TomAllen/Experimental15

$\begin{gather*} \begin{align*} \intertext {\texttt {... curvature tensor ... } } \begin{split} B_{.\,jkm}^i = & + \frac { \partial\Gamma_{jm}^i } { \partial c^k } \end{split} \\ \begin{split} & - \frac { \partial\Gamma_{jk}^i } { \partial c^m } \end{split} \\ \begin{split} & + \Gamma_{jm}^a \Gamma_{ak}^{\,i} \end{split} \\ \begin{split} & - \Gamma_{jk}^a \Gamma_{am}^{\,i} \end{split} \\ \end{align*} \end{gather*}$

NB. ... script experimental.ijs ...

B2kt0=:   ((0 3 1|:])"4)@(Gamdc               )
B2kt1=:-@:((0 1 3|:])"4)@(Gamdc               )
B2kt2=:   ((0 2 1|:])"4)@(Gam([smx 0|:])"3 Gam)
B2kt3=:-@:((0 1 2|:])"4)@(Gam([smx 0|:])"3 Gam)

B2k=:B2kt0+B2kt1+B2kt2+B2kt3

$\begin{gather*} \begin{align*} \intertext {\texttt {... a tensor ... } } \begin{split} (B_x)_{.\,jkm}^i = & \frac { \partial x^i } { \partial y^a } \frac { \partial y^b } { \partial x^j } \frac { \partial y^c } { \partial x^k } \frac { \partial y^d } { \partial x^m } (B_y)_{\,.\,bcd}^a \end{split} \\ \end{align*} \end{gather*}$

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

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   xCpts=:_ 1 3 26,_ 0 1p1 26,_ 0 2p1 26,:_ 0 20 26"_

   vA =:[B2k xRef
   vB0=:((0|:[)smx])"2 4
   vB1=:((0|:[)smx])"2 4
   vB2=:((0|:[)smx])"2 4
   vB3=:((   [)smx])"2 4
   vB =:[(xdy vB3 ydx vB2 ydx vB1 ydx vB0 B2k)yRef

   (gXT([(vA(((2^_44);2^_25)qteq[;])"4 vB)hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

$\begin{gather*} \begin{align*} \intertext {\texttt {... derivative of the curvature tensor ... } } \begin{split} \frac { \partial B_{.\,jkm}^i } { \partial c^n } = & + \frac { \partial^2\Gamma_{jm}^i } { \partial c^k \partial c^n } \end{split} \\ \begin{split} & - \frac { \partial^2\Gamma_{jk}^i } { \partial c^m \partial c^n } \end{split} \\ \begin{split} & + \Gamma_{jm}^a \frac { \partial\Gamma_{ak}^{\,i} } { \partial c^n } \end{split} \\ \begin{split} & + \frac { \partial\Gamma_{jm}^a } { \partial c^n } \Gamma_{ak}^{\,i} \end{split} \\ \begin{split} & - \Gamma_{jk}^a \frac { \partial\Gamma_{am}^{\,i} } { \partial c^n } \end{split} \\ \begin{split} & - \frac { \partial\Gamma_{jk}^a } { \partial c^n } \Gamma_{am}^{\,i} \end{split} \\ \end{align*} \end{gather*}$

NB. ... script experimental.ijs ...

B2kdct0=:   ((0 3 1 4|:])"5)@(Gamdcdc                        )
B2kdct1=:-@:((0 1 3 4|:])"5)@(Gamdcdc                        )
B2kdct2=:   ((0 2 1 4|:])"5)@(Gam  ((   [)smx 0|:])"3 4 Gamdc)
B2kdct3=:   ((0 3 1 2|:])"5)@(Gamdc((2|:[)smx 0|:])"4 3 Gam  )
B2kdct4=:-@:((0 1 2 4|:])"5)@(Gam  ((   [)smx 0|:])"3 4 Gamdc)
B2kdct5=:-@:((0 1 3 2|:])"5)@(Gamdc((2|:[)smx 0|:])"4 3 Gam  )

B2kdc=:B2kdct0+B2kdct1+B2kdct2+B2kdct3+B2kdct4+B2kdct5

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

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   xCpts=:_ 1 3 26,_ 0 1p1 26,_ 0 2p1 26,:_ 0 20 26"_

   A=:(gXT([B2kdc xRef@hkxF)]) ((0.2;50"_)pTRandom xCpts)''
   B=:(gXT((0|:[:(gXT([(0{])@B2k xRef@hkxF),:@])D.1])"_ 1)]) ((0.2;50"_)pTRandom xCpts)''
   
   A (((2^_44);2^_05)qteq[;])"5 B
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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. ... execute (ijx) ...

