Essays/SpaceTime2D/SpaceTime2D02

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3 Derivatives

GST0200C.jpg


... document ...

   x1R1
┌───────────┬─┬─┐
│┌───┬─┬───┐│"│1│
││cos│@│cv2││ │ │
│└───┴─┴───┘│ │ │
└───────────┴─┴─┘

   x1R2
┌─────────────────────────┬─┬─┐
│┌───┬─┬─────────────────┐│"│1│
││cv1│*│┌─────────┬─┬───┐││ │ │
││   │ ││┌─┬─┬───┐│@│cv2│││ │ │
││   │ │││-│@│sin││ │   │││ │ │
││   │ ││└─┴─┴───┘│ │   │││ │ │
││   │ │└─────────┴─┴───┘││ │ │
│└───┴─┴─────────────────┘│ │ │
└─────────────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x1R1=:cos@cv2"1
x1R2=:(cv1*-@sin@cv2)"1

x1v1=:x1R1
x1v2=:x1R2

x1dv=:(x1v1,x1v2)"1


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

   p2aUd1=.0.25;0.25;2;11.62;1
   p2aUd2=.0 0.646447 30,:0 2p1 100

   p2a1d1=.p2aUd1 x1dv vGen(p2aUd2)
   p2a1d2=.p2aUd1([:(p2aUd1 x1])D.1])"1 vGen(p2aUd2)
   ((2^_22)&gXtsz p2a1d1)((2^_17)gXteq[;])((2^_22)&gXtsz p2a1d2)
1
   (p2a1d1=.0),p2a1d2=.0
0 0


GST0201C.jpg


... document ...

   x1R11
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x1R12
┌─────────────────┬─┬─┐
│┌─────────┬─┬───┐│"│1│
││┌─┬─┬───┐│@│cv2││ │ │
│││-│@│sin││ │   ││ │ │
││└─┴─┴───┘│ │   ││ │ │
│└─────────┴─┴───┘│ │ │
└─────────────────┴─┴─┘

   x1R22
┌─────────────────────────┬─┬─┐
│┌───┬─┬─────────────────┐│"│1│
││cv1│*│┌─────────┬─┬───┐││ │ │
││   │ ││┌─┬─┬───┐│@│cv2│││ │ │
││   │ │││-│@│cos││ │   │││ │ │
││   │ ││└─┴─┴───┘│ │   │││ │ │
││   │ │└─────────┴─┴───┘││ │ │
│└───┴─┴─────────────────┘│ │ │
└─────────────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x1R11=:0:"1
x1R12=:-@sin@cv2"1
x1R22=:(cv1*-@cos@cv2)"1

x1v1v1=:x1R11
x1v1v2=:x1R12
x1v2v1=:x1R12
x1v2v2=:x1R22

x1dvdv=:((x1v1v1,x1v1v2),:x1v2v1,x1v2v2)"1


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

   p2aVd1=.0.25;0.25;2;11.62;1
   p2aVd2=.0 0.646447 30,:0 2p1 100

   p2a2d1=.p2aVd1 x1dvdv vGen(p2aVd2)
   p2a2d2=.p2aVd1(0|:[:(p2aVd1 x1dv])D.1])"1 vGen(p2aVd2)
   ((2^_21)&gXtsz p2a2d1)((2^_17)gXteq[;])((2^_21)&gXtsz p2a2d2)
1
   (p2a2d1=.0),p2a2d2=.0
0 0


GST0202C.jpg


... document ...

   x1R111
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x1R112
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x1R122
┌─────────────────┬─┬─┐
│┌─────────┬─┬───┐│"│1│
││┌─┬─┬───┐│@│cv2││ │ │
│││-│@│cos││ │   ││ │ │
││└─┴─┴───┘│ │   ││ │ │
│└─────────┴─┴───┘│ │ │
└─────────────────┴─┴─┘

   x1R222
┌───────────────────┬─┬─┐
│┌───┬─┬───────────┐│"│1│
││cv1│*│┌───┬─┬───┐││ │ │
││   │ ││sin│@│cv2│││ │ │
││   │ │└───┴─┴───┘││ │ │
│└───┴─┴───────────┘│ │ │
└───────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x1R111=:0:"1
x1R112=:0:"1
x1R122=:-@cos@cv2"1
x1R222=:(cv1*sin@cv2)"1

x1v1v1v1=:x1R111
x1v1v1v2=:x1R112
x1v1v2v1=:x1R112
x1v1v2v2=:x1R122

x1v2v1v1=:x1R112
x1v2v1v2=:x1R122
x1v2v2v1=:x1R122
x1v2v2v2=:x1R222

x1dvdvdv=:(((x1v1v1v1,x1v1v1v2),:x1v1v2v1,x1v1v2v2),:(x1v2v1v1,x1v2v1v2),:x1v2v2v1,x1v2v2v2)"1


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

   p2aWd1=.0.25;0.25;2;11.62;1
   p2aWd2=.0 0.646447 30,:0 2p1 100

   p2a3d1=.p2aWd1 x1dvdvdv vGen(p2aWd2)
   p2a3d2=.p2aWd1(0|:[:(p2aWd1 x1dvdv])D.1])"1 vGen(p2aWd2)
   ((2^_22)&gXtsz p2a3d1)((2^_17)gXteq[;])((2^_22)&gXtsz p2a3d2)
1
   (p2a3d1=.0),p2a3d2=.0
0 0


GST0203C.jpg


... document ...

