ecterrab

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These are answers submitted by ecterrab

Hi

It is difficult to help without a worksheet where you show, e.g. what makes you say "I caught aome serious errors with Maple 22 calculating the Ricci tensor". Regarding "How do I tell Maple to contract thae alpha and beta indices in above expression", maybe you mean multiplying by the metric to contract? If so, you can use * or . that includes simplification - all these things are explained in detail and with examples on the help page ?Physics,Tensors.

Anyway, if you post your worksheet (see the green arrow to upload it), it becomes possible to understand what you are saying and, from there, provide more specific feedback.

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



Download Total_differential.mw

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

In short: what you are saying is not correct. The solution y(x) = 3 is a particular solution, you get it taking c__1 = infinity.

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

This is your ODE and dsolve's solution, all tests OK

ode := diff(y(x), x)-cot(x)*(y(x)^(1/2)-y(x)) = 0

sol := dsolve(ode)

y(x)^(1/2)-(Int((1/2)*sin(x)^(1/2)*cot(x), x)+c__1)/sin(x)^(1/2) = 0

(1)

odetest(sol, ode)

0

(2)

dsolve's solution computing the integral also tests OK

sol := dsolve(ode, useint)

y(x)^(1/2)-1-c__1/sin(x)^(1/2) = 0

(3)

odetest(sol, ode)

0

(4)

dsolve's solution in explicit form requires RootOf and also tests OK

sol := dsolve(ode, useint, explicit)

y(x) = RootOf(-_Z^(1/2)*sin(x)^(1/2)+sin(x)^(1/2)+c__1)

(5)

odetest(sol, ode)

0

(6)

Now, this is your solution, nm,

sol_NM := y(x) = (exp(RootOf(-sin(x)*tanh((1/2)*_Z+(1/2)*c__1)^2+sin(x)+exp(_Z)))+sin(x))/sin(x)

y(x) = (exp(RootOf(-sin(x)*tanh((1/2)*_Z+(1/2)*c__1)^2+sin(x)+exp(_Z)))+sin(x))/sin(x)

(7)

I don't know from where you got this expression, but to my eyes it is not a solution to ODE. Remove the RootOf to test the actual expression behind

DEtools:-remove_RootOf(sol_NM)

-sin(x)*tanh((1/2)*ln(y(x)*sin(x)-sin(x))+(1/2)*c__1)^2+y(x)*sin(x) = 0

(8)

odetest(-sin(x)*tanh((1/2)*ln(y(x)*sin(x)-sin(x))+(1/2)*c__1)^2+y(x)*sin(x) = 0, ode)

`odetest/PIECEWISE`([0, -ln(y(x)*sin(x)-sin(x))+2*arctanh(y(x)^(1/2)) = c__1], [2*(sin(x)*exp(c__1)*(exp(-c__1)*(exp(c__1)-csc(x)+2*exp((1/2)*c__1)*(-sin(x))^(1/2)*csc(x)))^(1/2)-exp(c__1)*sin(x)-2*exp((1/2)*c__1)*(-sin(x))^(1/2)+1)*exp(-c__1)*cot(x)*csc(x)/((exp(-c__1)*(exp(c__1)-csc(x)+2*exp((1/2)*c__1)*(-sin(x))^(1/2)*csc(x)))^(1/2)-1), -ln(y(x)*sin(x)-sin(x))-2*arctanh(y(x)^(1/2)) = c__1])

(9)

Regarding testing your expression (not a solution) explicitly as if it were a solution, you hit a bug, but not in odetest; it is in radnormal, this is the expression, resulting from your not-a-solution, that goes into an infinite loop within radnormal

NULL

ee := (((2*I)*RootOf(_Z^3*_a^(2*c__1)+2*_Z^2*_a^c__1+((2*I)*_a^(c__1-I*x)-(2*I)*_a^(c__1+I*x)+1)*_Z)*_a^(I*x)+_a^((2*I)*x)-1)/(_a^((2*I)*x)-1))^(1/2)

radnormal(ee);radnormal(ee)

