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MathGroup Archive 2000

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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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