\begin{gather*} \intertext {\texttt {... } } \begin{align*} \begin{split} \Phi^i = & + h^{ij} \Phi_j \end{split} \\ \begin{split} \frac { \partial \Phi^i } { \partial x^k } = & + h^{ij} \frac { \partial \Phi_j } { \partial x^k } \\ & + \frac { \partial h^{ij} } { \partial x^k } \Phi_j \end{split} \\ \begin{split} \frac { \partial^2 \Phi^i } { \partial x^k \partial x^m } = & + h^{ij} \frac { \partial^2 \Phi_j } { \partial x^k \partial x^m } \\ & + \frac { \partial h^{ij} } { \partial x^m } \frac { \partial \Phi_j } { \partial x^k } \\ & + \frac { \partial h^{ij} } { \partial x^k } \frac { \partial \Phi_j } { \partial x^m } \\ & + \frac { \partial^2 h^{ij} } { \partial x^k \partial x^m } \Phi_j \end{split} \\ \end{align*} \end{gather*}

NB. ... script torsiontensor.ijs ...

Phicn=:hcn smx"2 1 Phicv

Phicndxt1=:hcn  ((   [)smx 0|:])"2 2 Phicvdx
Phicndxt2=:hcndx((1|:[)smx    ])"3 1 Phicv
Phicndx  =:Phicndxt1+Phicndxt2

Phicndxdxt1=:hcn           ((   [)smx 0|:])"2 3 Phicvdxdx
Phicndxdxt2=:hcndx  (1|:])@((1|:[)smx 0|:])"3 2 Phicvdx
Phicndxdxt3=:hcndx         ((1|:[)smx 0|:])"3 2 Phicvdx
Phicndxdxt4=:hcndxdx       ((1|:[)smx    ])"4 1 Phicv
Phicndxdx  =:Phicndxdxt1+Phicndxdxt2+Phicndxdxt3+Phicndxdxt4

NB. ... execute (ijx) ...

   gXcme =:( 1.5    0.45   0.754 )"_
   gXpAll=:( 6     _1.4    1.7   )"_
   gXqAll=:( 1      1     _1     )"_
   gXwAll=:( 1.25   _      _     )"_

   yCpts=:0.3 2 50,(0.1,(1p1-0.1),28),0 2p1 50,:1 8 50"_

   (gXT(([Phicndx   hkAy)(((2^_22);2^_14)qteq[;])"2(0|:[:(gXT([Phicn   hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   (gXT(([Phicndxdx hkAy)(((2^_18);2^_14)qteq[;])"3(0|:[:(gXT([Phicndx hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

\begin{gather*} \intertext {\texttt {... a rank 2 tensor ... \newline ... covariant derivative of $\Omega_i\,$ w.r.t.$\,$ Christoffel symbols ... } } \begin{align*} \begin{split} \Omega_{i,j} = & + \frac { \partial \Omega_i } { \partial x^j } \\ & - \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} \Omega_p \end{split} \\ \begin{split} \frac { \partial \Omega_{i,j} } { \partial x^k } = & + \frac { \partial^2 \Omega_i } { \partial x^j \partial x^k } \\ & - \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} \frac { \partial \Omega_p } { \partial x^k } \\ & - \frac { \partial \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} } { \partial x^k } \Omega_p \end{split} \\ \begin{split} \frac { \partial^2 \Omega_{i,j} } { \partial x^k \partial x^m } = & + \frac { \partial^3 \Omega_i } { \partial x^j \partial x^k \partial x^m } \\ & - \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} \frac { \partial^2 \Omega_p } { \partial x^k \partial x^m } \\ & - \frac { \partial \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} } { \partial x^m } \frac { \partial \Omega_p } { \partial x^k } \\ & - \frac { \partial \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} } { \partial x^k } \frac { \partial \Omega_p } { \partial x^m } \\ & - \frac { \partial^2 \Bigl\{ \begin{matrix} p \\ ij \end{matrix} \Bigr\} } { \partial x^k \partial x^m } \Omega_p \end{split} \\ \end{align*} \end{gather*}

NB. ... script torsiontensor.ijs ...

Omgcv=:Omgdx+-@(ch2k smx"3 1 Omg)

Omgcvdxt1=:Omgdxdx
Omgcvdxt2=:ch2k  -@((   [)smx 0|:])"3 2 Omgdx
Omgcvdxt3=:ch2kdx-@((2|:[)smx    ])"4 1 Omg
Omgcvdx  =:Omgcvdxt1+Omgcvdxt2+Omgcvdxt3

Omgcvdxdxt1=:Omgdxdxdx
Omgcvdxdxt2=:ch2k           -@((   [)smx 0|:])"3 3 Omgdxdx
Omgcvdxdxt3=:ch2kdx  -@(2|:])@((2|:[)smx 0|:])"4 2 Omgdx
Omgcvdxdxt4=:ch2kdx         -@((2|:[)smx 0|:])"4 2 Omgdx
Omgcvdxdxt5=:ch2kdxdx       -@((2|:[)smx    ])"5 1 Omg
Omgcvdxdx  =:Omgcvdxdxt1+Omgcvdxdxt2+Omgcvdxdxt3+Omgcvdxdxt4+Omgcvdxdxt5

NB. ... execute (ijx) ...

   gXcme =:( 1.5    0.45   0.754 )"_
   gXpAll=:( 6     _1.4    1.7   )"_
   gXqAll=:( 1      1     _1     )"_
   gXwAll=:( 1.25   _      _     )"_

   yCpts=:0.3 2 50,(0.1,(1p1-0.1),28),0 2p1 50,:1 8 50"_

NB. ... Omgcv is a tensor ...

