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Help: Tensor Operators
*To*: mathgroup at smc.vnet.net
*Subject*: [mg22259] Help: Tensor Operators
*From*: "Imran Haq" <imran_haq at mail.com>
*Date*: Sat, 19 Feb 2000 01:34:08 -0500 (EST)
*Organization*: Ye 'Ol Disorganized NNTPCache groupie
*Sender*: owner-wri-mathgroup at wolfram.com
Hi all,
I'm trying to write a Mathematica notebook to return the generalized
coordinates form
of the Navier-Stokes equations. These equations have a term:
Table[Sqrt[g[i,i]],{i,d}] Laplacian[Table[Sqrt[g[i,i]] U[i][X],{i,d}]]
where d is the spatial dimension of the coordinate system, and X are the
coordinates
(ie. X = Sequence[x,y] for Cartesian 2D coordinates, X=Sequence[r,theta] for
polar coordinates.).
U is a vector.
If I implement the above formula, and substitute in the metric and scale
factors for
polar cylindrical coordinates, I don't seem to get the correct answer which
should be:
i = 1: D[U[1],{r,2}] + 1/r/r D[U[1],{theta,2}] + 1/r D[U[1],r] - 2/r/r
D[U[2],theta] - U[1]/r/r
i = 2: D[U[2],{r,2}] + 1/r/r D[U[2],{theta,2}] + 1/r D[U[2],r] + 2/r/r
D[U[1],theta] - U[2]/r/r
Am I missing something, or do I need to use some tensor identity?
(Also: D[U[1],r] + U[1]/r + D[U[2],theta] == 0 is given. ie. Div[U] == 0)
Anyone can help?
Imran
imran_haq at mail.com
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