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ecterrab

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The problem and its workaround...

ecterrab 13431
September 30 2015
1 1

I actually like and use 2D Math input, but it has its problems, frequently invisible, and this one you mention looks like that kind of thing.

The simple workaround, that more than that it is a saving of your time all around, is to not rewrite the commands found in this site you mention, but instead use these commands as already written by Maple. I mean:

  1. Open the help page by typing ?Poincare,examples (you see I am entering the question mark "?" to invoke the help page) then pressing Enter.
  2. The page that opens is actually a worksheet with everything written, you can reproduce things, play around, change, etc. and more than nothing: you don't need to spend your time rewriting :) 

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

don't use spacetime contracted indices a...

ecterrab 13431
September 29 2015
3 9

It is not, at all, that a loop could change or corrupt in this way a tetrad definition.

The answer actually is that you shall not use spacetime contracted indices, that enter your definition of RicciT, I am talking of a (see the line that starts with 'Define'), also as a loop dummy variable, because in this way you corrupt your definition (because the repeated spacetime index a ends up having the value of the loop index, that actually refers to a free, not repeated index).

So the correction in this case is to use something else for the loop dummies, for instance i and j, that do not enter as repeated indices in the definition of your tensors or, to avoid having to read things by eye to discard letters, once you choose using lowercase latin to represent spacetime indices, then, in general, use some other kind of letter for for-do-od loops, for instance alpha and beta, or A and B.

 

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

You can use the Physics package...

ecterrab 13431
September 25 2015
2 1

Hi Damon,
This is also the kind of operation you do directly by using the Physics package; in this example using its Vectors subpackage; please give a look at the help page ?Physics,Vectors

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

The Physics package...

ecterrab 13431
September 25 2015
3 0

Tensor algebra using standard tensor indicial notation, including the LeviCivita and KroneckerDelta tensors (your epsilon and delta) is implemented in the Physics package, please give a look at the help page ?Physics.

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

Yes...

ecterrab 13431
September 22 2015
1 0

Yes the output is correct. Recalling, differentiation with respect to a 'function' is not an ambiguous operation but just differentiation with respect to a jet variable of a jet space. Skipping technical details you can think of a jet space as one where independent variables, dependent variables (what we call 'functions') and their partial derivatives are all "independent objects in equal footing". That is, q and dq/dt are independent objects, and so  d/dq (dq/dt) = 0. See the help page for PDEtools:-ToJet. Another page you can give a look for a concrete example of the form you are posting (a bit more general) is ?Physics,diff, in the Examples section, starting at equation number (6).

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

The error message...

ecterrab 13431
September 19 2015
3 1

Hi

The error message tells what the problem is. Change, in your definition of a (first equation), f(x) by f(x(t)) and everything will work as you expect.

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

You need to take PDE1 into account......

ecterrab 13431
September 19 2015
2 1

 

There are two issues: first, in each DetSys you use a different name for the infinitesimals - you needto use the same name in order to compare; second and more important: you forgot to include the information of PDE1 in DetSys2, in that way it is not possible to compare things. I adjusted these two things below and showed how to compare the resulting DetSys: they are both correct and equivalent (i.e. they have the same solutions).

 

Below I intercalate comments all italized, as well as adjust a bit a few input lines to make things more straightforward.

 

> 

with(PDEtools):
> 

DepVars := [u(x, t), a(t), b(t)]

[u(x, t), a(t), b(t)]

 (1)
> 

declare(u(x, t), a(t), b(t))

u(x, t)*`will now be displayed as`*u

  

a(t)*`will now be displayed as`*a

  

b(t)*`will now be displayed as`*b

 (2)
> 

U, A, B := diff_table(u(x, t)), diff_table(a(t)), diff_table(b(t))

table( [( ) = u(x, t) ] ), table( [( ) = a(t) ] ), table( [( ) = b(t) ] )

 (3)
> 

PDE1 := U[t]+u(x, t)*U[x]+a(t)*u(x, t)+b(t)*U[x, x, x] = 0

diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x))+a(t)*u(x, t)+b(t)*(diff(diff(diff(u(x, t), x), x), x)) = 0

 (4)

In this worksheet you have used _xi and _eta (with underscor) for DetSys and xi and eta (without underscore) for DetSys2, so to be sure you use the same names for the infinitesimals, set G right here and use it for both determining systems

> 

G := [seq(xi[j](x, t, u), j = [x, t]), seq(eta[j](x, t, u), j = [u])]

[xi[x](x, t, u), xi[t](x, t, u), eta[u](x, t, u)]

 (5)
> 

PDEtools:-declare(G)

eta(x, t, u)*`will now be displayed as`*eta

  

xi(x, t, u)*`will now be displayed as`*xi

 (6)
> 

DetSys := DeterminingPDE(PDE1, G, integrabilityconditions = false)

{3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*b(t)-3*(diff(xi[x](x, t, u), u)), 3*(diff(diff(diff(xi[x](x, t, u), u), u), x))-(diff(diff(diff(eta[u](x, t, u), u), u), u)), -3*(diff(diff(eta[u](x, t, u), u), x))+3*(diff(diff(xi[x](x, t, u), x), x)), 9*(diff(diff(xi[x](x, t, u), u), x))-3*(diff(diff(eta[u](x, t, u), u), u)), 3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*b(t)-3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*b(t)-3*(diff(xi[x](x, t, u), u))*u, u*(diff(xi[t](x, t, u), x))*b(t)-3*(diff(xi[x](x, t, u), x))*b(t)+(diff(diff(diff(xi[t](x, t, u), x), x), x))*b(t)^2+(diff(b(t), t))*xi[t](x, t, u)+(diff(xi[t](x, t, u), t))*b(t)-a(t)*u*(diff(xi[t](x, t, u), u))*b(t), -(diff(diff(diff(eta[u](x, t, u), x), x), x))*b(t)^2+b(t)*(diff(eta[u](x, t, u), u))*a(t)*u-b(t)*(diff(eta[u](x, t, u), t))-(diff(eta[u](x, t, u), x))*b(t)*u-3*b(t)*(diff(xi[x](x, t, u), x))*a(t)*u+u*(a(t)*(diff(b(t), t))-b(t)*(diff(a(t), t)))*xi[t](x, t, u)-eta[u](x, t, u)*b(t)*a(t), -2*b(t)*(diff(xi[x](x, t, u), x))*u+(diff(b(t), t))*xi[t](x, t, u)*u-3*(diff(diff(diff(eta[u](x, t, u), u), x), x))*b(t)^2+(diff(diff(diff(xi[x](x, t, u), x), x), x))*b(t)^2-eta[u](x, t, u)*b(t)+b(t)*(diff(xi[x](x, t, u), t))-4*b(t)*(diff(xi[x](x, t, u), u))*a(t)*u, diff(diff(diff(xi[t](x, t, u), u), u), u), diff(diff(diff(xi[t](x, t, u), u), u), x), diff(diff(diff(xi[x](x, t, u), u), u), u), diff(diff(xi[t](x, t, u), u), u), diff(diff(xi[t](x, t, u), u), x), diff(diff(xi[t](x, t, u), x), x), diff(diff(xi[x](x, t, u), u), u), diff(xi[t](x, t, u), u), diff(xi[t](x, t, u), x), diff(xi[x](x, t, u), u)}

 (7)

Instead of copying each equation by hand in order to list them, you can list the equations in this way

> 

for EQ in sort([op(DetSys)], length) do EQ = 0 end do

diff(xi[t](x, t, u), u) = 0

  

diff(xi[t](x, t, u), x) = 0

  

diff(xi[x](x, t, u), u) = 0

  

diff(diff(xi[t](x, t, u), u), u) = 0

  

diff(diff(xi[t](x, t, u), u), x) = 0

  

diff(diff(xi[t](x, t, u), x), x) = 0

  

diff(diff(xi[x](x, t, u), u), u) = 0

  

diff(diff(diff(xi[t](x, t, u), u), u), u) = 0

  

diff(diff(diff(xi[t](x, t, u), u), u), x) = 0

  

diff(diff(diff(xi[x](x, t, u), u), u), u) = 0

  

-3*(diff(diff(eta[u](x, t, u), u), x))+3*(diff(diff(xi[x](x, t, u), x), x)) = 0

  

9*(diff(diff(xi[x](x, t, u), u), x))-3*(diff(diff(eta[u](x, t, u), u), u)) = 0

  

3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*b(t)-3*(diff(xi[x](x, t, u), u)) = 0

  

3*(diff(diff(diff(xi[x](x, t, u), u), u), x))-(diff(diff(diff(eta[u](x, t, u), u), u), u)) = 0

  

3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*b(t)-3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*b(t)-3*(diff(xi[x](x, t, u), u))*u = 0

  

u*(diff(xi[t](x, t, u), x))*b(t)-3*(diff(xi[x](x, t, u), x))*b(t)+(diff(diff(diff(xi[t](x, t, u), x), x), x))*b(t)^2+(diff(b(t), t))*xi[t](x, t, u)+(diff(xi[t](x, t, u), t))*b(t)-a(t)*u*(diff(xi[t](x, t, u), u))*b(t) = 0

  

-2*b(t)*(diff(xi[x](x, t, u), x))*u+(diff(b(t), t))*xi[t](x, t, u)*u-3*(diff(diff(diff(eta[u](x, t, u), u), x), x))*b(t)^2+(diff(diff(diff(xi[x](x, t, u), x), x), x))*b(t)^2-eta[u](x, t, u)*b(t)+b(t)*(diff(xi[x](x, t, u), t))-4*b(t)*(diff(xi[x](x, t, u), u))*a(t)*u = 0

  

-(diff(diff(diff(eta[u](x, t, u), x), x), x))*b(t)^2+b(t)*(diff(eta[u](x, t, u), u))*a(t)*u-b(t)*(diff(eta[u](x, t, u), t))-(diff(eta[u](x, t, u), x))*b(t)*u-3*b(t)*(diff(xi[x](x, t, u), x))*a(t)*u+u*(a(t)*(diff(b(t), t))-b(t)*(diff(a(t), t)))*xi[t](x, t, u)-eta[u](x, t, u)*b(t)*a(t) = 0

 (8)
> 

> 

> 

DepVars := [u(x, t)]

[u(x, t)]

 (9)

G was already entered lines above, so I comment your input line here

 

> 

> 

S := PDEtools:-InfinitesimalGenerator(G, DepVars, expanded, prolongation = 3):

 

You do not need to enter the equation in Jet notation by hand (prone to mistakes); you can use ToJet instead