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   yCpts=:__ 0.5 5 8,__ 0.5 5 8,__ 0.5 5 8,:_ 0 20 8"_

   A=:(gXT([B2kdc yRef@hkyF)]) ((0;50"_)pTRandom yCpts)''
   B=:(gXT((0|:[:(gXT([(0{])@B2k yRef@hkyF),:@])D.1])"_ 1)]) ((0;50"_)pTRandom yCpts)''

   A (((2^_44);2^_04)qteq[;])"5 B
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

$\begin{gather*} \begin{align*} \intertext {\texttt {... covariant derivative of the curvature tensor ... } } \begin{split} B_{\,.\,jkm\,,\,n}^i = & + \frac { \partial B_{\,.\,jkm}^i } { \partial c^n } \end{split} \\ \begin{split} & + \Gamma_{an}^{\,i} B_{\,.\,jkm}^a \end{split} \\ \begin{split} & - \Gamma_{jn}^a B_{\,.\,akm}^i \end{split} \\ \begin{split} & - \Gamma_{kn}^{\,a} B_{\,.\,jam}^i \end{split} \\ \begin{split} & - \Gamma_{mn}^{\,\,a} B_{\,.\,jka}^i \end{split} \\ \end{align*} \end{gather*}$

NB. ... script experimental.ijs ...

B2kcvt0=:                                             B2kdc
B2kcvt1=:   ((      0|:])"5)@(Gam((0|:[)smx 0|:])"3 4 B2k  )
B2kcvt2=:-@:((0 3 4 1|:])"5)@(Gam((   [)smx 1|:])"3 4 B2k  )
B2kcvt3=:-@:((  0 4 1|:])"5)@(Gam((   [)smx 2|:])"3 4 B2k  )
B2kcvt4=:-@:((    0 1|:])"5)@(Gam((   [)smx    ])"3 4 B2k  )

B2kcv=:B2kcvt0+B2kcvt1+B2kcvt2+B2kcvt3+B2kcvt4

$\begin{gather*} \begin{align*} \intertext {\texttt {... a tensor ... } } \begin{split} (B_x)_{.\,jkm\,,\,n}^i = & \frac { \partial x^i } { \partial y^a } \frac { \partial y^b } { \partial x^j } \frac { \partial y^c } { \partial x^k } \frac { \partial y^d } { \partial x^m } \frac { \partial y^e } { \partial x^n } (B_y)_{\,.\,bcd\,,\,e}^a \end{split} \\ \end{align*} \end{gather*}$

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

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   xCpts=:_ 1 3 26,_ 0 1p1 26,_ 0 2p1 26,:_ 0 20 26"_

   vA =:[B2kcv xRef
   vB0=:((0|:[)smx])"2 5
   vB1=:((0|:[)smx])"2 5
   vB2=:((0|:[)smx])"2 5
   vB3=:((0|:[)smx])"2 5
   vB4=:((   [)smx])"2 5
   vB =:[(xdy vB4 ydx vB3 ydx vB2 ydx vB1 ydx vB0 B2kcv)yRef

   (gXT([(vA(((2^_44);2^_24)qteq[;])"5 vB)hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

$\begin{gather*} \begin{align*} \intertext {\texttt {... } } \begin{split} (SB)_{\,.\,jkmn}^i = & S_{km\,.}^{\,\,\,\,\,\,\,\,a}\, B_{\,.\,jna}^i \end{split} \\ \intertext {\texttt {... } } \begin{split} \Lambda_{\,.\,jkmn}^i = & B_{\,.\,jkm\,,\,n}^i + B_{\,.\,jmn\,,\,k}^i + B_{\,.\,jnk\,,\,m}^i \end{split} \\ \begin{split} \Xi_{\,.\,jkmn}^{\,i} = & (SB)_{\,.\,jkmn}^i + (SB)_{\,.\,jmnk}^i + (SB)_{\,.\,jnkm}^i \end{split} \\ \end{align*} \end{gather*}$

NB. ... script experimental.ijs ...

