4*c__4-(4/3)*(y-3/2)*y^2*c__4*u^2+(4/9)*(y-3/2)^2*y^4*c__4+2*c__2)*x^2+(1/12)*c__4*v^6-(1/12)*(2*y^3-3*u^2-3*y^2)*c__4*v^4+(1/36)*(9*u^4*c__4-12*(y-3/2)*y^2*c__4*u^2+4*(y-3/2)^2*y^4*c__4+18*c__2)*v^2+(1/12)*c__4*u^6-(1/12)*(2*y-3)*y^2*c__4*u^4+(1/18)*(2*(y-3/2)^2*y^4*c__4+9*c__2)*u^2-(1/3)*c__2*y^3+(1/2)*c__2*y^2+c__1}$

(5)

(6)

This other one depends on 6 arbitrary constants; I omit the output to avoid cluttering.

The 6 arbitrary constants (you could also use suffixed(c__))

${c__1, c__2, c__3, c__4, c__5, c__6}$

(7)

Download particular_solutions.mw

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

The covariant derivative command is D_...

Answered: ecterrab 14370

July 14 2024

2 0

There are different things you may want regarding the covariant derivative of Einstein's tensor. Below you have a sample of them.

(1)

>	ds_2 := -A(r)dt^2+B(r)dr^2+dtheta^2+sin(theta)^2*dphi^2

(2)

$[metric = {(1, 1) = B(r), (2, 2) = 1, (3, 3) = sin(theta)^2, (4, 4) = -A(r)}, spaceindices = lowercaselatin_is]$

(3)

Note that after a call to CompactDisplay , derivatives are displayed indexed; to see what is behind this display, use show

$CompactDisplay([metric = {(1, 1) = B(r), (2, 2) = 1, (3, 3) = sin(theta)^2, (4, 4) = -A(r)}, spaceindices = lowercaselatin_is])$

(4)

Physics:-Einstein[mu, nu] = Matrix(%id = 36893488151945680044)

(5)

Physics:-Einstein[mu, nu] = Matrix(%id = 36893488151945680044)

(6)

The covariant derivative, you may want to compute with it symbolically (simplification of expressions involving it is invoked using Physics:-Simplify).

(7)

(8)

(7) has three indices, the default display is a slice

_rtable[36893488152498098532]

(9)

This is practical

${(1/4)*(2*(diff(diff(diff(A(r), r), r), r))*B(r)^2*A(r)^2-4*B(r)^2*A(r)*(diff(diff(A(r), r), r))*(diff(A(r), r))+2*B(r)^2*(diff(A(r), r))^3-3*B(r)*A(r)^2*(diff(diff(A(r), r), r))*(diff(B(r), r))-B(r)*A(r)^2*(diff(A(r), r))*(diff(diff(B(r), r), r))+2*B(r)*A(r)*(diff(B(r), r))*(diff(A(r), r))^2+2*A(r)^2*(diff(B(r), r))^2*(diff(A(r), r)))/(B(r)^3*A(r)^3) = 0, (1/4)*sin(theta)^2*(2*(diff(diff(diff(A(r), r), r), r))*B(r)^2*A(r)^2-4*B(r)^2*A(r)*(diff(diff(A(r), r), r))*(diff(A(r), r))+2*B(r)^2*(diff(A(r), r))^3-3*B(r)*A(r)^2*(diff(diff(A(r), r), r))*(diff(B(r), r))-B(r)*A(r)^2*(diff(A(r), r))*(diff(diff(B(r), r), r))+2*B(r)*A(r)*(diff(B(r), r))*(diff(A(r), r))^2+2*A(r)^2*(diff(B(r), r))^2*(diff(A(r), r)))/(B(r)^3*A(r)^3) = 0, -(1/8)*(2*(diff(diff(A(r), r), r))*B(r)*A(r)-(diff(A(r), r))^2*B(r)-(diff(A(r), r))*(diff(B(r), r))*A(r))*(2*cot(theta)*sin(theta)^2-sin(2*theta))/(B(r)^2*A(r)^2) = 0, -(1/2)*sin(theta)*(2*(diff(diff(A(r), r), r))*B(r)*A(r)-(diff(A(r), r))^2*B(r)-(diff(A(r), r))*(diff(B(r), r))*A(r))*(cot(theta)*sin(theta)-cos(theta))/(B(r)^2*A(r)^2) = 0}$