> 

> 

L := PDEtools:-ToJet(PDE1, DepVars)

u[t]+u*u[x]+a(t)*u+b(t)*u[x, x, x] = 0

 (10)
> 

S(L)

xi[t](x, t, u)*((diff(a(t), t))*u+(diff(b(t), t))*u[x, x, x])+eta[u](x, t, u)*(u[x]+a(t))+(-(diff(xi[x](x, t, u), u))*u[x]^2-(diff(xi[t](x, t, u), u))*u[x]*u[t]+u[x]*(diff(eta[u](x, t, u), u))-(diff(xi[x](x, t, u), x))*u[x]-(diff(xi[t](x, t, u), x))*u[t]+diff(eta[u](x, t, u), x))*u-(diff(xi[x](x, t, u), u))*u[t]*u[x]-(diff(xi[t](x, t, u), u))*u[t]^2+(diff(eta[u](x, t, u), u))*u[t]-(diff(xi[x](x, t, u), t))*u[x]-(diff(xi[t](x, t, u), t))*u[t]+diff(eta[u](x, t, u), t)+(-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]*u[t]*u[x, x]-3*(diff(xi[t](x, t, u), u))*u[x, x]*u[x, t]-4*(diff(xi[x](x, t, u), u))*u[x]*u[x, x, x]-(diff(xi[t](x, t, u), u))*u[t]*u[x, x, x]-3*(diff(xi[t](x, t, u), u))*u[x]*u[x, x, t]-(diff(diff(diff(xi[t](x, t, u), u), u), u))*u[x]^3*u[t]-6*(diff(diff(xi[x](x, t, u), u), u))*u[x]^2*u[x, x]-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]^2*u[x, t]+3*(diff(diff(eta[u](x, t, u), u), u))*u[x]*u[x, x]-3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*u[x]^2*u[t]-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*u[x]*u[t]-9*(diff(diff(xi[x](x, t, u), u), x))*u[x]*u[x, x]-6*(diff(diff(xi[t](x, t, u), u), x))*u[x]*u[x, t]-3*(diff(diff(xi[t](x, t, u), u), x))*u[t]*u[x, x]+diff(diff(diff(eta[u](x, t, u), x), x), x)-3*(diff(xi[x](x, t, u), u))*u[x, x]^2+(diff(eta[u](x, t, u), u))*u[x, x, x]-3*(diff(xi[x](x, t, u), x))*u[x, x, x]-3*(diff(xi[t](x, t, u), x))*u[x, x, t]-3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*u[x]^3-3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*u[x]^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*u[x]^2+3*u[x]*(diff(diff(diff(eta[u](x, t, u), u), x), x))-(diff(diff(diff(xi[x](x, t, u), x), x), x))*u[x]-(diff(diff(diff(xi[t](x, t, u), x), x), x))*u[t]+3*(diff(diff(eta[u](x, t, u), u), x))*u[x, x]-3*(diff(diff(xi[x](x, t, u), x), x))*u[x, x]-3*(diff(diff(xi[t](x, t, u), x), x))*u[x, t]-(diff(diff(diff(xi[x](x, t, u), u), u), u))*u[x]^4+(diff(diff(diff(eta[u](x, t, u), u), u), u))*u[x]^3)*b(t) = 0

 (11)
> 

> 

#In (14), on equating coefficients of various differentials of u(x,t) the determining equation are obtained which does not matches with obtained at (6), what could be the reason???

 

There is one essential issue here: the way you constructed DetSys2, it does not take into account the actual form of PDE1 anywhere. Hence your DetSys2 depends on u[x, x, x]

> 

indets(xi[t](x, t, u)*((diff(a(t), t))*u+(diff(b(t), t))*u[x, x, x])+eta[u](x, t, u)*(u[x]+a(t))+(-(diff(xi[x](x, t, u), u))*u[x]^2-(diff(xi[t](x, t, u), u))*u[x]*u[t]+u[x]*(diff(eta[u](x, t, u), u))-(diff(xi[x](x, t, u), x))*u[x]-(diff(xi[t](x, t, u), x))*u[t]+diff(eta[u](x, t, u), x))*u-(diff(xi[x](x, t, u), u))*u[t]*u[x]-(diff(xi[t](x, t, u), u))*u[t]^2+(diff(eta[u](x, t, u), u))*u[t]-(diff(xi[x](x, t, u), t))*u[x]-(diff(xi[t](x, t, u), t))*u[t]+diff(eta[u](x, t, u), t)+(-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]*u[t]*u[x, x]-3*(diff(xi[t](x, t, u), u))*u[x, x]*u[x, t]-4*(diff(xi[x](x, t, u), u))*u[x]*u[x, x, x]-(diff(xi[t](x, t, u), u))*u[t]*u[x, x, x]-3*(diff(xi[t](x, t, u), u))*u[x]*u[x, x, t]-(diff(diff(diff(xi[t](x, t, u), u), u), u))*u[x]^3*u[t]-6*(diff(diff(xi[x](x, t, u), u), u))*u[x]^2*u[x, x]-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]^2*u[x, t]+3*(diff(diff(eta[u](x, t, u), u), u))*u[x]*u[x, x]-3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*u[x]^2*u[t]-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*u[x]*u[t]-9*(diff(diff(xi[x](x, t, u), u), x))*u[x]*u[x, x]-6*(diff(diff(xi[t](x, t, u), u), x))*u[x]*u[x, t]-3*(diff(diff(xi[t](x, t, u), u), x))*u[t]*u[x, x]+diff(diff(diff(eta[u](x, t, u), x), x), x)-3*(diff(xi[x](x, t, u), u))*u[x, x]^2+(diff(eta[u](x, t, u), u))*u[x, x, x]-3*(diff(xi[x](x, t, u), x))*u[x, x, x]-3*(diff(xi[t](x, t, u), x))*u[x, x, t]-3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*u[x]^3-3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*u[x]^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*u[x]^2+3*u[x]*(diff(diff(diff(eta[u](x, t, u), u), x), x))-(diff(diff(diff(xi[x](x, t, u), x), x), x))*u[x]-(diff(diff(diff(xi[t](x, t, u), x), x), x))*u[t]+3*(diff(diff(eta[u](x, t, u), u), x))*u[x, x]-3*(diff(diff(xi[x](x, t, u), x), x))*u[x, x]-3*(diff(diff(xi[t](x, t, u), x), x))*u[x, t]-(diff(diff(diff(xi[x](x, t, u), u), u), u))*u[x]^4+(diff(diff(diff(eta[u](x, t, u), u), u), u))*u[x]^3)*b(t) = 0, specindex(u))

{u[t], u[x], u[x, t], u[x, x], u[x, x, t], u[x, x, x]}

 (12)

and therefore you cannot compare it with DetSys, that does take into account the actual form of PDE1; it does not depend on u[x, x, x].

 

So let's first remove this third derivative from DetSys2 introducing the information regarding the actual form of PDE1, and in that way you have something valid to be compared with DetSys

> 

isolate(L, u[x, x, x])

u[x, x, x] = (-u*u[x]-u[t]-a(t)*u)/b(t)

 (13)

Introduce this value of u[x, x, x] in the expression for the determining system you constructed step by step using the InfinitesimalGenerator then ToJet, then isolate, now subs:

> 

subs(u[x, x, x] = (-u*u[x]-u[t]-a(t)*u)/b(t), xi[t](x, t, u)*((diff(a(t), t))*u+(diff(b(t), t))*u[x, x, x])+eta[u](x, t, u)*(u[x]+a(t))+(-(diff(xi[x](x, t, u), u))*u[x]^2-(diff(xi[t](x, t, u), u))*u[x]*u[t]+u[x]*(diff(eta[u](x, t, u), u))-(diff(xi[x](x, t, u), x))*u[x]-(diff(xi[t](x, t, u), x))*u[t]+diff(eta[u](x, t, u), x))*u-(diff(xi[x](x, t, u), u))*u[t]*u[x]-(diff(xi[t](x, t, u), u))*u[t]^2+(diff(eta[u](x, t, u), u))*u[t]-(diff(xi[x](x, t, u), t))*u[x]-(diff(xi[t](x, t, u), t))*u[t]+diff(eta[u](x, t, u), t)+(-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]*u[t]*u[x, x]-3*(diff(xi[t](x, t, u), u))*u[x, x]*u[x, t]-4*(diff(xi[x](x, t, u), u))*u[x]*u[x, x, x]-(diff(xi[t](x, t, u), u))*u[t]*u[x, x, x]-3*(diff(xi[t](x, t, u), u))*u[x]*u[x, x, t]-(diff(diff(diff(xi[t](x, t, u), u), u), u))*u[x]^3*u[t]-6*(diff(diff(xi[x](x, t, u), u), u))*u[x]^2*u[x, x]-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]^2*u[x, t]+3*(diff(diff(eta[u](x, t, u), u), u))*u[x]*u[x, x]-3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*u[x]^2*u[t]-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*u[x]*u[t]-9*(diff(diff(xi[x](x, t, u), u), x))*u[x]*u[x, x]-6*(diff(diff(xi[t](x, t, u), u), x))*u[x]*u[x, t]-3*(diff(diff(xi[t](x, t, u), u), x))*u[t]*u[x, x]+diff(diff(diff(eta[u](x, t, u), x), x), x)-3*(diff(xi[x](x, t, u), u))*u[x, x]^2+(diff(eta[u](x, t, u), u))*u[x, x, x]-3*(diff(xi[x](x, t, u), x))*u[x, x, x]-3*(diff(xi[t](x, t, u), x))*u[x, x, t]-3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*u[x]^3-3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*u[x]^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*u[x]^2+3*u[x]*(diff(diff(diff(eta[u](x, t, u), u), x), x))-(diff(diff(diff(xi[x](x, t, u), x), x), x))*u[x]-(diff(diff(diff(xi[t](x, t, u), x), x), x))*u[t]+3*(diff(diff(eta[u](x, t, u), u), x))*u[x, x]-3*(diff(diff(xi[x](x, t, u), x), x))*u[x, x]-3*(diff(diff(xi[t](x, t, u), x), x))*u[x, t]-(diff(diff(diff(xi[x](x, t, u), u), u), u))*u[x]^4+(diff(diff(diff(eta[u](x, t, u), u), u), u))*u[x]^3)*b(t) = 0)

xi[t](x, t, u)*((diff(a(t), t))*u+(diff(b(t), t))*(-u*u[x]-u[t]-a(t)*u)/b(t))+eta[u](x, t, u)*(u[x]+a(t))+(-(diff(xi[x](x, t, u), u))*u[x]^2-(diff(xi[t](x, t, u), u))*u[x]*u[t]+u[x]*(diff(eta[u](x, t, u), u))-(diff(xi[x](x, t, u), x))*u[x]-(diff(xi[t](x, t, u), x))*u[t]+diff(eta[u](x, t, u), x))*u-(diff(xi[x](x, t, u), u))*u[t]*u[x]-(diff(xi[t](x, t, u), u))*u[t]^2+(diff(eta[u](x, t, u), u))*u[t]-(diff(xi[x](x, t, u), t))*u[x]-(diff(xi[t](x, t, u), t))*u[t]+diff(eta[u](x, t, u), t)+(-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]*u[t]*u[x, x]-3*(diff(xi[t](x, t, u), u))*u[x, x]*u[x, t]-3*(diff(xi[t](x, t, u), u))*u[x]*u[x, x, t]-(diff(diff(diff(xi[t](x, t, u), u), u), u))*u[x]^3*u[t]-6*(diff(diff(xi[x](x, t, u), u), u))*u[x]^2*u[x, x]-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]^2*u[x, t]+3*(diff(diff(eta[u](x, t, u), u), u))*u[x]*u[x, x]-3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*u[x]^2*u[t]-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*u[x]*u[t]-9*(diff(diff(xi[x](x, t, u), u), x))*u[x]*u[x, x]-6*(diff(diff(xi[t](x, t, u), u), x))*u[x]*u[x, t]-3*(diff(diff(xi[t](x, t, u), u), x))*u[t]*u[x, x]+diff(diff(diff(eta[u](x, t, u), x), x), x)-3*(diff(xi[x](x, t, u), u))*u[x, x]^2-3*(diff(xi[t](x, t, u), x))*u[x, x, t]-3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*u[x]^3-3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*u[x]^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*u[x]^2+3*u[x]*(diff(diff(diff(eta[u](x, t, u), u), x), x))-(diff(diff(diff(xi[x](x, t, u), x), x), x))*u[x]-(diff(diff(diff(xi[t](x, t, u), x), x), x))*u[t]+3*(diff(diff(eta[u](x, t, u), u), x))*u[x, x]-3*(diff(diff(xi[x](x, t, u), x), x))*u[x, x]-3*(diff(diff(xi[t](x, t, u), x), x))*u[x, t]-(diff(diff(diff(xi[x](x, t, u), u), u), u))*u[x]^4+(diff(diff(diff(eta[u](x, t, u), u), u), u))*u[x]^3+(diff(eta[u](x, t, u), u))*(-u*u[x]-u[t]-a(t)*u)/b(t)-3*(diff(xi[x](x, t, u), x))*(-u*u[x]-u[t]-a(t)*u)/b(t)-4*(diff(xi[x](x, t, u), u))*u[x]*(-u*u[x]-u[t]-a(t)*u)/b(t)-(diff(xi[t](x, t, u), u))*u[t]*(-u*u[x]-u[t]-a(t)*u)/b(t))*b(t) = 0