SBk=:(0 1 4|:])"5@(S smx"3 4 B2k)

Lam=:(]+(2 3|:])+2|:])"5@B2kcv
Xi =:(]+(2 3|:])+2|:])"5@SBk

$\begin{gather*} \begin{align*} \intertext {\texttt {... a tensor ... } } \begin{split} (\Lambda_x)_{.\,jkmn}^i = & \frac { \partial x^i } { \partial y^a } \frac { \partial y^b } { \partial x^j } \frac { \partial y^c } { \partial x^k } \frac { \partial y^d } { \partial x^m } \frac { \partial y^e } { \partial x^n } (\Lambda_y)_{\,.\,bcde}^a \end{split} \\ \end{align*} \end{gather*}$

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

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   xCpts=:_ 1 3 26,_ 0 1p1 26,_ 0 2p1 26,:_ 0 20 26"_

   vA =:[Lam xRef
   vB0=:((0|:[)smx])"2 5
   vB1=:((0|:[)smx])"2 5
   vB2=:((0|:[)smx])"2 5
   vB3=:((0|:[)smx])"2 5
   vB4=:((   [)smx])"2 5
   vB =:[(xdy vB4 ydx vB3 ydx vB2 ydx vB1 ydx vB0 Lam)yRef

   (gXT([(vA(((2^_44);2^_24)qteq[;])"5 vB)hkxF)]) ((0.2;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

$\begin{gather*} \begin{align*} \intertext {\texttt {... a tensor ... } } \begin{split} (\Xi_x)_{.\,jkmn}^i = & \frac { \partial x^i } { \partial y^a } \frac { \partial y^b } { \partial x^j } \frac { \partial y^c } { \partial x^k } \frac { \partial y^d } { \partial x^m } \frac { \partial y^e } { \partial x^n } (\Xi_y)_{\,.\,bcde}^a \end{split} \\ \end{align*} \end{gather*}$

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

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   xCpts=:_ 1 3 26,_ 0 1p1 26,_ 0 2p1 26,:_ 0 20 26"_

   vA =:[Xi xRef
   vB0=:((0|:[)smx])"2 5
   vB1=:((0|:[)smx])"2 5
   vB2=:((0|:[)smx])"2 5
   vB3=:((0|:[)smx])"2 5
   vB4=:((   [)smx])"2 5
   vB =:[(xdy vB4 ydx vB3 ydx vB2 ydx vB1 ydx vB0 Xi)yRef

   (gXT([(vA(((2^_44);2^_29)qteq[;])"5 vB)hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

$\begin{gather*} \begin{align*} \intertext {\texttt {... Bianchi Identity ... } } \begin{split} & \Lambda_{.\,jkmn}^i + \Xi_{.\,jkmn}^i = 0 \end{split} \\ \end{align*} \end{gather*}$

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

   gXcw  =: 1.2  0.3  _    _    _  "_
   gXpAll=: 1.1  1.2  1.3  1.4  1.5"_

   xCpts=:_ 1 3 26,_ 0 1p1 26,_ 0 2p1 26,:_ 0 20 26"_

NB. ... for Bianchi Identity ...

   *./^:5"5 (0=]) ((2^_43)qtsz]) (gXT([(Lam+Xi)xRef@hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   *./^:5"5 (0=]) ((2^_44)qtsz]) (gXT([(Lam+Xi)yRef@hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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. ... or ...

   (gXT([(Lam(((2^_43);2^_22)qteq[;])"5 -@Xi)xRef@hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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([(Lam(((2^_44);2^_25)qteq[;])"5 -@Xi)yRef@hkxF)]) ((0.1;50"_)pTRandom xCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1