(10)

$simplify({(1/4)*(2*(diff(diff(diff(A(r), r), r), r))*B(r)^2*A(r)^2-4*B(r)^2*A(r)*(diff(diff(A(r), r), r))*(diff(A(r), r))+2*B(r)^2*(diff(A(r), r))^3-3*B(r)*A(r)^2*(diff(diff(A(r), r), r))*(diff(B(r), r))-B(r)*A(r)^2*(diff(A(r), r))*(diff(diff(B(r), r), r))+2*B(r)*A(r)*(diff(B(r), r))*(diff(A(r), r))^2+2*A(r)^2*(diff(B(r), r))^2*(diff(A(r), r)))/(B(r)^3*A(r)^3) = 0, (1/4)*sin(theta)^2*(2*(diff(diff(diff(A(r), r), r), r))*B(r)^2*A(r)^2-4*B(r)^2*A(r)*(diff(diff(A(r), r), r))*(diff(A(r), r))+2*B(r)^2*(diff(A(r), r))^3-3*B(r)*A(r)^2*(diff(diff(A(r), r), r))*(diff(B(r), r))-B(r)*A(r)^2*(diff(A(r), r))*(diff(diff(B(r), r), r))+2*B(r)*A(r)*(diff(B(r), r))*(diff(A(r), r))^2+2*A(r)^2*(diff(B(r), r))^2*(diff(A(r), r)))/(B(r)^3*A(r)^3) = 0, -(1/8)*(2*(diff(diff(A(r), r), r))*B(r)*A(r)-(diff(A(r), r))^2*B(r)-(diff(A(r), r))*(diff(B(r), r))*A(r))*(2*cot(theta)*sin(theta)^2-sin(2*theta))/(B(r)^2*A(r)^2) = 0, -(1/2)*sin(theta)*(2*(diff(diff(A(r), r), r))*B(r)*A(r)-(diff(A(r), r))^2*B(r)-(diff(A(r), r))*(diff(B(r), r))*A(r))*(cot(theta)*sin(theta)-cos(theta))/(B(r)^2*A(r)^2) = 0})$

${0 = 0, (1/4)*(2*(diff(diff(diff(A(r), r), r), r))*B(r)^2*A(r)^2-4*B(r)^2*A(r)*(diff(diff(A(r), r), r))*(diff(A(r), r))+2*B(r)^2*(diff(A(r), r))^3-3*B(r)*A(r)^2*(diff(diff(A(r), r), r))*(diff(B(r), r))-B(r)*A(r)^2*(diff(A(r), r))*(diff(diff(B(r), r), r))+2*B(r)*A(r)*(diff(B(r), r))*(diff(A(r), r))^2+2*A(r)^2*(diff(B(r), r))^2*(diff(A(r), r)))/(B(r)^3*A(r)^3) = 0, (1/4)*sin(theta)^2*(2*(diff(diff(diff(A(r), r), r), r))*B(r)^2*A(r)^2-4*B(r)^2*A(r)*(diff(diff(A(r), r), r))*(diff(A(r), r))+2*B(r)^2*(diff(A(r), r))^3-3*B(r)*A(r)^2*(diff(diff(A(r), r), r))*(diff(B(r), r))-B(r)*A(r)^2*(diff(A(r), r))*(diff(diff(B(r), r), r))+2*B(r)*A(r)*(diff(B(r), r))*(diff(A(r), r))^2+2*A(r)^2*(diff(B(r), r))^2*(diff(A(r), r)))/(B(r)^3*A(r)^3) = 0}$

(11)

This gives you a slice display

(12)

Physics:-D_[1](Physics:-Einstein[alpha, beta](r, theta), [X]) = Matrix(%id = 36893488151928256020), Physics:-D_[2](Physics:-Einstein[alpha, beta](r, theta), [X]) = Matrix(%id = 36893488152498095876), Physics:-D_[3](Physics:-Einstein[alpha, beta](r, theta), [X]) = Matrix(%id = 36893488152498080212), Physics:-D_[4](Physics:-Einstein[alpha, beta](r, theta), [X]) = Matrix(%id = 36893488152498057212)