Essays/Christoffel/Christoffel04

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10.5 Covariant Differentiation (ISS Section 33)
GCH0400C.jpg


NB. ... script (ijs) ...

f=:(*:@x1**:@sin@x2*x3)"1


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

   $f xpgen''
8000


GCH0401C.jpg


NB. ... script (ijs) ...

A=:((2*x1**:@sin@x2*x3),(2**:@x1*sin@x2*cos@x2*x3),*:@x1**:@sin@x2)"1


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

   $A xpgen''
8000 3


...

... check using the 'D.' primitive ...

...


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

   +/@:((A((2^_21)teq[;])f D.1)"1) xpgen''
8000


GCH0403C.jpg


NB. ... script (ijs) ...

B=:axs@((0;dxdy),0;A)"1


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

   $B xpgen''
8000 3


...

... check using the 'D.' primitive ...

...


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

NB. ... example ...

   nzmin B xpgen''
0.00220366
   nzmin (2^_21)&tsz@(f@cxx D.1@cyy)"1 xpgen''
0.00220367
   nzmax B xpgen''
29.1275
   nzmax (2^_21)&tsz@(f@cxx D.1@cyy)"1 xpgen''
29.1275
   nzcount B xpgen''
16000
   nzcount (2^_21)&tsz@(f@cxx D.1@cyy)"1 xpgen''
16000

NB. ... so ...

   +/@:((B((2^_19)teq[;])(2^_21)&tsz@(f@cxx D.1@cyy))"1) xpgen''
8000


GCH0405C.jpg


NB. ... script (ijs) ...

dAdx0=:(2**:@sin@x2*x3),(4*x1*(sin@x2*cos@x2)*x3),2*x1**:@sin@x2
dAdx1=:(4*x1*(sin@x2*cos@x2)*x3),(2**:@x1*(*:@cos@x2-*:@sin@x2)*x3),2**:@x1*sin@x2*cos@x2
dAdx2=:(2*x1**:@sin@x2),(2**:@x1*sin@x2*cos@x2),0:
dAdx =:(3 3$dAdx0,dAdx1,dAdx2)"1


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

   $dAdx xpgen''
8000 3 3


...

... check using the 'D.' primitive ...

...


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

   +/@:((dAdx((2^_19)teq[;])0|:A D.1)"1) xpgen''
8000


GCH0407C.jpg


NB. ... script (ijs) ...

d2Adxdx00=:0,(4*(sin@x2*cos@x2)*x3),2**:@sin@x2
d2Adxdx01=:(4*(sin@x2*cos@x2)*x3),(4*x1*(*:@cos@x2-*:@sin@x2)*x3),4*x1*sin@x2*cos@x2
d2Adxdx02=:(2**:@sin@x2),(4*x1*sin@x2*cos@x2),0:

d2Adxdx10=:(4*(sin@x2*cos@x2)*x3),(4*x1*(*:@cos@x2-*:@sin@x2)*x3),4*x1*sin@x2*cos@x2
d2Adxdx11=:(4*x1*(*:@cos@x2-*:@sin@x2)*x3),(-@(8**:@x1*(sin@x2*cos@x2)*x3)),2**:@x1**:@cos@x2-*:@sin@x2
d2Adxdx12=:(4*x1*sin@x2*cos@x2),(2**:@x1**:@cos@x2-*:@sin@x2),0:

d2Adxdx20=:(2**:@sin@x2),(4*x1*sin@x2*cos@x2),0:
d2Adxdx21=:(4*x1*sin@x2*cos@x2),(2**:@x1**:@cos@x2-*:@sin@x2),0:
d2Adxdx22=:0,0,0:

d2Adxdx0=:d2Adxdx00,d2Adxdx01,d2Adxdx02
d2Adxdx1=:d2Adxdx10,d2Adxdx11,d2Adxdx12
d2Adxdx2=:d2Adxdx20,d2Adxdx21,d2Adxdx22
d2Adxdx =:(3 3 3$d2Adxdx0,d2Adxdx1,d2Adxdx2)"1


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

   $d2Adxdx xpgen''
8000 3 3 3


...

... check using the 'D.' primitive ...

...


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

   +/@:((d2Adxdx((2^_19)teq[;])0|:dAdx D.1)"1) xpgen''
8000


GCH0409C.jpg


NB. ... script (ijs) ...

dBdyt1p0=:axs@((0;dxdy),1;dAdx)
dBdyt1  =:axs@((0;dxdy),1;dBdyt1p0)
dBdyt2p0=:axs@((2;d2xdydx),0;dxdy)
dBdyt2  =:axs@((0;dBdyt2p0),0;A)
dBdy    =:(dBdyt1(+tz)dBdyt2)"1


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

   $dBdy xpgen''
8000 3 3


...

... check using the 'D.' primitive ...

...


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

   +/@:((dBdy((2^_19)teq[;])(2^_23)&tsz@(0|:B@cxx D.1@cyy))"1) xpgen''
8000


GCH0411C.jpg


NB. ... script (ijs) ...