 (14)

 

Now, to compare with DetSys you still need to take the coefficients of the indexed u (here representing derivatives in Jet notation)

> 

du := indets(xi[t](x, t, u)*((diff(a(t), t))*u+(diff(b(t), t))*(-u*u[x]-u[t]-a(t)*u)/b(t))+eta[u](x, t, u)*(u[x]+a(t))+(-(diff(xi[x](x, t, u), u))*u[x]^2-(diff(xi[t](x, t, u), u))*u[x]*u[t]+u[x]*(diff(eta[u](x, t, u), u))-(diff(xi[x](x, t, u), x))*u[x]-(diff(xi[t](x, t, u), x))*u[t]+diff(eta[u](x, t, u), x))*u-(diff(xi[x](x, t, u), u))*u[t]*u[x]-(diff(xi[t](x, t, u), u))*u[t]^2+(diff(eta[u](x, t, u), u))*u[t]-(diff(xi[x](x, t, u), t))*u[x]-(diff(xi[t](x, t, u), t))*u[t]+diff(eta[u](x, t, u), t)+(-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]*u[t]*u[x, x]-3*(diff(xi[t](x, t, u), u))*u[x, x]*u[x, t]-3*(diff(xi[t](x, t, u), u))*u[x]*u[x, x, t]-(diff(diff(diff(xi[t](x, t, u), u), u), u))*u[x]^3*u[t]-6*(diff(diff(xi[x](x, t, u), u), u))*u[x]^2*u[x, x]-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]^2*u[x, t]+3*(diff(diff(eta[u](x, t, u), u), u))*u[x]*u[x, x]-3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*u[x]^2*u[t]-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*u[x]*u[t]-9*(diff(diff(xi[x](x, t, u), u), x))*u[x]*u[x, x]-6*(diff(diff(xi[t](x, t, u), u), x))*u[x]*u[x, t]-3*(diff(diff(xi[t](x, t, u), u), x))*u[t]*u[x, x]+diff(diff(diff(eta[u](x, t, u), x), x), x)-3*(diff(xi[x](x, t, u), u))*u[x, x]^2-3*(diff(xi[t](x, t, u), x))*u[x, x, t]-3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*u[x]^3-3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*u[x]^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*u[x]^2+3*u[x]*(diff(diff(diff(eta[u](x, t, u), u), x), x))-(diff(diff(diff(xi[x](x, t, u), x), x), x))*u[x]-(diff(diff(diff(xi[t](x, t, u), x), x), x))*u[t]+3*(diff(diff(eta[u](x, t, u), u), x))*u[x, x]-3*(diff(diff(xi[x](x, t, u), x), x))*u[x, x]-3*(diff(diff(xi[t](x, t, u), x), x))*u[x, t]-(diff(diff(diff(xi[x](x, t, u), u), u), u))*u[x]^4+(diff(diff(diff(eta[u](x, t, u), u), u), u))*u[x]^3+(diff(eta[u](x, t, u), u))*(-u*u[x]-u[t]-a(t)*u)/b(t)-3*(diff(xi[x](x, t, u), x))*(-u*u[x]-u[t]-a(t)*u)/b(t)-4*(diff(xi[x](x, t, u), u))*u[x]*(-u*u[x]-u[t]-a(t)*u)/b(t)-(diff(xi[t](x, t, u), u))*u[t]*(-u*u[x]-u[t]-a(t)*u)/b(t))*b(t) = 0, specindex(u))

{u[t], u[x], u[x, t], u[x, x], u[x, x, t]}

 (15)

 

Note that for coeff to work, you need to apply it to the left-hand side of the equation (14), and I also take the numerator of this expression - in this way there is no need to call expand

> 

DetSys3 := {coeffs(numer(lhs(xi[t](x, t, u)*((diff(a(t), t))*u+(diff(b(t), t))*(-u*u[x]-u[t]-a(t)*u)/b(t))+eta[u](x, t, u)*(u[x]+a(t))+(-(diff(xi[x](x, t, u), u))*u[x]^2-(diff(xi[t](x, t, u), u))*u[x]*u[t]+u[x]*(diff(eta[u](x, t, u), u))-(diff(xi[x](x, t, u), x))*u[x]-(diff(xi[t](x, t, u), x))*u[t]+diff(eta[u](x, t, u), x))*u-(diff(xi[x](x, t, u), u))*u[t]*u[x]-(diff(xi[t](x, t, u), u))*u[t]^2+(diff(eta[u](x, t, u), u))*u[t]-(diff(xi[x](x, t, u), t))*u[x]-(diff(xi[t](x, t, u), t))*u[t]+diff(eta[u](x, t, u), t)+(-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]*u[t]*u[x, x]-3*(diff(xi[t](x, t, u), u))*u[x, x]*u[x, t]-3*(diff(xi[t](x, t, u), u))*u[x]*u[x, x, t]-(diff(diff(diff(xi[t](x, t, u), u), u), u))*u[x]^3*u[t]-6*(diff(diff(xi[x](x, t, u), u), u))*u[x]^2*u[x, x]-3*(diff(diff(xi[t](x, t, u), u), u))*u[x]^2*u[x, t]+3*(diff(diff(eta[u](x, t, u), u), u))*u[x]*u[x, x]-3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*u[x]^2*u[t]-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*u[x]*u[t]-9*(diff(diff(xi[x](x, t, u), u), x))*u[x]*u[x, x]-6*(diff(diff(xi[t](x, t, u), u), x))*u[x]*u[x, t]-3*(diff(diff(xi[t](x, t, u), u), x))*u[t]*u[x, x]+diff(diff(diff(eta[u](x, t, u), x), x), x)-3*(diff(xi[x](x, t, u), u))*u[x, x]^2-3*(diff(xi[t](x, t, u), x))*u[x, x, t]-3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*u[x]^3-3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*u[x]^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*u[x]^2+3*u[x]*(diff(diff(diff(eta[u](x, t, u), u), x), x))-(diff(diff(diff(xi[x](x, t, u), x), x), x))*u[x]-(diff(diff(diff(xi[t](x, t, u), x), x), x))*u[t]+3*(diff(diff(eta[u](x, t, u), u), x))*u[x, x]-3*(diff(diff(xi[x](x, t, u), x), x))*u[x, x]-3*(diff(diff(xi[t](x, t, u), x), x))*u[x, t]-(diff(diff(diff(xi[x](x, t, u), u), u), u))*u[x]^4+(diff(diff(diff(eta[u](x, t, u), u), u), u))*u[x]^3+(diff(eta[u](x, t, u), u))*(-u*u[x]-u[t]-a(t)*u)/b(t)-3*(diff(xi[x](x, t, u), x))*(-u*u[x]-u[t]-a(t)*u)/b(t)-4*(diff(xi[x](x, t, u), u))*u[x]*(-u*u[x]-u[t]-a(t)*u)/b(t)-(diff(xi[t](x, t, u), u))*u[t]*(-u*u[x]-u[t]-a(t)*u)/b(t))*b(t) = 0)), du)}

{-(diff(diff(diff(xi[t](x, t, u), u), u), u))*b(t)^2, -3*(diff(diff(diff(xi[t](x, t, u), u), u), x))*b(t)^2, -(diff(diff(diff(xi[x](x, t, u), u), u), u))*b(t)^2, -3*(diff(diff(xi[t](x, t, u), u), u))*b(t)^2, -6*(diff(diff(xi[t](x, t, u), u), x))*b(t)^2, -3*(diff(diff(xi[t](x, t, u), u), x))*b(t)^2, -3*(diff(diff(xi[t](x, t, u), x), x))*b(t)^2, -6*(diff(diff(xi[x](x, t, u), u), u))*b(t)^2, -3*(diff(xi[t](x, t, u), u))*b(t)^2, -3*(diff(xi[t](x, t, u), x))*b(t)^2, -3*(diff(xi[x](x, t, u), u))*b(t)^2, -3*(diff(diff(diff(xi[x](x, t, u), u), u), x))*b(t)^2+(diff(diff(diff(eta[u](x, t, u), u), u), u))*b(t)^2, 3*(diff(diff(eta[u](x, t, u), u), u))*b(t)^2-9*(diff(diff(xi[x](x, t, u), u), x))*b(t)^2, 3*(diff(diff(eta[u](x, t, u), u), x))*b(t)^2-3*(diff(diff(xi[x](x, t, u), x), x))*b(t)^2, 3*(diff(xi[x](x, t, u), u))*b(t)-3*(diff(diff(diff(xi[t](x, t, u), u), x), x))*b(t)^2, -3*(diff(diff(diff(xi[x](x, t, u), u), x), x))*b(t)^2+3*(diff(diff(diff(eta[u](x, t, u), u), u), x))*b(t)^2+3*b(t)*(diff(xi[x](x, t, u), u))*u, -(diff(diff(diff(xi[t](x, t, u), x), x), x))*b(t)^2-(diff(b(t), t))*xi[t](x, t, u)-(diff(xi[t](x, t, u), t))*b(t)+3*(diff(xi[x](x, t, u), x))*b(t)+a(t)*u*(diff(xi[t](x, t, u), u))*b(t)-u*(diff(xi[t](x, t, u), x))*b(t), 3*(diff(diff(diff(eta[u](x, t, u), u), x), x))*b(t)^2-(diff(diff(diff(xi[x](x, t, u), x), x), x))*b(t)^2-b(t)*(diff(xi[x](x, t, u), t))+eta[u](x, t, u)*b(t)+4*b(t)*(diff(xi[x](x, t, u), u))*a(t)*u+2*b(t)*(diff(xi[x](x, t, u), x))*u-(diff(b(t), t))*xi[t](x, t, u)*u, (diff(eta[u](x, t, u), x))*b(t)*u+eta[u](x, t, u)*b(t)*a(t)+(diff(diff(diff(eta[u](x, t, u), x), x), x))*b(t)^2+b(t)*(diff(eta[u](x, t, u), t))-b(t)*(diff(eta[u](x, t, u), u))*a(t)*u+3*b(t)*(diff(xi[x](x, t, u), x))*a(t)*u+xi[t](x, t, u)*(diff(a(t), t))*b(t)*u-xi[t](x, t, u)*(diff(b(t), t))*a(t)*u}

 (16)

Now it is time to compare. The simplest way to do it is: "Reduce one system using the other and you should obtain 0 for all of its equations". If that is happens, then both systems have the same solution, regardless of whether one system has some equations further simplified removing constant factors etc.

