Mathematica
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Tensorial5

Packages  TensorCalculus5 Documents_and_Setting AppData Mathematica Mathematica MathematicaVer9 Unprotect[SyntaxInformation]; Unprotect[Symmetric]; Unprotect[TensorSymmetry]; Needs["TensorCalculus5`Tensorial`"] (*Needs["TensorCalculus4V6`Tensorial`"] *) (*Needs["TContinuumMechanics21`TContinuumMechanics`"] *) Tensorial5 Tensrial 4 DeclareBaseIndices[{0, 1, 2, 3}] DefineTensor[x,x',a,b,v,η] metric = DiagonalMatrix[{-1, 1, 1}]; % // MatrixForm
（ 0 0 0 1 Tensorial SetTensorValues[η@dd[i, j], metric] ToArrayValues[] n@dd[i, j] % // ToArrayValues[] // MatrixForm切り/ -1 0 0 0 0 0 0 1 EinsteinSum[] x' x EinsteinSum[] x'@u[μ] = a@ud[μ, v] x@u[v] +b@u[μ] // EinsteinSum[]; % // MatrixForm x'@u[v] = a@ud[v, μ] x@u[μ] + b@u[v] // EinsteinSum[]; % // MatrixForm b" + a" o x" + a" x 1 + a" 2 x" + a" 3 x 3 b¥ + a¥ 0 x 0 + a¥ 1 x 1 + a¥ 2 x 2 + a¥ 3 x 3 α“ ν l 2 η@dd[μ, v] x@u[μ] x@u[v] % // ToArrayValues[] η@dd[μ, v] x'@u[μ] x'@u[v] x" x" ημν - (x o )² + (x¹)² + (x²)² + (x³)² (b ′ + a ′ o x ′ + a ′ 1 x ¹ + a ′ 2 x ² + a ′ 3 x ³ ) (b ′ + a ′ o x ′ + a ′ 1 x ² + a ′ 2 x ² + a ′ 3 x ³ ) ημ ν ημν = ηρσ αρ μ αρ ν ημν = ηρσ α ρ μ α ρ ν 0 1 0 0 0 l 0 0 (2)
η@dd[ρ, σ] a@ud[ρ, μ] a@ud[σ, ν] fg1 = % // ToArrayValues[] // Simplify // MatrixForm + (a ³ o ) ² -a 0 a 0 1 + a 1 0 a 1 + a 2 0 a 2 1 + a 3 0 a 3 1 -a o a o 2 + (a 1 ) a³oa³ı (a o i )   + (a² ı a 1 a 2 + a 1 a 1 2 a30 a32 , a 3 + a  1 a - a o = (3) a 0 ≥ 1 or α 0 ≤ −1 (4) |a|=± l mation) (poper Lorentz transfor- Mathematica (5) a o ≥  l   | a | = l Tnsorial Mathemtia Tensorial β=V/c {1,0,0} LorentzTransformation[β, {1, 0, 0}] // MatrixForm 7 {1,1,1} LorentzTransformation[.7, {1, 1, 1}] // MatrixForm {nl,n2,n3} B 2 β  LT Mathematica V LorentzTransformation[β, {n1, n2, n3}] ; LT = Simplify[%, 0 < β < 1八{n1, n2, n3} ∈ Reals]; % // MatrixForm 2 Tensor η@dd[i,j] Tensorial ToArrayValues[] YT = η@dd[i, j] // ToArrayValues[]; Transpose[LT].YT.LT // Simplify // MatrixForm – 1 0 0 0 0 0 1 Mathematica Manipulate Manipulate r0 = {{1, x, y, z}}; kc = 1.4; g1 = Graphics3D[{Arrow[{{-2, 0, 0}, {2, 0, 0}}], Arrow[{{0, -2, 0}, {0, 2, 0}}], Arrow[{{0, 0, -2}, {0, 0, 2}}], Arrow[Tube[{kc {nx v, ny v, nz v}, 2 kc {nx v, ny v, nz v}}]}]; g2 = ContourPlot3D[x^2+y^2+z^2 = 1, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ContourStyle → Opacity[0.3], Mesh → None]; g3 = ContourPlot3D[ (Part[r0.Lz[v, nx, ny, nz] , 1][[2]])^2 + (Part[r0.Lz[v, nx, ny, nz] , 1][[3]])^2 + (Part[r0.Lz[v, nx, ny, nz], 1][[4]])^2 = 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Mesh → None]; Show[g1, g2, g3], {{v, 0, " "}, 0, 0.99, 0.01, Appearance → "Labeled"}, {{nx, 0.5, "x "}, -1, 1, 0.1, Appearance → "Labeled" {{ny, 0.5, "y "}, -1, 1, 0.1, Appearance → "Labeled" {{nz, 0.5, "z "}, -1, 1, 0.1, Appearance → "Labeled"} Initialization : Lz [ β n3 ] := n1 β 1 - B2 Abs [n1]² + Abs [n2]² + Abs [n3]² n2 β [n2]² + Abs [n3]² Abs [n1]² + Abs [n2]² + Abs [n3]² n1 β 1 Abs [n1]² + Abs [n2]² + Abs [n3]² Abs [n1]² + Abs [n2]² + Abs [n3]² 0 1 0 0 0 0 1 LT
