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Question: How to define components of the tensor

Question:How to define components of the tensor

EugeneKalentev 85 Maple 2015
tensor transformation
April 01 2015
0

Hello. Earlier, I asked about it, (see http://www.mapleprimes.com/questions/203573-How-To-Do-Simple-Operations-On-Tensors). However, not all I was able to understand. Below I will give a try, and maybe you'll show me where I'm wrong.

Also, I'm interested in how you can determine the components of the tensor in a different coordinate system connected with the original in any conversion. Thank for your help.

restart; with(Physics); with(DifferentialGeometry)

ds := Physics:-`^`(dx__1, 2)+Physics:-`^`(dx__2, 2)+Physics:-`^`(dx__3, 2)

dx__1^2+dx__2^2+dx__3^2

 (1)

Physics:-Setup(coordinates = (X = [x__1, x__2, x__3]), dimension = 3, metric = ds, quiet)

[coordinatesystems = {X}, dimension = 3, metric = {(1, 1) = 1, (2, 2) = 1, (3, 3) = 1}]

 (2)

g_[]

g_[mu, nu] = (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]))

 (3)

``

u__1 := Physics:-`*`(Physics:-`*`(P, Physics:-`^`(Physics:-`*`(Physics:-`*`(4, Pi), G), -1)), Physics:-`*`(x__3, Physics:-`*`(x__1, Physics:-`^`(Physics:-`^`(r, 3), -1)))-Physics:-`*`(Physics:-`*`(1-Physics:-`*`(2, nu), x__1), Physics:-`^`(Physics:-`*`(r, r+x__3), -1))):u__2 := Physics:-`*`(Physics:-`*`(P, Physics:-`^`(Physics:-`*`(Physics:-`*`(4, Pi), G), -1)), Physics:-`*`(x__2, Physics:-`*`(x__3, Physics:-`^`(Physics:-`^`(r, 3), -1)))-Physics:-`*`(Physics:-`*`(1-Physics:-`*`(2, nu), x__2), Physics:-`^`(Physics:-`*`(r, r+x__3), -1))):u__3 := Physics:-`*`(Physics:-`*`(P, Physics:-`^`(Physics:-`*`(Physics:-`*`(4, Pi), G), -1)), Physics:-`*`(Physics:-`*`(2, 1-nu), Physics:-`^`(r, -1))+Physics:-`*`(Physics:-`^`(x__3, 2), Physics:-`^`(Physics:-`^`(r, 3), -1))):

`e__1,1` := diff(u__1, x__1):`e__2,2` := diff(u__2, x__2):`e__3,3` := diff(u__3, x__3):

`e__1,2` := Physics:-`*`(Physics:-`^`(2, -1), diff(u__1, x__2)+diff(u__2, x__1)):`e__1,3` := Physics:-`*`(Physics:-`^`(2, -1), diff(u__1, x__3)+diff(u__3, x__1)):`e__2,3` := Physics:-`*`(Physics:-`^`(2, -1), diff(u__2, x__3)+diff(u__3, x__2)):

`e__2,1` := `e__1,2`:

`e__3,1` := `e__1,3`:

`e__3,2` := `e__2,3`:

  E := matrix(3, 3, proc (i, j) options operator, arrow; e[i, j] end proc)

Matrix(3, 3, {(1, 1) = e[1, 1], (1, 2) = e[1, 2], (1, 3) = e[1, 3], (2, 1) = e[2, 1], (2, 2) = e[2, 2], (2, 3) = e[2, 3], (3, 1) = e[3, 1], (3, 2) = e[3, 2], (3, 3) = e[3, 3]})

 (4)

Physics:-Define(E[i, j])

{gamma[mu], E[i, j], sigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], delta[mu, nu], epsilon[alpha, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (5)

Physics:-TensorArray(%)

{E[i, j], Array(1..3, 1..3, 1..3, {(1, 1, 1) = 0, (1, 1, 2) = 0, (1, 1, 3) = 0, (1, 2, 1) = 0, (1, 2, 2) = 0, (1, 2, 3) = 0, (1, 3, 1) = 0, (1, 3, 2) = 0, (1, 3, 3) = 0, (2, 1, 1) = 0, (2, 1, 2) = 0, (2, 1, 3) = 0, (2, 2, 1) = 0, (2, 2, 2) = 0, (2, 2, 3) = 0, (2, 3, 1) = 1, (2, 3, 2) = 1, (2, 3, 3) = 1, (3, 1, 1) = 0, (3, 1, 2) = 0, (3, 1, 3) = 0, (3, 2, 1) = -1, (3, 2, 2) = -1, (3, 2, 3) = -1, (3, 3, 1) = 0, (3, 3, 2) = 0, (3, 3, 3) = 0}), Array(1..3, {(1) = x__1, (2) = x__2, (3) = x__3}), Array(1..3, {(1) = Physics:-Psigma[1], (2) = Physics:-Psigma[2], (3) = Physics:-Psigma[3]}), Array(1..3, {(1) = Physics:-d_[1], (2) = Physics:-d_[2], (3) = Physics:-d_[3]}), Array(1..3, {(1) = Physics:-Dgamma[1], (2) = Physics:-Dgamma[2], (3) = Physics:-Dgamma[3]}), Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1}), Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})}

 (6)

``

Physics:-Setup(dimension)

[dimension = 3]

 (7)

NULL

Physics:-Define(E[i, j], query)

[E, [0, 0, 0], 0]

 (8)

DifferentialGeometry:-DGsetup([y__1, y__2, y__3], M):

Phi1 := DifferentialGeometry:-Transformation(N, M, [y__1 = Physics:-`*`(Physics:-`^`(sqrt(6), -1), x__1)+Physics:-`*`(Physics:-`*`(2, Physics:-`^`(sqrt(6), -1)), x__2)+Physics:-`*`(Physics:-`^`(sqrt(6), -1), x__3), y__2 = Physics:-`*`(Physics:-`^`(sqrt(2), -1), x__1)-Physics:-`*`(Physics:-`^`(sqrt(3), -1), x__2)+Physics:-`*`(Physics:-`^`(sqrt(3), -1), x__3), y__3 = Physics:-`*`(Physics:-`^`(sqrt(2), -1), x__1)-Physics:-`*`(Physics:-`^`(sqrt(2), -1), x__3)]):

NULL

 

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