Question: Reducing a tensor expression and then expanding it

January 12 2014 maneeshkm 5
1

I am trying to reduce a tensor expression: ωiωjUi,j 

For which I have tried the following

restart; with(Physics)

Setup(dimension = [3, `+`], coordinatesystems = X, spacetimeindices = lowercaselatin):

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

 

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

 

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

 

`Defined objects with tensor properties`

(1)

omega[i] := d_[j](U[k](X))*ep_[i, j, k];

Physics:-LeviCivita[i, j, k]*Physics:-d_[`~j`](U[`~k`](X), [X])

 

Physics:-LeviCivita[l, m, n]*Physics:-d_[`~m`](U[`~n`](X), [X])

 

Physics:-LeviCivita[i, j, k]*Physics:-LeviCivita[l, m, n]*Physics:-d_[`~j`](U[`~k`](X), [X])*Physics:-d_[`~m`](U[`~n`](X), [X])*Physics:-d_[`~l`](U[`~i`](X), [X])

 

-(-Physics:-d_[n](U[i](X), [X])*Physics:-d_[k](U[`~n`](X), [X])+Physics:-d_[m](U[i](X), [X])*Physics:-d_[`~m`](U[k](X), [X])-Physics:-d_[i](U[k](X), [X])*Physics:-d_[l](U[`~l`](X), [X])+Physics:-d_[k](U[i](X), [X])*Physics:-d_[l](U[`~l`](X), [X]))*Physics:-d_[`~i`](U[`~k`](X), [X])

(2)

continuity := [D_[l](U[l](X)) = 0]

[Physics:-d_[l](U[`~l`](X), [X]) = 0]

(3)

red_eq := subs(continuity, expr)

-(-Physics:-d_[n](U[i](X), [X])*Physics:-d_[k](U[`~n`](X), [X])+Physics:-d_[m](U[i](X), [X])*Physics:-d_[`~m`](U[k](X), [X]))*Physics:-d_[`~i`](U[`~k`](X), [X])

(4)

Question 1. Am I using the continuity condition correctly? How do I use this condition correctly? If I change the index for the expression, substitution does not work correctly. 

Question 2. How do I expand the red_eq term in terms of the basis to give out the full expression? 

Question 3. I would like to eventually replace U by (A-Amean) in the current expression. How do I implement this? 

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