 

To reduce systems you can use the ReducedForm command. Start with detecting the variables here - the unknown of these determining systems, these are the infinitesimals

> 

Vars := remove(has, indets(DetSys3, unknown), {a, b})

{eta[u](x, t, u), xi[t](x, t, u), xi[x](x, t, u)}

 (17)

Reduce now DetSys3, obtained going one step at a time using InfinitesimalGenerator, using DetSys obtained in one go using DeterminingPDE

> 

PDEtools:-ReducedForm(DetSys3, DetSys, Vars)

`casesplit/ans`([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [])

 (18)

That by itself tells that all the solutions of DetSys are also solutions of DetSys3. Let's see now whether the reversed statement is also true

> 

PDEtools:-ReducedForm(DetSys, DetSys3, Vars)

`casesplit/ans`([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [])

 (19)

Hence, all solutions of DetSys3 are also solutions of DetSys, from where these two systems just have the same solution, given by

> 

pdsolve(DetSys, Vars)

{eta[u](x, t, u) = exp(Int(-a(t), t))*_C2, xi[t](x, t, u) = 0, xi[x](x, t, u) = _C1+(Int(exp(Int(-a(t), t)), t))*_C2}

 (20)
> 

pdsolve(DetSys3, Vars)

{eta[u](x, t, u) = exp(Int(-a(t), t))*_C2, xi[t](x, t, u) = 0, xi[x](x, t, u) = _C1+(Int(exp(Int(-a(t), t)), t))*_C2}

 (21)

By the way you can get the solution also in one go, and with the integration constants already specialized

> 

PDEtools:-Infinitesimals(PDE1)

[_xi[x](x, t, u) = 1, _xi[t](x, t, u) = 0, _eta[u](x, t, u) = 0], [_xi[x](x, t, u) = Int(exp(Int(-a(t), t)), t), _xi[t](x, t, u) = 0, _eta[u](x, t, u) = exp(Int(-a(t), t))]

 (22)

Or you can use the method to reduce the independent variables of PDE1 as much as possible in one go, also displaying the intermediate solutions; you can also specify the dependency ofthe invariant solutions you are interested on, etc (give a look at the help page for this command, it has a large number of options).

> 

PDEtools:-InvariantSolutions(PDE1, display)

invariants = [t, (u(x, t)*(Int(exp(-(Int(a(t), t))), t))-exp(-(Int(a(t), t)))*x)/(Int(exp(-(Int(a(t), t))), t))]

  

solutions = [{u(x, t) = (_C1*exp(-(Int((a(t)*(Int(exp(-(Int(a(t), t))), t))+exp(-(Int(a(t), t))))/(Int(exp(-(Int(a(t), t))), t)), t)))*(Int(exp(-(Int(a(t), t))), t))+exp(-(Int(a(t), t)))*x)/(Int(exp(-(Int(a(t), t))), t))}]

  

_________________________________________________

  

invariants = [t, u(x, t)]

  

solutions = [{u(x, t) = exp(-(Int(a(t), t)))*_C2}]

  

_________________________________________________

  

u(x, t) = exp(-(Int(a(t), t)))*_C1, u(x, t) = (_C1*exp(-(Int((a(t)*(Int(exp(-(Int(a(t), t))), t))+exp(-(Int(a(t), t))))/(Int(exp(-(Int(a(t), t))), t)), t)))*(Int(exp(-(Int(a(t), t))), t))+exp(-(Int(a(t), t)))*x)/(Int(exp(-(Int(a(t), t))), t))

 (23)

So two invariant solutions can be derived from these infinitesimals. Let's test these solutions

> 

map(pdetest, [u(x, t) = exp(-(Int(a(t), t)))*_C1, u(x, t) = (_C1*exp(-(Int((a(t)*(Int(exp(-(Int(a(t), t))), t))+exp(-(Int(a(t), t))))/(Int(exp(-(Int(a(t), t))), t)), t)))*(Int(exp(-(Int(a(t), t))), t))+exp(-(Int(a(t), t)))*x)/(Int(exp(-(Int(a(t), t))), t))], PDE1)

[0, 0]

 (24)

I.e. the solutions are correct

 

Download [411]_Determing_Equations_for_Variable_coefficient_KdV_(reviewed).mw

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

mu and ~mu are different...

ecterrab 13431
September 17 2015
1 0

@escorpsy 

Just gave a look to your worksheet (BTW why do you ask your question with an exclamation mark `!`, or you intended `?` and the `!` is just a typo?).

The problem in your worksheet is that you use eval (ok...) but then in the input you pass to it you do not distinguish between mu (the covariant index) and~mu (the contravariant index). If you want to use eval, or for the case subs, to replace ~mu, then use, for instance following what I see in your worksheet, eval(expression, ~mu = ~2) and in doing so be sure that there is a blank space between = and ~2, because without that space, =~ has a different meaning (more like the zip command).

Alternatively, you may prefer to jump over these syntactic subtleties of eval, subs, etc. and just use the Physics command SubstituteTensorIndices, which will do the substitution exactly as you intended, that is, you input SubstituteTensorIndice(mu = 2, your_expression) and it will automatically replace mu = 2 or~mu = ~2. Note that SubstituteTensorIndices accepts, among others, the optional arguments evaluatetensor and evaluateexpression, so that you can use this command to work as you would expect from eval, or from subs (these two behave different, if you were not aware of that, please give a look to the corresponding help pages), including some extra flexibility that none of them have.

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

Use Multi-Index summation...

ecterrab 13431
September 11 2015
4 11

 

Hi shzan,

Besides Carl's suggestion of a loop, you can also enter your problem directly as you suggested, so as a double sum, where one of the summation index is j+k = m-1. For that purpose you need to redefinesum through the Physics  package (it just extends the syntax to what you want plus a few other enhancements). So,

> 

Physics:-Setup(redefinesum = true)

[redefinesum = true]

 (1)

Now go ahead as you intended, in one go, for instance,

> 

Sum(Sum(2^m*(D@@k)(f)*(D@@j)(f), j+k = m-1), m = 1 .. 5)

Sum(Sum(2^m*(D@@k)(f)*(D@@j)(f), j+k = m-1), m = 1 .. 5)

 (2)
> 

value(Sum(Sum(2^m*(D@@k)(f)*(D@@j)(f), j+k = m-1), m = 1 .. 5))

2*f^2+8*D(f)*f+16*(D@@2)(f)*f+8*D(f)^2+32*(D@@3)(f)*f+32*(D@@2)(f)*D(f)+64*(D@@4)(f)*f+64*(D@@3)(f)*D(f)+32*(D@@2)(f)^2

 (3)

You see this is the same result as you get using Carl's loop, or his suggested rewriting of your problem in terms of a single index j (possible in this example, but not always possible)

> 

add(add(2^m*(D@@j)(f)*(D@@(m-1-j))(f), j = 0 .. m-1), m = 1 .. 5);

2*f^2+8*D(f)*f+16*(D@@2)(f)*f+8*D(f)^2+32*(D@@3)(f)*f+32*(D@@2)(f)*D(f)+64*(D@@4)(f)*f+64*(D@@3)(f)*D(f)+32*(D@@2)(f)^2

 (4)

To know more about redefinesum, check the help page of what's new in Physics in Maple 18

 

Download Multi-Index_summation.mw

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

The results look correct to me....

ecterrab 13431
September 11 2015
2 2

Hi

The results you get in your worksheet look correct to me. I corrected anyway one or two things in your input lines, added comments, and verifications. I note that you also do not say that something is wrong anywhere, you just ask whether the results are correct. Anyway, I imagine you got in doubt due to a subtletly motivating this answer below: when you compute a covariant derivaive passing a tensor component as argument, if that tensor component gets evaluated into a scalar before arriving at D_, the command receives a scalar, so the output does not contain a Christoffel symbol. It is different when the tensor component e.g. 'N[1]' arrives at D_. The details are as follows (and I suggest you to update to the latest version of Physics in order to reproduce what you see next):


> 

restart
> 

Physics:-Version()[2]

`2015, September 11, 0:4 hours`

 (1)
> 

with(Physics):
> 

Physics:-Setup(math = true)

`* Partial match of 'math' against keyword 'mathematicalnotation'`

  

[mathematicalnotation = true]

 (2)
> 

g_[]

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

 (3)
> 

PDEtools:-declare(Physics:-g_[mu, nu] = Matrix(%id = 18446744078341407198), Phi(r), N(r))

A(r)*`will now be displayed as`*A

  

B(r)*`will now be displayed as`*B

  

C(r)*`will now be displayed as`*C

  

Phi(r)*`will now be displayed as`*Phi

  

N(r)*`will now be displayed as`*N

 (4)
> 

N[nu] = [-Physics:-`^`(A(r), Physics:-`^`(2, -1)), 0, 0, 0]

N[nu] = [-A(r)^(1/2), 0, 0, 0]

 (5)
> 

Physics:-Define(%)

`Defined objects with tensor properties`

  

{Physics:-D_[mu], Physics:-Dgamma[mu], N[nu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (6)

You have already defined N, no need to define it again (changes nothing)

> 

 

This definition of Lambda could be problematic if F contains mu because the index would appear repeated more than once - just have in mind this.

> 

Lambda := proc (F) options operator, arrow; Physics:-D_[mu](Physics:-D_[`~mu`](F)) end proc

proc (F) options operator, arrow; Physics:-D_[mu](Physics:-D_[`~mu`](F)) end proc

 (7)

Here there is a mistake: you defined N[nu] in a way that depends on (r). But then you add again that functionality, so depending on the operation you do you will have N[1](r) = (-A(r)^(1/2))(r) = (A(r)(r))^(1/2). I correct here that mistake

> 

Lambda(N[nu])

Physics:-D_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])

 (8)
> 

T[nu] = %

T[nu] = Physics:-D_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])

 (9)
> 

Physics:-Define(%)

`Defined objects with tensor properties`

  

{Physics:-D_[mu], Physics:-Dgamma[mu], N[nu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], T[nu], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (10)
> 

T[definition]

T[nu] = Physics:-D_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])

 (11)

In addition, for testing purposes I will define a H[mu,nu] tensor:

> 

H[mu, nu] = Physics:-D_[mu](N[nu])

H[mu, nu] = Physics:-D_[mu](N[nu], [X])

 (12)
> 

Physics:-Define(H[mu, nu] = Physics:-D_[mu](N[nu], [X]))

`Defined objects with tensor properties`

  

{Physics:-D_[mu], Physics:-Dgamma[mu], H[mu, nu], N[nu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], T[nu], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (13)
> 

Physics:-Setup(auto = true)

`* Partial match of 'auto' against keyword 'automaticsimplification'`

  

[automaticsimplification = true]

 (14)
> 

T[1]

(1/4)*(diff(A(r), r))^2/(B(r)*A(r)^(3/2))

 (15)
> 

NULL

 

So your first question is about correctness of this result (15) But, is there something that makes you think is wrong?

 

In any case what follows below is what I would do to verify this result for correctness, but first let's agree on the meaning of this expression:

 

`𝒟`[mu](N[1]) = eval(`𝒟`[mu](N[nu]), nu = 1)NULL

The relevant thing here is that the computation of the covariant derivative having a tensor component for argument will include a Christoffel term. So the operation is performed in two steps: first compute with nu, then replace nu by the number 1. Because Lambda = `𝒟`[mu]*`𝒟`[`~mu`], what I am saying is also that this is the meaning of the expression you are asking:

 

Lambda(N[1]) = eval(Lambda(N[nu]), nu = 1)

 

I will check now that the result (15) for T[1] is correct, through four different ways.

 

1) Expand the covariant derivatives in  T[nu] = `𝒟`[mu](`𝒟`[`~mu`](N[nu](r))) which is the same as rewriting it in terms of Christoffel symbols and noncovariant derivatives, then make nu = 1. This is the general approach, and the important thing is that you remove all covariant derivatives D_ before introducing any tensor component.