Tensrial 4 δ DeclareBaseIndices[{1, -1, -1}] DeclareBaseIndices[{0, 1, 2, 3}] labs = {x, δ, g, r}; DefineTensor[a, b, x, x', j, ξ, g, f, ef, bf, a, δ, η, a, F, r] SetTensorValues[ δ@ud[i, j], IdentityMatrix[NDim]] Tensorial xAct  MathTensor 4
(cmetric = DiagonalMatrix[{-1, 1, 1}]) // MatrixForm; (metric = cmetric // CoordinatesToTensors[{t, x, y, z}]) // MatrixForm -1 0 0 0 1 0 0 o 0 0 l 0 0 0 1 0 Mathematica  MapThread g MapThread[SetTensorValues[#1, #2] &, {{g@dd[a, b], g@uu[a, b]}, {metric, Inverse[metric] // Simplify}}]; ToArrayValues[] g�uu[i,j] % // ToArrayValues[] // MatrixForm g�dd[i, j] fg = % // ToArrayValues[] // MatrixForm g@uu[i, j] g@dd[i, j] // ToArrayValues[] gij -1 0 0 0 0 0 1 911 0 0 1 0 MetricSimplify[ MetricSimplify[] g�dd[i, k] g�uu[k, j] % // MetricSimplify[g] g   gix a」 Ef 4 0 t,l x,2 y,3 z EinsteinArray[ { 1,2,3 } ] Bf 0 1 0 0 0 l 0 0 0 0 -1 0 0 0 4 / F 4 4 EF Ef@d[α] == PartialD[labs][A@d[0], x@u[α]] - PartialD[labs][A@d[α], x@u[0]] % // EinsteinArray[{1, 2, 3}] // MatrixForm
Efa
0A, Ox'
Bf@dd[β, γ] == PartialD[labs][A@d[γ], x@u[β]] - PartialD[labs][A@d[β], x@u[γ]] % // EinsteinArray[{1, 2, 3}] // MatrixForm
d A ∂x' ∂ x k
F@dd[β, γ] == PartialD[labs][A@d[γ], x@u[β]] - PartialD[labs][A@d[β], x@u[γ]] % // ToArrayValues[] // MatrixForm
9 A
Tensorial
SetTensorValueRules[ TensorValueRules[ Tensorial TensorValueRules[] SetTensorValueRules[] SetTensorValueRules[A@ud[α, β] , trans] SetTensorValueRules[F@uu[α, β], fbasis] SetTensorValueRules[F@dd[ɣ, ŋ], finverse] SetTensorValueRules[F @uu[α, β] , fdual] TensorValueRules[A] TensorValueRules[F] TensorValueRules[F] {^° o → y, ^° 1 → -β y, ^° 2 → 0, ^° 3 → 0, ^¹ o → -β y, ^¹¹ → y, ^¹ 2 → 0, ∧²³ → 0, ∧² o → 0, ∧²¹ → 0, ∧²² → 1, ∧²³ → 0, ∧³ o → 0, ∧³ 1 → 0, ∧³ 2 → 0, ∧³ 3 {F00 → 0, F01 → -E1, F02 → -E2, F03 → -E3, F10 → E1, F11 → 0, F12 → -B3, F13 → B2, F20 → E2, F21 → B3, F22 → 0, F23 → -B1, F30 → E3, F31 → -B2, F32 → B1, F33 → 0, Foo → 0, Foi → E1, Foi → E2, Foi → E3, Fio → − E1, Fii → 0, Fii 2 → − B3, Fii → B2, F 2 o → −E2, F 2 1 → B3, F 2 2 → 0, F 2 3 → −B1, F 3 o → −E3, F 3 1 → −B2, F 3 2 → B1, F 3 3 → 0 } {Foo → 0, For → -B1, For → -B2, Fos → -B3, F10 → B1, F11 → 0, F12 → E3, F13 → -E2, F20 → B2, F21 → −E3, F22 → 0, F23 → E1, F30 → B3, F31 → E2, F32 → −E1, F33 → 0} / / Λ�ud[i,j] % // ToArrayValues[] // MatrixFormへシ
^@ud[i, j] x@u[j] % // ToArrayValues[] // MatrixForm x」 ∧ , (  y x o - β y x 1 -ßyx°+yx° x² x 3 F�uu[i, j] % // ToArrayValues[] // MatrixForm Fi | 0  -E1 -E2 -E3
l  8ij F ij g�dd[α, ɤ] g�dd[η, β] F�uu[ɣ, η] % // MetricSimplify[g] %% // ToArrayValues[] // MatrixForm F@dd[α, β] // ToArrayValues[] // MatrixForm F 7 gay 97 p Fαβ
le g@dd[α, γ] g@dd[η, β] F@uu[ণ, η] F@uu[α, β] % // MetricSimplify[g] %% // ToArrayValues[] T1 = Simplify[%] Fa b  Fy  J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J   J  J   Fyn Fyn 2 B1² + 2 B2² + 2 B3² - 2 E1² - 2 E2² - 2 E3² 2 (B1² + B2² + B3² - E1² - E2² - E3²) 4  g@dd[0, 0] g@dd[η, β] F@uu[0, η] F@uu[0, β] % // MetricSimplify[g] %% // ToArrayValues[] T2 = Simplify[%] Simplify[T1 / 4 - T2] Fo ç Fo 7 goo gn p F η F η goo –E1² – E2² – E3² -E1² - E2² - E32 1
− (B1² + B2² + B3² + E1² + E2² + E3²)
2