Error, (in anonymous procedure called from depends) too many levels of recursion |lib/depends/src/depends.mpl:80|

 

NULL


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

Download ode_solved_and_tested_OK.mw

This is resolved; the integration constants shown by ODESteps are now the new ones (nicer), as the ones returned by default by dsolve. For this change to be active, as usual, install the latest update by opening Maple 2024 and entering Physics:-Version(latest)

Regarding your question in the worksheet, as explained in ?dsolve,details, by default, dsolve and pdsolve return integration constants and functions as c__n and f__n(...), where n is an integer and is displayed as a subscript (and the `__` is not visible, it just produces the subscript), but you can set these commands to work as in the past (2 releases ago), i.e. using _Cn and _Fn.

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

Thanks for the report, @nm. This one is fixed in the Updates v.1724 or newer. As usual, to install, open Maple and input Physics:-Version(latest);

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

Hi @nm, good catch. It's was a subtle issue. It is fixed in the Udpates v.1723 for Maple 2024. As usual, to install, open Maple 2024 and input Physics:-Version(latest).

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

So maybe I do not understand your question.

You define R[a, b] = P, where P is a tensorial expression with a and b as free indices. OK.

Next, you compute the Taylor expansion of R[1, 1] and equate that to 0. But nowhere in your worksheet do I see why R[1,1] would be equal to 0. Anyway, that implies an ODE for f(r).

Next, you compute the Taylor expansion of R[~1, 1] and equate that to 0.  Naturally, R[1, 1] <> R[~1, 1] because R[~1, 1] = g_[~1, ~a]*R[a, 1] (including there the sum over the repeated index a) and g_[~1, ~a] <> 1. So, naturally, the ODE you get for f(r) from the Taylor expansion of R[~1, 1] is different from the one you got from the expansion of R[1, 1].

Why are you equating two different things to 0, and then expecting them to result in the same equation for f(r)? Or even why are you suggesting that R[1, 1] = R[~1, 1]?

Guessing a bit, if you want to determine the value of f(r) such that R[1, 1] = R[~1, 1], then do this:
dsolve(Gtaylor(R[~1, 1] - R[1, 1] = 0, b, 2), f(r));
                                    f(r) = -1


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

This is Maple TTY running with no initialization whatsoever, and with the latest Maplesoft Physics Updates: there is no crash. I suggest you try removing any initialization file.

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

The Physics Updates do work. What didn't work was the download mechanism. It looks fixed now in v.1714 - I just downloaded it (input Physics:-Version(latest), as usual)

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

Hi @nm
I removed all initialization files to be sure, and run it directly in TTY to be sure also that there was no GUI element interfering. Even so, I cannot reproduce your problem, and I do have, as you, the Maplesoft Physics Updates installed, as you see in this pic. The first line, anames(); shows there is no initialization file or previous assignment. I'd suggest you check that in your computer.

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

You can define differential operators and operate with them algebraically, with multiplication understood as the operator's application. You go from multiplication to application using Physics:-Library:-ApplyProductsOfDifferentialOperators. All that is explained in the help page of Physics:-Setup, the section for differential operators.

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

Hi
A fix for this is distributed for everybody using Maple 2024 within the Maplesoft Physics Updates v.1705 or newer. As usual, to install the Updates, open Maple and input Physics:-Version(latest) So now we have:

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

Yes, you can construct these operators and compute with them algebraically, check the help page ?Physics,Setup, the section on differentialoperators. You can see an illustration of how to work with these operators in the post on "Quantum Commutation Rules - Basics"

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

Hi
As hinted by @acer, this was a problem in the simplifier, an undesired side effect of recent developments in simplifying trigonometric functions. The issue is fixed, and the fix is distributed for everybody using Maple 2024 within the Maplesoft Physics Updates v.1702 or newer. As usual, to install the Updates, open Maple and input Physics:-Version(latest)

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

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