> 

expand(D_[mu](D_[`~mu`](N[nu], [X]), [X]))

Physics:-g_[`~beta`, `~mu`]*(Physics:-d_[mu](Physics:-d_[beta](N[nu], [X]), [X])-Physics:-Christoffel[`~sigma`, beta, mu]*Physics:-d_[sigma](N[nu], [X])-Physics:-Christoffel[`~sigma`, mu, nu]*Physics:-d_[beta](N[sigma], [X])-N[alpha]*Physics:-d_[mu](Physics:-Christoffel[`~alpha`, beta, nu], [X])+N[alpha]*Physics:-Christoffel[`~lambda`, beta, mu]*Physics:-Christoffel[`~alpha`, lambda, nu]+N[alpha]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, lambda]-N[alpha]*Physics:-Christoffel[`~alpha`, lambda, mu]*Physics:-Christoffel[`~lambda`, beta, nu]-Physics:-Christoffel[`~alpha`, beta, nu]*Physics:-d_[mu](N[alpha], [X])+Physics:-Christoffel[`~alpha`, beta, nu]*Physics:-Christoffel[`~kappa`, alpha, mu]*N[kappa])

 (16)
> 

ee := eval(Physics[g_][`~beta`, `~mu`]*(Physics[d_][mu](Physics[d_][beta](N[nu], [X]), [X])-Physics[Christoffel][`~sigma`, beta, mu]*Physics[d_][sigma](N[nu], [X])-Physics[Christoffel][`~sigma`, mu, nu]*Physics[d_][beta](N[sigma], [X])-N[alpha]*Physics[d_][mu](Physics[Christoffel][`~alpha`, beta, nu], [X])+N[alpha]*Physics[Christoffel][`~lambda`, beta, mu]*Physics[Christoffel][`~alpha`, lambda, nu]+N[alpha]*Physics[Christoffel][`~lambda`, mu, nu]*Physics[Christoffel][`~alpha`, beta, lambda]-N[alpha]*Physics[Christoffel][`~alpha`, lambda, mu]*Physics[Christoffel][`~lambda`, beta, nu]-Physics[Christoffel][`~alpha`, beta, nu]*Physics[d_][mu](N[alpha], [X])+Physics[Christoffel][`~alpha`, beta, nu]*Physics[Christoffel][`~kappa`, alpha, mu]*N[kappa]), nu = 1)

Physics:-g_[`~beta`, `~mu`]*(-(1/2)*Physics:-g_[beta, `~2`]*(-(1/2)*(diff(A(r), r))^2*Physics:-g_[mu, `~2`]/A(r)^(3/2)+(diff(diff(A(r), r), r))*Physics:-g_[mu, `~2`]/A(r)^(1/2))+(1/2)*Physics:-Christoffel[`~sigma`, beta, mu]*(diff(A(r), r))*Physics:-g_[sigma, `~2`]/A(r)^(1/2)-Physics:-Christoffel[`~sigma`, 1, mu]*Physics:-d_[beta](N[sigma], [X])-N[alpha]*Physics:-d_[mu](Physics:-Christoffel[`~alpha`, 1, beta], [X])+N[alpha]*Physics:-Christoffel[`~lambda`, beta, mu]*Physics:-Christoffel[`~alpha`, 1, lambda]+N[alpha]*Physics:-Christoffel[`~lambda`, 1, mu]*Physics:-Christoffel[`~alpha`, beta, lambda]-N[alpha]*Physics:-Christoffel[`~alpha`, lambda, mu]*Physics:-Christoffel[`~lambda`, 1, beta]-Physics:-Christoffel[`~alpha`, 1, beta]*Physics:-d_[mu](N[alpha], [X])+Physics:-Christoffel[`~alpha`, 1, beta]*Physics:-Christoffel[`~kappa`, alpha, mu]*N[kappa])

 (17)

Sum over the repeated indices

> 

SumOverRepeatedIndices(Physics[g_][`~beta`, `~mu`]*(-(1/2)*Physics[g_][beta, `~2`]*(-(1/2)*(diff(A(r), r))^2*Physics[g_][mu, `~2`]/A(r)^(3/2)+(diff(diff(A(r), r), r))*Physics[g_][mu, `~2`]/A(r)^(1/2))+(1/2)*Physics[Christoffel][`~sigma`, beta, mu]*(diff(A(r), r))*Physics[g_][sigma, `~2`]/A(r)^(1/2)-Physics[Christoffel][`~sigma`, 1, mu]*Physics[d_][beta](N[sigma], [X])-N[alpha]*Physics[d_][mu](Physics[Christoffel][`~alpha`, 1, beta], [X])+N[alpha]*Physics[Christoffel][`~lambda`, beta, mu]*Physics[Christoffel][`~alpha`, 1, lambda]+N[alpha]*Physics[Christoffel][`~lambda`, 1, mu]*Physics[Christoffel][`~alpha`, beta, lambda]-N[alpha]*Physics[Christoffel][`~alpha`, lambda, mu]*Physics[Christoffel][`~lambda`, 1, beta]-Physics[Christoffel][`~alpha`, 1, beta]*Physics[d_][mu](N[alpha], [X])+Physics[Christoffel][`~alpha`, 1, beta]*Physics[Christoffel][`~kappa`, alpha, mu]*N[kappa]))

(1/4)*(diff(A(r), r))^2/(B(r)*A(r)^(3/2))

 (18)

That is the same you got in one go in (15) for T[1]. By the way note that you can also simplify (16), then evaluate

> 

Physics:-Simplify(Physics:-g_[`~beta`, `~mu`]*(Physics:-d_[mu](Physics:-d_[beta](N[nu], [X]), [X])-Physics:-Christoffel[`~sigma`, beta, mu]*Physics:-d_[sigma](N[nu], [X])-Physics:-Christoffel[`~sigma`, mu, nu]*Physics:-d_[beta](N[sigma], [X])-N[alpha]*Physics:-d_[mu](Physics:-Christoffel[`~alpha`, beta, nu], [X])+N[alpha]*Physics:-Christoffel[`~lambda`, beta, mu]*Physics:-Christoffel[`~alpha`, lambda, nu]+N[alpha]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, lambda]-N[alpha]*Physics:-Christoffel[`~alpha`, lambda, mu]*Physics:-Christoffel[`~lambda`, beta, nu]-Physics:-Christoffel[`~alpha`, beta, nu]*Physics:-d_[mu](N[alpha], [X])+Physics:-Christoffel[`~alpha`, beta, nu]*Physics:-Christoffel[`~kappa`, alpha, mu]*N[kappa]))

N[alpha]*Physics:-Christoffel[`~alpha`, beta, `~kappa`]*Physics:-Christoffel[`~beta`, kappa, nu]+N[alpha]*Physics:-Christoffel[`~alpha`, kappa, nu]*Physics:-Christoffel[`~kappa`, mu, `~mu`]-N[alpha]*Physics:-d_[`~beta`](Physics:-Christoffel[`~alpha`, beta, nu], [X])-2*Physics:-Christoffel[`~beta`, kappa, nu]*Physics:-d_[`~kappa`](N[beta], [X])-Physics:-Christoffel[sigma, mu, `~mu`]*Physics:-d_[`~sigma`](N[nu], [X])+Physics:-dAlembertian(N[nu], [X])

 (19)
> 

Physics:-SumOverRepeatedIndices(eval(N[alpha]*Physics:-Christoffel[`~alpha`, beta, `~kappa`]*Physics:-Christoffel[`~beta`, kappa, nu]+N[alpha]*Physics:-Christoffel[`~alpha`, kappa, nu]*Physics:-Christoffel[`~kappa`, mu, `~mu`]-N[alpha]*Physics:-d_[`~beta`](Physics:-Christoffel[`~alpha`, beta, nu], [X])-2*Physics:-Christoffel[`~beta`, kappa, nu]*Physics:-d_[`~kappa`](N[beta], [X])-Physics:-Christoffel[sigma, mu, `~mu`]*Physics:-d_[`~sigma`](N[nu], [X])+Physics:-dAlembertian(N[nu], [X]), nu = 1))

(1/4)*(diff(A(r), r))^2/(B(r)*A(r)^(3/2))

 (20)

 

2) A different path: sum first, then make nu = 1

> 

T[definition]

T[nu] = Physics:-D_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])

 (21)
> 

Physics:-SumOverRepeatedIndices(T[definition])

T[nu] = -(diff(B(r), r))*(Physics:-diff(N[nu], r)+Physics:-Christoffel[`~1`, 2, nu]*A(r)^(1/2))/B(r)^2+(Physics:-diff(Physics:-diff(N[nu], r), r)+Physics:-diff(Physics:-Christoffel[`~1`, 2, nu], r)*A(r)^(1/2)+(1/2)*Physics:-Christoffel[`~1`, 2, nu]*(diff(A(r), r))/A(r)^(1/2))/B(r)+Physics:-diff(Physics:-Christoffel[`~1`, 3, nu], theta)*A(r)^(1/2)/(C(r)*r^2)+(1/2)*Physics:-Christoffel[`~2`, 1, nu]*(diff(A(r), r))/A(r)^(3/2)+((1/2)*(diff(A(r), r))/A(r)+(1/2)*(diff(B(r), r))/B(r)+((diff(C(r), r))*r+2*C(r))/(r*C(r)))*(Physics:-diff(N[nu], r)+Physics:-Christoffel[`~1`, 2, nu]*A(r)^(1/2))/B(r)+cos(theta)*Physics:-Christoffel[`~1`, 3, nu]*A(r)^(1/2)/(sin(theta)*C(r)*r^2)

 (22)
> 

eval(T[nu] = -(diff(B(r), r))*(Physics:-diff(N[nu], r)+Physics:-Christoffel[`~1`, 2, nu]*A(r)^(1/2))/B(r)^2+(Physics:-diff(Physics:-diff(N[nu], r), r)+Physics:-diff(Physics:-Christoffel[`~1`, 2, nu], r)*A(r)^(1/2)+(1/2)*Physics:-Christoffel[`~1`, 2, nu]*(diff(A(r), r))/A(r)^(1/2))/B(r)+Physics:-diff(Physics:-Christoffel[`~1`, 3, nu], theta)*A(r)^(1/2)/(C(r)*r^2)+(1/2)*Physics:-Christoffel[`~2`, 1, nu]*(diff(A(r), r))/A(r)^(3/2)+((1/2)*(diff(A(r), r))/A(r)+(1/2)*(diff(B(r), r))/B(r)+((diff(C(r), r))*r+2*C(r))/(r*C(r)))*(Physics:-diff(N[nu], r)+Physics:-Christoffel[`~1`, 2, nu]*A(r)^(1/2))/B(r)+cos(theta)*Physics:-Christoffel[`~1`, 3, nu]*A(r)^(1/2)/(sin(theta)*C(r)*r^2), nu = 1)

(1/4)*(diff(A(r), r))^2/(B(r)*A(r)^(3/2)) = (1/4)*(diff(A(r), r))^2/(B(r)*A(r)^(3/2))

 (23)

3) This is a third way, one step at a time so that you can verify the results by eye. For that purposes use the first derivative represented by H

> 

Physics:-D_[mu](H[mu, nu])

Physics:-D_[mu](H[`~mu`, nu], [X])

 (24)
> 

H[definition]

H[mu, nu] = Physics:-D_[mu](N[nu], [X])

 (25)
> 

Physics:-D_[mu](H[`~mu`, nu], [X]) = Physics:-SubstituteTensor(H[definition], Physics:-D_[mu](H[`~mu`, nu], [X]))

Physics:-D_[mu](H[`~mu`, nu], [X]) = Physics:-D_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])

 (26)

So work with (24), start expanding

> 

expand(Physics:-D_[mu](H[`~mu`, nu], [X]))

Physics:-d_[mu](H[`~mu`, nu], [X])-Physics:-Christoffel[`~alpha`, mu, nu]*H[`~mu`, alpha]+Physics:-Christoffel[`~mu`, alpha, mu]*H[`~alpha`, nu]