   OmgcvBox=:([Omgcv hkAx);([(0|:])"2@:ycdx xfromy);[Omgcv hkAy
   OmgcvChk=:[(>@(0{])(((2^_44);2^_44)qteq[;])"2>@(1{])([smx"2 smx"2)>@(2{]))OmgcvBox

   (gXT OmgcvChk]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

NB. ... derivatives ...

   (gXT(([Omgcvdx   hkAy)(((2^_23);2^_18)qteq[;])"3(0|:[:(gXT([Omgcv   hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   (gXT(([Omgcvdxdx hkAy)(((2^_19);2^_14)qteq[;])"4(0|:[:(gXT([Omgcvdx hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

\begin{gather*} \intertext {\texttt {... a rank 2 skew-symmetric tensor ... } } \begin{align*} \begin{split} \Upsilon_{ij} = & \Omega_{j,i} - \Omega_{i,j} \end{split} \\ \begin{split} \frac { \partial \Upsilon_{ij} } { \partial x^k } = & \frac { \partial \Omega_{j,i} } { \partial x^k } - \frac { \partial \Omega_{i,j} } { \partial x^k } \end{split} \\ \begin{split} \frac { \partial^2 \Upsilon_{ij} } { \partial x^k \partial x^m } = & \frac { \partial^2 \Omega_{j,i} } { \partial x^k \partial x^m } - \frac { \partial^2 \Omega_{i,j} } { \partial x^k \partial x^m } \end{split} \\ \end{align*} \end{gather*}

NB. ... script torsiontensor.ijs ...

Ups    =:((      0|:])-])"2@Omgcv
Upsdx  =:((  1 0 2|:])-])"3@Omgcvdx
Upsdxdx=:((1 0 2 3|:])-])"4@Omgcvdxdx

NB. ... execute (ijx) ...

   gXcme =:( 1.5    0.45   0.754 )"_
   gXpAll=:( 6     _1.4    1.7   )"_
   gXqAll=:( 1      1     _1     )"_
   gXwAll=:( 1.25   _      _     )"_

   yCpts=:0.3 2 50,(0.1,(1p1-0.1),28),0 2p1 50,:1 8 50"_

NB. ... Ups is a tensor ...

   UpsBox=:([Ups hkAx);([(0|:])"2@:ycdx xfromy);[Ups hkAy
   UpsChk=:[(>@(0{])(((2^_44);2^_44)qteq[;])"2>@(1{])([smx"2 smx"2)>@(2{]))UpsBox

   (gXT UpsChk]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

NB. ... derivatives ...

   (gXT(([Upsdx   hkAy)(((2^_24);2^_22)qteq[;])"3(0|:[:(gXT([Ups   hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   (gXT(([Upsdxdx hkAy)(((2^_24);2^_22)qteq[;])"4(0|:[:(gXT([Upsdx hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

\begin{gather*} \intertext {\texttt {... the torsion tensor $S_{ij\,.}^{\,\,\,\,\,k}$ ... } } \begin{align*} \begin{split} S_{ij\,.}^{\,\,\,\,\,k} = & + \Upsilon_{ij} \Phi^k \end{split} \\ \begin{split} \frac { \partial S_{ij\,.}^{\,\,\,\,\,k} } { \partial x^m } = & + \Upsilon_{ij} \frac { \partial \Phi^k } { \partial x^m } \\ & + \frac { \partial \Upsilon_{ij} } { \partial x^m } \Phi^k \end{split} \\ \begin{split} \frac { \partial^2 S_{ij\,.}^{\,\,\,\,\,k} } { \partial x^m \partial x^p } = & + \Upsilon_{ij} \frac { \partial^2 \Phi^k } { \partial x^m \partial x^p } \\ & + \frac { \partial \Upsilon_{ij} } { \partial x^p } \frac { \partial \Phi^k } { \partial x^m } \\ & + \frac { \partial \Upsilon_{ij} } { \partial x^m } \frac { \partial \Phi^k } { \partial x^p } \\ & + \frac { \partial^2 \Upsilon_{ij} } { \partial x^m \partial x^p } \Phi^k \end{split} \\ \end{align*} \end{gather*}

NB. ... script torsiontensor.ijs ...