Acv=:(dAdx(-tz)axs@((2;hC2k),0;A))"1


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

   $Acv xpgen''
8000 3 3


GCH0412C.jpg


NB. ... script (ijs) ...

Bcv=:(dBdy(-tz)axs@((2;gC2k),0;B))"1


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

   $Bcv xpgen''
8000 3 3


GCH0413C.jpg


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

   s0t1p0=:axs@((0;dxdy),1;Acv)
   s0t1  =:axs@((0;dxdy),1;s0t1p0)
   +/@:((Bcv-:s0t1)"1) xpgen''
8000


GCH0414C.jpg


GCH0415C.jpg


GCH0416C.jpg


NB. ... script (ijs) ...

dAcvdx=:(d2Adxdx(-tz)axs@((2;dhC2kdx),0;A)(+tz)axs@((2;hC2k),0;dAdx))"1


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

   $dAcvdx xpgen''
8000 3 3 3


...

... check using the 'D.' primitive ...

...


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

   +/@:((dAcvdx((2^_19)teq[;])0|:Acv D.1)"1) xpgen''
8000


GCH0418C.jpg


NB. ... script (ijs) ...

Acvcvp0=:(dAcvdx(-tz)(1|:axs@((2;hC2k),0;Acv))(+tz)0|:0|:axs@((2;hC2k),1;Acv))"1
Acvcv  =:(2^_42)&tsz@Acvcvp0      NB. special


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

   $Acvcv xpgen''
8000 3 3 3

   nzmin Acvcvp0 xpgen''
5.68434e_14
   nzmin Acvcv xpgen''
0.0244717


GCH0419C.jpg


NB. ... script (ijs) ...

Bcvcvp0=:axs@((0;dxdy),2;Acvcv)
Bcvcvp1=:axs@((0;dxdy),2;Bcvcvp0)
Bcvcv  =:axs@((0;dxdy),2;Bcvcvp1)"1


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

   $Bcvcv xpgen''
8000 3 3 3


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

NB. ... for this xpgen ...
NB. ... values calculated are within a reasonable range ...

   nf0=:cxx;cyy;dxdy;dydx;d2xdydx;d2ydxdx;h20;h02;dh20dx;hC1k;hC2k;dhC2kdx
   nf1=:f;A;B;dAdx;d2Adxdx;dBdy;Acv;Bcv;dAcvdx;Acvcv;Bcvcv

   (0j_3&":@nzmin;0j_3&":@nzmax;6&":@nzcount)&>@(nf0,nf1) xpgen''
┌────────┬───────┬──────┐
│1.571e_2│1.020e1│ 24000│
├────────┼───────┼──────┤
│1.224e_2│9.877e0│ 24000│
├────────┼───────┼──────┤
│2.447e_3│1.263e1│ 64000│
├────────┼───────┼──────┤
│1.224e_2│9.877e0│ 64000│
├────────┼───────┼──────┤
│2.447e_4│7.973e1│152000│
├────────┼───────┼──────┤
│1.224e_2│9.877e0│152000│
├────────┼───────┼──────┤
│6.118e_3│1.000e2│ 24000│
├────────┼───────┼──────┤
│1.000e_2│1.635e2│ 24000│
├────────┼───────┼──────┤
│2.447e_2│9.978e1│ 24000│
├────────┼───────┼──────┤
│1.224e_2│4.989e1│ 72000│
├────────┼───────┼──────┤
│1.224e_2│1.000e1│ 72000│
├────────┼───────┼──────┤
│1.000e_2│4.086e1│ 80000│
├────────┼───────┼──────┤
│9.610e_4│1.379e2│  8000│
├────────┼───────┼──────┤
│3.844e_3│1.411e2│ 24000│
├────────┼───────┼──────┤
│2.204e_3│2.913e1│ 16000│
├────────┼───────┼──────┤
│5.191e_3│2.689e2│ 64000│
├────────┼───────┼──────┤
│2.076e_2│5.643e2│152000│
├────────┼───────┼──────┤
│5.142e_3│3.136e0│ 32000│
├────────┼───────┼──────┤
│1.922e_3│2.758e2│ 72000│
├────────┼───────┼──────┤
│5.142e_3│3.136e0│ 32000│
├────────┼───────┼──────┤
│7.688e_3│2.821e2│176000│
├────────┼───────┼──────┤
│2.447e_2│3.902e2│  8000│
├────────┼───────┼──────┤
│1.550e_3│4.927e1│ 64000│
└────────┴───────┴──────┘



Download: File:LCH0400C.txt
Download: File:LCH0401C.txt
Download: File:LCH0403C.txt
Download: File:LCH0405C.txt
Download: File:LCH0407C.txt
Download: File:LCH0409C.txt
Download: File:LCH0411C.txt
Download: File:LCH0412C.txt
Download: File:LCH0413C.txt
Download: File:LCH0414C.txt
Download: File:LCH0415C.txt
Download: File:LCH0416C.txt
Download: File:LCH0418C.txt
Download: File:LCH0419C.txt




Download MoinMoin source: File:Christoffel04.ijs




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Contributed by Tom Allen