 (27)

By eye this result is correct. Next:

> 

Physics:-SubstituteTensor(H[definition], Physics:-d_[mu](H[`~mu`, nu], [X])-Physics:-Christoffel[`~alpha`, mu, nu]*H[`~mu`, alpha]+Physics:-Christoffel[`~mu`, alpha, mu]*H[`~alpha`, nu])

Physics:-d_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])-Physics:-Christoffel[`~alpha`, mu, nu]*Physics:-D_[`~mu`](N[alpha], [X])+Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-D_[`~alpha`](N[nu], [X])

 (28)
> 

expand(Physics:-d_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])-Physics:-Christoffel[`~alpha`, mu, nu]*Physics:-D_[`~mu`](N[alpha], [X])+Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-D_[`~alpha`](N[nu], [X]))

Physics:-d_[mu](Physics:-g_[`~kappa`, `~mu`]*(Physics:-d_[kappa](N[nu], [X])-Physics:-Christoffel[`~beta`, kappa, nu]*N[beta]), [X])-Physics:-Christoffel[`~alpha`, mu, nu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-d_[upsilon](N[alpha], [X])+Physics:-Christoffel[`~alpha`, mu, nu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-Christoffel[`~tau`, alpha, upsilon]*N[tau]+Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*Physics:-d_[sigma](N[nu], [X])-Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*Physics:-Christoffel[`~lambda`, nu, sigma]*N[lambda]

 (29)

This result above is also correct by eye, you can check all the signs in the Christoffel symbols. Next:

> 

eval(Physics:-d_[mu](Physics:-g_[`~kappa`, `~mu`]*(Physics:-d_[kappa](N[nu], [X])-Physics:-Christoffel[`~beta`, kappa, nu]*N[beta]), [X])-Physics:-Christoffel[`~alpha`, mu, nu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-d_[upsilon](N[alpha], [X])+Physics:-Christoffel[`~alpha`, mu, nu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-Christoffel[`~tau`, alpha, upsilon]*N[tau]+Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*Physics:-d_[sigma](N[nu], [X])-Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*Physics:-Christoffel[`~lambda`, nu, sigma]*N[lambda], nu = 1)

-Physics:-d_[mu](Physics:-g_[`~kappa`, `~mu`], [X])*((1/2)*(diff(A(r), r))*Physics:-g_[kappa, `~2`]/A(r)^(1/2)+Physics:-Christoffel[`~beta`, 1, kappa]*N[beta])-(-(1/4)*Physics:-g_[kappa, `~2`]*(diff(A(r), r))^2*Physics:-g_[mu, `~2`]/A(r)^(3/2)+(1/2)*Physics:-g_[kappa, `~2`]*(diff(diff(A(r), r), r))*Physics:-g_[mu, `~2`]/A(r)^(1/2)+Physics:-d_[mu](Physics:-Christoffel[`~beta`, 1, kappa], [X])*N[beta]+Physics:-Christoffel[`~beta`, 1, kappa]*Physics:-d_[mu](N[beta], [X]))*Physics:-g_[`~kappa`, `~mu`]-Physics:-Christoffel[`~alpha`, 1, mu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-d_[upsilon](N[alpha], [X])+Physics:-Christoffel[`~alpha`, 1, mu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-Christoffel[`~tau`, alpha, upsilon]*N[tau]-(1/2)*Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*(diff(A(r), r))*Physics:-g_[sigma, `~2`]/A(r)^(1/2)-Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*Physics:-Christoffel[`~lambda`, 1, sigma]*N[lambda]

 (30)
> 

Physics:-SumOverRepeatedIndices(-Physics:-d_[mu](Physics:-g_[`~kappa`, `~mu`], [X])*((1/2)*(diff(A(r), r))*Physics:-g_[kappa, `~2`]/A(r)^(1/2)+Physics:-Christoffel[`~beta`, 1, kappa]*N[beta])-(-(1/4)*Physics:-g_[kappa, `~2`]*(diff(A(r), r))^2*Physics:-g_[mu, `~2`]/A(r)^(3/2)+(1/2)*Physics:-g_[kappa, `~2`]*(diff(diff(A(r), r), r))*Physics:-g_[mu, `~2`]/A(r)^(1/2)+Physics:-d_[mu](Physics:-Christoffel[`~beta`, 1, kappa], [X])*N[beta]+Physics:-Christoffel[`~beta`, 1, kappa]*Physics:-d_[mu](N[beta], [X]))*Physics:-g_[`~kappa`, `~mu`]-Physics:-Christoffel[`~alpha`, 1, mu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-d_[upsilon](N[alpha], [X])+Physics:-Christoffel[`~alpha`, 1, mu]*Physics:-g_[`~mu`, `~upsilon`]*Physics:-Christoffel[`~tau`, alpha, upsilon]*N[tau]-(1/2)*Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*(diff(A(r), r))*Physics:-g_[sigma, `~2`]/A(r)^(1/2)-Physics:-Christoffel[`~mu`, alpha, mu]*Physics:-g_[`~alpha`, `~sigma`]*Physics:-Christoffel[`~lambda`, 1, sigma]*N[lambda])

(1/4)*(diff(A(r), r))^2/(B(r)*A(r)^(3/2))

 (31)

Again the same result.

 

4) Try computing directly a TensorArray for this definition

> 

T[definition]

T[nu] = Physics:-D_[mu](Physics:-D_[`~mu`](N[nu], [X]), [X])

 (32)
> 

TensorArray(T[nu] = D_[mu](D_[`~mu`](N[nu], [X]), [X]))

Array(%id = 18446744078362242094)

 (33)

So we have arrived to the same result going through four different paths.  Is there something concrete making you think this result would be incorrect?

 

Next you ask

> 

NULL

 

To answer this other question, again, it is key here that the covariant derivative operator knows whether its argument is or not a tensor. So Let's first agree on the meaning:

 

Lambda(Phi(r)*N[1]*N[1]) = eval(Lambda(Phi(r)*N[alpha]*N[beta]), {alpha = 1, beta = 1})

Then start with

> 

Lambda(Physics:-`*`(Physics:-`*`(Phi(r), N[alpha]), N[beta]))

Physics:-g_[`~kappa`, `~mu`]*(Phi(r)*Physics:-D_[mu](Physics:-d_[kappa](N[alpha], [X]), [X])*N[beta]+Phi(r)*N[alpha]*Physics:-D_[mu](Physics:-d_[kappa](N[beta], [X]), [X])+(Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[beta]-Physics:-Christoffel[`~nu`, beta, kappa]*N[nu]*Phi(r)+Phi(r)*Physics:-d_[kappa](N[beta], [X]))*Physics:-D_[mu](N[alpha], [X])+(Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[alpha]-Physics:-Christoffel[`~nu`, alpha, kappa]*N[nu]*Phi(r)+Phi(r)*Physics:-d_[kappa](N[alpha], [X]))*Physics:-D_[mu](N[beta], [X])+(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*Physics:-d_[kappa](N[alpha], [X])*N[beta]+(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*N[alpha]*Physics:-d_[kappa](N[beta], [X])-Phi(r)*(N[alpha]*Physics:-Christoffel[`~nu`, beta, kappa]+N[beta]*Physics:-Christoffel[`~nu`, alpha, kappa])*Physics:-D_[mu](N[nu], [X])-Physics:-D_[mu](Physics:-Christoffel[`~nu`, alpha, kappa], [X])*Phi(r)*N[nu]*N[beta]-Physics:-D_[mu](Physics:-Christoffel[`~nu`, beta, kappa], [X])*Phi(r)*N[alpha]*N[nu]+Physics:-g_[mu, `~2`]*(Physics:-g_[kappa, `~2`]*(diff(diff(Phi(r), r), r))*N[beta]*N[alpha]-(diff(Phi(r), r))*N[nu]*(N[alpha]*Physics:-Christoffel[`~nu`, beta, kappa]+N[beta]*Physics:-Christoffel[`~nu`, alpha, kappa])))

 (34)
> 

expand(Physics:-g_[`~kappa`, `~mu`]*(Phi(r)*Physics:-D_[mu](Physics:-d_[kappa](N[alpha], [X]), [X])*N[beta]+Phi(r)*N[alpha]*Physics:-D_[mu](Physics:-d_[kappa](N[beta], [X]), [X])+(Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[beta]-Physics:-Christoffel[`~nu`, beta, kappa]*N[nu]*Phi(r)+Phi(r)*Physics:-d_[kappa](N[beta], [X]))*Physics:-D_[mu](N[alpha], [X])+(Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[alpha]-Physics:-Christoffel[`~nu`, alpha, kappa]*N[nu]*Phi(r)+Phi(r)*Physics:-d_[kappa](N[alpha], [X]))*Physics:-D_[mu](N[beta], [X])+(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*Physics:-d_[kappa](N[alpha], [X])*N[beta]+(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*N[alpha]*Physics:-d_[kappa](N[beta], [X])-Phi(r)*(N[alpha]*Physics:-Christoffel[`~nu`, beta, kappa]+N[beta]*Physics:-Christoffel[`~nu`, alpha, kappa])*Physics:-D_[mu](N[nu], [X])-Physics:-D_[mu](Physics:-Christoffel[`~nu`, alpha, kappa], [X])*Phi(r)*N[nu]*N[beta]-Physics:-D_[mu](Physics:-Christoffel[`~nu`, beta, kappa], [X])*Phi(r)*N[alpha]*N[nu]+Physics:-g_[mu, `~2`]*(Physics:-g_[kappa, `~2`]*(diff(diff(Phi(r), r), r))*N[beta]*N[alpha]-(diff(Phi(r), r))*N[nu]*(N[alpha]*Physics:-Christoffel[`~nu`, beta, kappa]+N[beta]*Physics:-Christoffel[`~nu`, alpha, kappa]))))