S=:Ups*/"2 1 Phicn

Sdxt1=:Ups         (*/)"2 2 Phicndx
Sdxt2=:Upsdx(2|:])@(*/)"3 1 Phicn
Sdx  =:Sdxt1+Sdxt2

Sdxdxt1=:Ups             (*/)"2 3 Phicndxdx
Sdxdxt2=:Upsdx  (  2|:])@(*/)"3 2 Phicndx
Sdxdxt3=:Upsdx  (2 4|:])@(*/)"3 2 Phicndx
Sdxdxt4=:Upsdxdx(2 3|:])@(*/)"4 1 Phicn
Sdxdx  =:Sdxdxt1+Sdxdxt2+Sdxdxt3+Sdxdxt4

NB. ... execute (ijx) ...

   gXcme =:( 1.5    0.45   0.754 )"_
   gXpAll=:( 6     _1.4    1.7   )"_
   gXqAll=:( 1      1     _1     )"_
   gXwAll=:( 1.25   _      _     )"_

   yCpts=:0.3 2 50,(0.1,(1p1-0.1),28),0 2p1 50,:1 8 50"_

NB. ... S is a tensor ...

   SBox=:([S hkAx);xcdy;([(0|:])"2@:ycdx xfromy);[S hkAy
   SChk=:[(>@(0{])(((2^_44);2^_44)qteq[;])"3>@(2{])([smx"2 3 smx"2 3)>@(1{])smx"2 3>@(3{]))SBox

   (gXT SChk]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 
NB. ... example ...
NB.
NB. ... S in frame of reference which is not rotating ... 
NB. ... S in frame of reference which is rotating ...

   (gXT(([S hkAy),:([S hkAx))]) (1 yCrandom yCpts)''
       0 0        0        0
       0 0        0        0
       0 0        0        0
_25.9184 0        0  11.9925

       0 0        0        0
       0 0        0        0
       0 0        0        0
       0 0        0        0

       0 0        0        0
       0 0        0        0
       0 0        0        0
       0 0        0        0

 25.9184 0        0 _11.9925
       0 0        0        0
       0 0        0        0
       0 0        0        0



       0 0        0        0
       0 0        0        0
       0 0        0        0
_25.9184 0 _14.9906  11.9925

       0 0        0        0
       0 0        0        0
       0 0        0        0
       0 0        0        0

       0 0        0        0
       0 0        0        0
       0 0        0        0
       0 0        0        0

 25.9184 0  14.9906 _11.9925
       0 0        0        0
       0 0        0        0
       0 0        0        0

NB. ... derivatives ...

   (gXT(([Sdx   hkAy)(((2^_19);2^_14)qteq[;])"4(0|:[:(gXT([S   hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   (gXT(([Sdxdx hkAy)(((2^_16);2^_14)qteq[;])"5(0|:[:(gXT([Sdx hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

\begin{gather*} \intertext {\texttt {... the connection ... } } \begin{align*} \begin{split} \Gamma_{\beta\gamma}^{\,\alpha} = & \Bigl\{ \begin{matrix} \alpha \\ \beta\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} \end{split} \\ \begin{split} \frac { \partial \Gamma_{\beta\gamma}^{\,\alpha} } { \partial x^\delta } = & \frac { \partial \Bigl\{ \begin{matrix} \alpha \\ \beta\gamma \end{matrix} \Bigr\} } { \partial x^\delta } + \frac{1}{2} \frac { \partial S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} } { \partial x^\delta } \end{split} \\ \begin{split} \frac { \partial^2 \Gamma_{\beta\gamma}^{\,\alpha} } { \partial x^\delta \partial x^\epsilon } = & \frac { \partial^2 \Bigl\{ \begin{matrix} \alpha \\ \beta\gamma \end{matrix} \Bigr\} } { \partial x^\delta \partial x^\epsilon } + \frac{1}{2} \frac { \partial^2 S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} } { \partial x^\delta \partial x^\epsilon } \end{split} \\ \end{align*} \end{gather*}

NB. ... script torsiontensor.ijs ...

Gam    =:ch2k    +0.5*S
Gamdx  =:ch2kdx  +0.5*Sdx
Gamdxdx=:ch2kdxdx+0.5*Sdxdx

NB. ... execute (ijx) ...

   gXcme =:( 1.5    0.45   0.754 )"_
   gXpAll=:( 6     _1.4    1.7   )"_
   gXqAll=:( 1      1     _1     )"_
   gXwAll=:( 1.25   _      _     )"_

   yCpts=:0.3 2 50,(0.1,(1p1-0.1),28),0 2p1 50,:1 8 50"_

   (gXT(([Gamdx   hkAy)(((2^_20);2^_14)qteq[;])"4(0|:[:(gXT([Gam   hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   (gXT(([Gamdxdx hkAy)(((2^_17);2^_14)qteq[;])"5(0|:[:(gXT([Gamdx hkAy)])D.1])"_ 1)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


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Contributed by TomAllen

Essays/Torsion Tensor/TorsionTensor03 (last edited 2012-02-06 01:02:27 by TomAllen)