-(-Phi(r)*N[beta]*Physics:-d_[mu](Physics:-d_[kappa](N[alpha], [X]), [X])-Phi(r)*N[alpha]*Physics:-d_[mu](Physics:-d_[kappa](N[beta], [X]), [X])+(Phi(r)*Physics:-Christoffel[`~sigma`, beta, mu]*N[sigma]-(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*N[beta]-Phi(r)*Physics:-d_[mu](N[beta], [X]))*Physics:-d_[kappa](N[alpha], [X])+(-(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*N[alpha]+Phi(r)*Physics:-Christoffel[`~tau`, alpha, mu]*N[tau]-Phi(r)*Physics:-d_[mu](N[alpha], [X]))*Physics:-d_[kappa](N[beta], [X])+(-Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[beta]+Physics:-Christoffel[`~nu`, beta, kappa]*N[nu]*Phi(r))*Physics:-d_[mu](N[alpha], [X])+(-Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[alpha]+Physics:-Christoffel[`~nu`, alpha, kappa]*N[nu]*Phi(r))*Physics:-d_[mu](N[beta], [X])+Phi(r)*(N[alpha]*Physics:-Christoffel[`~nu`, beta, kappa]+N[beta]*Physics:-Christoffel[`~nu`, alpha, kappa])*Physics:-d_[mu](N[nu], [X])+Phi(r)*N[nu]*N[beta]*Physics:-d_[mu](Physics:-Christoffel[`~nu`, alpha, kappa], [X])+Phi(r)*N[alpha]*N[nu]*Physics:-d_[mu](Physics:-Christoffel[`~nu`, beta, kappa], [X])+Phi(r)*N[beta]*Physics:-Christoffel[`~omega`, alpha, mu]*Physics:-d_[kappa](N[omega], [X])+Phi(r)*N[beta]*Physics:-Christoffel[`~omega`, kappa, mu]*Physics:-d_[omega](N[alpha], [X])+Phi(r)*N[alpha]*Physics:-Christoffel[`~psi`, beta, mu]*Physics:-d_[kappa](N[psi], [X])+Phi(r)*N[alpha]*Physics:-Christoffel[`~psi`, kappa, mu]*Physics:-d_[psi](N[beta], [X])-(diff(diff(Phi(r), r), r))*Physics:-g_[mu, `~2`]*Physics:-g_[kappa, `~2`]*N[alpha]*N[beta]+((N[nu]*Physics:-g_[mu, `~2`]*Physics:-Christoffel[`~nu`, beta, kappa]+N[sigma]*Physics:-g_[kappa, `~2`]*Physics:-Christoffel[`~sigma`, beta, mu])*N[alpha]+N[beta]*(N[nu]*Physics:-g_[mu, `~2`]*Physics:-Christoffel[`~nu`, alpha, kappa]+N[tau]*Physics:-g_[kappa, `~2`]*Physics:-Christoffel[`~tau`, alpha, mu]))*(diff(Phi(r), r))-Phi(r)*(((Physics:-Christoffel[`~nu`, beta, upsilon]*Physics:-Christoffel[`~upsilon`, kappa, mu]+Physics:-Christoffel[`~nu`, kappa, upsilon]*Physics:-Christoffel[`~upsilon`, beta, mu]-Physics:-Christoffel[`~nu`, mu, upsilon]*Physics:-Christoffel[`~upsilon`, beta, kappa])*N[nu]+N[lambda]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~nu`, beta, kappa])*N[alpha]+((-Physics:-Christoffel[`~chi`, alpha, kappa]*Physics:-Christoffel[`~nu`, chi, mu]+Physics:-Christoffel[`~chi`, alpha, mu]*Physics:-Christoffel[`~nu`, chi, kappa]+Physics:-Christoffel[`~chi`, kappa, mu]*Physics:-Christoffel[`~nu`, alpha, chi])*N[nu]+N[lambda]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~nu`, alpha, kappa])*N[beta]+N[nu]*(N[sigma]*Physics:-Christoffel[`~nu`, alpha, kappa]*Physics:-Christoffel[`~sigma`, beta, mu]+N[tau]*Physics:-Christoffel[`~nu`, beta, kappa]*Physics:-Christoffel[`~tau`, alpha, mu])))*Physics:-g_[`~kappa`, `~mu`]

 (35)
> 

eval(-(-Phi(r)*N[beta]*Physics:-d_[mu](Physics:-d_[kappa](N[alpha], [X]), [X])-Phi(r)*N[alpha]*Physics:-d_[mu](Physics:-d_[kappa](N[beta], [X]), [X])+(Phi(r)*Physics:-Christoffel[`~sigma`, beta, mu]*N[sigma]-(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*N[beta]-Phi(r)*Physics:-d_[mu](N[beta], [X]))*Physics:-d_[kappa](N[alpha], [X])+(-(diff(Phi(r), r))*Physics:-g_[mu, `~2`]*N[alpha]+Phi(r)*Physics:-Christoffel[`~tau`, alpha, mu]*N[tau]-Phi(r)*Physics:-d_[mu](N[alpha], [X]))*Physics:-d_[kappa](N[beta], [X])+(-Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[beta]+Physics:-Christoffel[`~nu`, beta, kappa]*N[nu]*Phi(r))*Physics:-d_[mu](N[alpha], [X])+(-Physics:-g_[kappa, `~2`]*(diff(Phi(r), r))*N[alpha]+Physics:-Christoffel[`~nu`, alpha, kappa]*N[nu]*Phi(r))*Physics:-d_[mu](N[beta], [X])+Phi(r)*(N[alpha]*Physics:-Christoffel[`~nu`, beta, kappa]+N[beta]*Physics:-Christoffel[`~nu`, alpha, kappa])*Physics:-d_[mu](N[nu], [X])+Phi(r)*N[nu]*N[beta]*Physics:-d_[mu](Physics:-Christoffel[`~nu`, alpha, kappa], [X])+Phi(r)*N[alpha]*N[nu]*Physics:-d_[mu](Physics:-Christoffel[`~nu`, beta, kappa], [X])+Phi(r)*N[beta]*Physics:-Christoffel[`~omega`, alpha, mu]*Physics:-d_[kappa](N[omega], [X])+Phi(r)*N[beta]*Physics:-Christoffel[`~omega`, kappa, mu]*Physics:-d_[omega](N[alpha], [X])+Phi(r)*N[alpha]*Physics:-Christoffel[`~psi`, beta, mu]*Physics:-d_[kappa](N[psi], [X])+Phi(r)*N[alpha]*Physics:-Christoffel[`~psi`, kappa, mu]*Physics:-d_[psi](N[beta], [X])-(diff(diff(Phi(r), r), r))*Physics:-g_[mu, `~2`]*Physics:-g_[kappa, `~2`]*N[alpha]*N[beta]+((N[nu]*Physics:-g_[mu, `~2`]*Physics:-Christoffel[`~nu`, beta, kappa]+N[sigma]*Physics:-g_[kappa, `~2`]*Physics:-Christoffel[`~sigma`, beta, mu])*N[alpha]+N[beta]*(N[nu]*Physics:-g_[mu, `~2`]*Physics:-Christoffel[`~nu`, alpha, kappa]+N[tau]*Physics:-g_[kappa, `~2`]*Physics:-Christoffel[`~tau`, alpha, mu]))*(diff(Phi(r), r))-Phi(r)*(((Physics:-Christoffel[`~nu`, beta, upsilon]*Physics:-Christoffel[`~upsilon`, kappa, mu]+Physics:-Christoffel[`~nu`, kappa, upsilon]*Physics:-Christoffel[`~upsilon`, beta, mu]-Physics:-Christoffel[`~nu`, mu, upsilon]*Physics:-Christoffel[`~upsilon`, beta, kappa])*N[nu]+N[lambda]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~nu`, beta, kappa])*N[alpha]+((-Physics:-Christoffel[`~chi`, alpha, kappa]*Physics:-Christoffel[`~nu`, chi, mu]+Physics:-Christoffel[`~chi`, alpha, mu]*Physics:-Christoffel[`~nu`, chi, kappa]+Physics:-Christoffel[`~chi`, kappa, mu]*Physics:-Christoffel[`~nu`, alpha, chi])*N[nu]+N[lambda]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~nu`, alpha, kappa])*N[beta]+N[nu]*(N[sigma]*Physics:-Christoffel[`~nu`, alpha, kappa]*Physics:-Christoffel[`~sigma`, beta, mu]+N[tau]*Physics:-Christoffel[`~nu`, beta, kappa]*Physics:-Christoffel[`~tau`, alpha, mu])))*Physics:-g_[`~kappa`, `~mu`], [alpha = 1, beta = 1])

Physics:-g_[`~kappa`, `~mu`]*(2*Phi(r)*N[nu]*A(r)*Physics:-d_[mu](Physics:-Christoffel[`~nu`, 1, kappa], [X])+Phi(r)*A(r)*Physics:-Christoffel[`~omega`, 1, mu]*Physics:-d_[kappa](N[omega], [X])+Phi(r)*A(r)*Physics:-Christoffel[`~psi`, 1, mu]*Physics:-d_[kappa](N[psi], [X])+2*Phi(r)*A(r)*Physics:-Christoffel[`~nu`, 1, kappa]*Physics:-d_[mu](N[nu], [X])+(diff(diff(Phi(r), r), r))*Physics:-g_[mu, `~2`]*Physics:-g_[kappa, `~2`]*A(r)^(3/2)+Phi(r)*(diff(diff(A(r), r), r))*A(r)^(1/2)*Physics:-g_[kappa, `~2`]*Physics:-g_[mu, `~2`]+((2*(diff(Phi(r), r))*Physics:-g_[kappa, `~2`]*Physics:-g_[mu, `~2`]-(1/2)*Phi(r)*(Physics:-g_[omega, `~2`]*Physics:-Christoffel[`~omega`, kappa, mu]+Physics:-g_[psi, `~2`]*Physics:-Christoffel[`~psi`, kappa, mu]))*(diff(A(r), r))+Physics:-Christoffel[`~nu`, 1, kappa]*N[nu]*Phi(r)*(N[sigma]*Physics:-Christoffel[`~sigma`, 1, mu]+N[tau]*Physics:-Christoffel[`~tau`, 1, mu]))*A(r)^(1/2)+(1/2)*(2*N[nu]*Physics:-g_[mu, `~2`]*Physics:-Christoffel[`~nu`, 1, kappa]+Physics:-g_[kappa, `~2`]*(N[sigma]*Physics:-Christoffel[`~sigma`, 1, mu]+N[tau]*Physics:-Christoffel[`~tau`, 1, mu]))*Phi(r)*(diff(A(r), r))+A(r)*((2*N[nu]*Physics:-g_[mu, `~2`]*Physics:-Christoffel[`~nu`, 1, kappa]+Physics:-g_[kappa, `~2`]*(N[sigma]*Physics:-Christoffel[`~sigma`, 1, mu]+N[tau]*Physics:-Christoffel[`~tau`, 1, mu]))*(diff(Phi(r), r))-2*(((1/2)*Physics:-Christoffel[`~chi`, 1, mu]*Physics:-Christoffel[`~nu`, chi, kappa]+(1/2)*Physics:-Christoffel[`~chi`, kappa, mu]*Physics:-Christoffel[`~nu`, 1, chi]-(1/2)*Physics:-Christoffel[`~chi`, 1, kappa]*Physics:-Christoffel[`~nu`, chi, mu]+(1/2)*Physics:-Christoffel[`~nu`, 1, upsilon]*Physics:-Christoffel[`~upsilon`, kappa, mu]+(1/2)*Physics:-Christoffel[`~nu`, kappa, upsilon]*Physics:-Christoffel[`~upsilon`, 1, mu]-(1/2)*Physics:-Christoffel[`~nu`, mu, upsilon]*Physics:-Christoffel[`~upsilon`, 1, kappa])*N[nu]+N[lambda]*Physics:-Christoffel[`~lambda`, mu, nu]*Physics:-Christoffel[`~nu`, 1, kappa])*Phi(r)))/A(r)^(1/2)

 (36)
> 

Physics:-SumOverRepeatedIndices(%)

(1/2)*(-Phi(r)*(diff(A(r), r))^2+2*A(r)^2*(diff(diff(Phi(r), r), r)))/(B(r)*A(r))

 (37)

Now, you will not get the same result if you enter

> 

Lambda(Phi(r)*N[1]*N[1])

(B(r)*((diff(A(r), r))*Phi(r)+A(r)*(diff(Phi(r), r)))*Physics:-d_[mu](Physics:-g_[`~2`, `~mu`], [X])+(diff(diff(Phi(r), r), r))*A(r)+(diff(diff(A(r), r), r))*Phi(r)+(Physics:-Christoffel[`~mu`, mu, nu]*A(r)*B(r)*Physics:-g_[`~2`, `~nu`]+2*(diff(A(r), r)))*(diff(Phi(r), r))+Physics:-Christoffel[`~mu`, mu, nu]*Phi(r)*(diff(A(r), r))*Physics:-g_[`~2`, `~nu`]*B(r))/B(r)

 (38)
> 

Physics:-SumOverRepeatedIndices((B(r)*((diff(A(r), r))*Phi(r)+A(r)*(diff(Phi(r), r)))*Physics:-d_[mu](Physics:-g_[`~2`, `~mu`], [X])+(diff(diff(Phi(r), r), r))*A(r)+(diff(diff(A(r), r), r))*Phi(r)+(Physics:-Christoffel[`~mu`, mu, nu]*A(r)*B(r)*Physics:-g_[`~2`, `~nu`]+2*(diff(A(r), r)))*(diff(Phi(r), r))+Physics:-Christoffel[`~mu`, mu, nu]*Phi(r)*(diff(A(r), r))*Physics:-g_[`~2`, `~nu`]*B(r))/B(r))

(1/2)*(2*(diff(diff(Phi(r), r), r))*A(r)^2*B(r)*r*C(r)+2*(diff(diff(A(r), r), r))*Phi(r)*A(r)*B(r)*r*C(r)+(diff(A(r), r))^2*C(r)*B(r)*Phi(r)*r+5*A(r)*(r*B(r)*C(r)*(diff(Phi(r), r))+(2/5)*(-(1/2)*r*C(r)*(diff(B(r), r))+B(r)*((diff(C(r), r))*r+2*C(r)))*Phi(r))*(diff(A(r), r))+2*A(r)^2*(-(1/2)*r*C(r)*(diff(B(r), r))+B(r)*((diff(C(r), r))*r+2*C(r)))*(diff(Phi(r), r)))/(A(r)*B(r)^2*r*C(r))

 (39)

And why is that? Because when you input Lambda(Phi(r)*N[1]*N[1]), N[1] gets evaluated into a scalar function before D_[mu] could perceive it is a tensor:

> 

N[1]

-A(r)^(1/2)

 (40)

You can check this with a simpler example:

> 

Upsilon := proc (F) options operator, arrow; D_[mu](F) end proc

proc (F) options operator, arrow; Physics:-D_[mu](F) end proc

 (41)
> 

debug(D_)

Physics:-D_

 (42)
> 

Upsilon(N[1])

{--> enter Physics:-D_[mu], args = -A(r)^(1/2)

<-- exit Physics:-D_[mu] (now in Upsilon) = -(1/2)*(diff(A(r), r))*Physics:-g_[mu, ~2]/A(r)^(1/2)}

  

You see: a scalar function entered D_, and so the result has no Christoffel symbols. N[1] gets evaluated into a scalar function before D_[mu] can touch it. Compare with passing N[1] to D_[mu]

> 

D_[mu](N[1])

{--> enter Physics:-D_[mu], args = N[1]

<-- exit Physics:-D_[mu] (now at top level) = -(1/2)*(diff(A(r), r))*Physics:-g_[mu, ~2]/A(r)^(1/2)-Physics:-Christoffel[~alpha, 1, mu]*N[alpha]}

  

Now you see N[1] arrives at D_[mu], allowing for perceiving that it is the first component of a tensor, so that there is a Christoffel symbol in the output, and in this way we have the correct meaning, as said lines above, repeating here:

 

 `&Dscr;`[mu](N[1]) = eval(`&Dscr;`[mu](N[nu]), nu = 1)

Summarizing: all the results you get look correct to me, and you need to remember, when passing tensor components to procedures that then call D_: those procedures (e.g. your Lambda) will evaluate the tensor component (e.g you N[1]) before forwarding it to D_, so that D_ will receive a scalar instead of the component of a tensor and you will miss the Christoffel symbol.


Download tensor_correct_(reviewed).mw

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

simplify(expression, size)...

ecterrab 13431
September 10 2015
4 2

Hi Rouben

Try please simplify(your_expression, size) and you get what you want.

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

PDEtools:-dcoeffs...

ecterrab 13431
August 31 2015
3 0

Check the help page of PDEtools:-dcoeffs, it does what you are looking for, just enter 

PDEtools:-dcoeffs(PDE, u(x,y)).

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

Ordering of free indices - improvement...

ecterrab 13431
August 19 2015
3 0


Hi

The issue: in a tensorial expression, there is no way to tell which free index comes first. The TransformCoordinates command was using Maple's sort command, and sort([~a, b]) returns [b, ~a]. Therfore, the result you obtained coincides with the transpose of what you expected. This doesn't mean you are wrong in what you were expecting but rather that TransformCoordiantes was in need of a new optional argument, to indicate the ordering of the freeindices, as in freeindices = [~a, b]. It is also true that when you are transforming a tensor, not a tensorial expression, one can assume that the ordering of the indices is exactly the one found in the tensor, so for a case like the one you posted, it should not  be necessary to indicate the ordering of the free indices: transforming A[~a, b] should automatically take [~a, b] as the ordering of the free indices.

 

All this is now implemented, see my comments below, italized, with input in 2D format to distinguish from yours that is in 1D format. As usual, the improvement is available for download at the Maplesoft R&D Physics webpage.

 

> 

restart
> 

with(Physics):
> 

Setup(mathematicalnotation = true):
> 

ds := dx[1]^2+dx[2]^2+dx[3]^2:
> 

Setup(coordinates = (X = [x[1], x[2], x[3]]), dimension = 3, metric = ds, spacetimeindices = lowercaselatin, quiet):
> 

g_[]

g[a, b] = (Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 2) = 1, (2, 3) = 0, (3, 3) = 1}, storage = triangular[upper], shape = [symmetric]))

 (1)
> 

A[a, b] = Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1})

A[a, b] = (Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1}))

 (2)
> 

Define(%):

`Defined objects with tensor properties`

 (3)
> 

A[]

A[a, b] = (Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1}))

 (4)
> 

A[`~`]

A[`~a`, `~b`] = (Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1}))

 (5)
> 

A[`~a`,b,matrix]

A[`~a`, b] = (Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1}))

 (6)
> 

A[`a`,~b,matrix]

A[a, `~b`] = (Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1}))

 (7)
> 

> 

[y[1] = x[1]-x[2], y[2] = x[2]-x[3], y[3] = x[3]]

[y[1] = x[1]-x[2], y[2] = x[2]-x[3], y[3] = x[3]]

 (8)
> 

solve((8), {x[1], x[2], x[3]})

{x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}

 (9)

OK;   #  note however that you only need to pass these fewer arguments to perform the computation

> 

B[a, b] = TransformCoordinates({x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}, A[a, b], [y[1], y[2], y[3]])

B[a, b] = Matrix(%id = 18446744078235230446)

 (10)

I comment here your 1D input

> 

# B[a,b] = TransformCoordinates((9), A[a, b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))

OK

> 

B[`~a`, `~b`] = TransformCoordinates({x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}, A[`~a`, `~b`], [y[1], y[2], y[3]])

B[`~a`, `~b`] = Matrix(%id = 18446744078235279598)

 (11)
> 

# C[a,b] = TransformCoordinates((9), A[~a,~b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))

OK

> 

B[a, `~b`] = TransformCoordinates({x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}, A[a, `~b`], [y[1], y[2], y[3]])

B[a, `~b`] = Matrix(%id = 18446744078235269590)

 (12)
> 

# D[a,b] = TransformCoordinates((9), A[a,~b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))``
> 

Was "unable to decide the ordering of the free indices", now it is deterministic when the object being transformed it is a tensor - itself- i.e. not a tensorial expression and there is a new optional argument, freeindices of type list, so that you can not only specify the ordering of the new indices but also indicate less of them, to only transform a restricted number of them, perhaps one at a time, a useful new feature.
> 

B[`~a`, b] = TransformCoordinates({x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}, A[`~a`, b], [y[1], y[2], y[3]])

B[`~a`, b] = Matrix(%id = 18446744078235257782)

 (13)
> 

# E[a, b] = TransformCoordinates((9), A[~a,b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))NULL``

should be
> LinearAlgebra:-Transpose(rhs())

Matrix(3, 3, {(1, 1) = 0, (1, 2) = -2, (1, 3) = -3, (2, 1) = 1, (2, 2) = 2, (2, 3) = 5, (3, 1) = 1, (3, 2) = 3, (3, 3) = 4})

Check the new functionality, transform only one index at a time

> 

TransformCoordinates({x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}, A[`~a`, b], [y[1], y[2], y[3]], freeindices = [a])

Vector[column](%id = 18446744078272589566)

 (14)
> 

TransformCoordinates({x[1] = y[1]+y[2]+y[3], x[2] = y[2]+y[3], x[3] = y[3]}, A[`~a`, b], [y[1], y[2], y[3]], freeindices = [b])

Vector[column](%id = 18446744078272580414)

 (15)
> 

 

> 

``


Download Transformation_tensor_components_(reviewed).mw

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

plots:-plotcompare(function_or_expresssi...

ecterrab 13431
August 16 2015
3 1

In my opinion, the best visualization you can have of a complex function, or for the case an arbitrary complex expression, is the one that you get using plotcompare; check its help page, use the option 'expression_plot'. In brief, you will see two 3D plots side by side: the first one shows the real part of the function on the vertical axis versus the real (x) and imaginary (y) parts of the complex variable (z) on the horizontal (complex) plane, while the second 3D plot shows you the imaginary part of the function on the vertical axis against x and y.

I find these two 3D plots much easier to rapidly decipher with my brain than the corresponding plots you can get using conformal3d, complexplot3d.

The options provided by plotcompare to zoom, shift or scale ranges, manipulate each plot and automatic assignment to a variable _P for further manipulations without having to recompute things are also valuable extra features. The way this command interacts with assuming (real or imaginary) is also very handy. Finally, the possibilty to in addition compare complex expressions visually (in which case you receive four 3D plots instead of two) is for me fantastic.

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

Definitions are equivalent and bug fixed...

ecterrab 13431
August 05 2015
3 0

Hi Jschulzb

Defining tensors as arrays is equivalent to defining them as algebraic expressions that, in turn, frequently involve other tensors, as in the example you posted. There was however a bug, subtle, not less of a bug though. This bug is fixed, and the fix is available for download for everybody in the usual place, the Maplesoft R&D Physics webpage.

Details: the Lie derivative of h[a, b], involving three covariant derivatives with all of their indices covariant, is defined as LKh. Then your expression rho (the one within the worksheet you posted) involves some of the derivatives with contravariant indices in the derivand. All that is OK. But then to compute the actual value of these expressions you need to expand these covariant derivatives into the corresponding non-covariant derivatives d_[mu](...) plus Christoffel terms, you know. These Christoffel terms, in turn, have a sign that depends on the character of the indices (covariant or contravariant) of the derivand. The bug: there was one place where these covariant derivatives (in LKh) were first expanded, so that the terms involving christoffel symbols were appearing with a sign corresponding to the covariant character found in the definition of LKh, and then they were evaluate at contravariant values of some of these indices (those of the derivand), but at that point the sign of the Christoffel terms were the ones corresponding to covariant instead of contravariant indices of the derivand.

Not so easy to cross with this bug. Nicely spotted, thanks for posting about the problem. As said the fix is available for download for everybody in the usual place, the Maplesoft R&D Physics webpage.

Regarding the actual difference betrween the two ways of defining tensors, using algebraic expressions:

  • the defintion of tensors is more natural (similar to paper and pencil computations)
  • you can define tensors in terms of tensorial expressions that involve other tensors, possibly defined in terms of other algebraic tensorial expressions involving yet other tensors and so on. This is a very flexible way of defining tensors.
  • if you change the definition of a tensor entering any of these nested levels of definitions, the change is taken into account everywhere.
  • You can compute with the algebraic definitions, including simpification taking Einstein's sum rule into account, in ways you cannot do with Array definitions.

Depending on what you are working on, these features of the definition of tensors in terms of algebraic tensorial expressions can be relevant and simplify your work in significant ways.

By the way: I see you introduced a Killing Vector. Maybe you already know, maybe not: within Physics there is already a command very easy to use to compute Killing vectors, also with a number of features; if you were not aware, it is worth giving a look at the corresponding help page. (Physics:-KillingVectors).

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

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