   x1R1111
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x1R1112
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x1R1122
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x1R1222
┌───────────┬─┬─┐
│┌───┬─┬───┐│"│1│
││sin│@│cv2││ │ │
│└───┴─┴───┘│ │ │
└───────────┴─┴─┘

   x1R2222
┌───────────────────┬─┬─┐
│┌───┬─┬───────────┐│"│1│
││cv1│*│┌───┬─┬───┐││ │ │
││   │ ││cos│@│cv2│││ │ │
││   │ │└───┴─┴───┘││ │ │
│└───┴─┴───────────┘│ │ │
└───────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x1R1111=:0:"1
x1R1112=:0:"1
x1R1122=:0:"1
x1R1222=:sin@cv2"1
x1R2222=:(cv1*cos@cv2)"1

x1v1v1v1v1=:x1R1111
x1v1v1v1v2=:x1R1112
x1v1v1v2v1=:x1R1112
x1v1v1v2v2=:x1R1122

x1v1v2v1v1=:x1R1112
x1v1v2v1v2=:x1R1122
x1v1v2v2v1=:x1R1122
x1v1v2v2v2=:x1R1222


x1v2v1v1v1=:x1R1112
x1v2v1v1v2=:x1R1122
x1v2v1v2v1=:x1R1122
x1v2v1v2v2=:x1R1222

x1v2v2v1v1=:x1R1122
x1v2v2v1v2=:x1R1222
x1v2v2v2v1=:x1R1222
x1v2v2v2v2=:x1R2222

x1dvdvdvdvp1=:((x1v1v1v1v1,x1v1v1v1v2),:x1v1v1v2v1,x1v1v1v2v2),:(x1v1v2v1v1,x1v1v2v1v2),:x1v1v2v2v1,x1v1v2v2v2
x1dvdvdvdvp2=:((x1v2v1v1v1,x1v2v1v1v2),:x1v2v1v2v1,x1v2v1v2v2),:(x1v2v2v1v1,x1v2v2v1v2),:x1v2v2v2v1,x1v2v2v2v2
x1dvdvdvdv  =:(x1dvdvdvdvp1,:x1dvdvdvdvp2)"1


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

   p2aXd1=.0.25;0.25;2;11.62;1
   p2aXd2=.0 0.646447 30,:0 2p1 100

   p2a4d1=.p2aXd1 x1dvdvdvdv vGen(p2aXd2)
   p2a4d2=.p2aXd1(0|:[:(p2aXd1 x1dvdvdv])D.1])"1 vGen(p2aXd2)
   ((2^_21)&gXtsz p2a4d1)((2^_17)gXteq[;])((2^_21)&gXtsz p2a4d2)
1
   (p2a4d1=.0),p2a4d2=.0
0 0


GST0204C.jpg


... document ...

   x2R1
┌───────────┬─┬─┐
│┌───┬─┬───┐│"│1│
││sin│@│cv2││ │ │
│└───┴─┴───┘│ │ │
└───────────┴─┴─┘

   x2R2
┌───────────────────┬─┬─┐
│┌───┬─┬───────────┐│"│1│
││cv1│*│┌───┬─┬───┐││ │ │
││   │ ││cos│@│cv2│││ │ │
││   │ │└───┴─┴───┘││ │ │
│└───┴─┴───────────┘│ │ │
└───────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x2R1=:sin@cv2"1
x2R2=:(cv1*cos@cv2)"1

x2v1=:x2R1
x2v2=:x2R2

x2dv=:(x2v1,x2v2)"1


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

   p2bUd1=.0.25;0.25;2;11.62;1
   p2bUd2=.0 0.646447 30,:0 2p1 100

   p2b1d1=.p2bUd1 x2dv vGen(p2bUd2)
   p2b1d2=.p2bUd1([:(p2bUd1 x2])D.1])"1 vGen(p2bUd2)
   ((2^_22)&gXtsz p2b1d1)((2^_18)gXteq[;])((2^_22)&gXtsz p2b1d2)
1
   (p2b1d1=.0),p2b1d2=.0
0 0


GST0205C.jpg


... document ...

   x2R11
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x2R12
┌───────────┬─┬─┐
│┌───┬─┬───┐│"│1│
││cos│@│cv2││ │ │
│└───┴─┴───┘│ │ │
└───────────┴─┴─┘

   x2R22
┌─────────────────────────┬─┬─┐
│┌───┬─┬─────────────────┐│"│1│
││cv1│*│┌─────────┬─┬───┐││ │ │
││   │ ││┌─┬─┬───┐│@│cv2│││ │ │
││   │ │││-│@│sin││ │   │││ │ │
││   │ ││└─┴─┴───┘│ │   │││ │ │
││   │ │└─────────┴─┴───┘││ │ │
│└───┴─┴─────────────────┘│ │ │
└─────────────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x2R11=:0:"1
x2R12=:cos@cv2"1
x2R22=:(cv1*-@sin@cv2)"1

x2v1v1=:x2R11
x2v1v2=:x2R12
x2v2v1=:x2R12
x2v2v2=:x2R22

x2dvdv=:((x2v1v1,x2v1v2),:x2v2v1,x2v2v2)"1


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

   p2bVd1=.0.25;0.25;2;11.62;1
   p2bVd2=.0 0.646447 30,:0 2p1 100

   p2b2d1=.p2bVd1 x2dvdv vGen(p2bVd2)
   p2b2d2=.p2bVd1(0|:[:(p2bVd1 x2dv])D.1])"1 vGen(p2bVd2)
   ((2^_22)&gXtsz p2b2d1)((2^_17)gXteq[;])((2^_22)&gXtsz p2b2d2)
1
   (p2b2d1=.0),p2b2d2=.0
0 0


GST0206C.jpg


... document ...

   x2R111
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x2R112
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x2R122
┌─────────────────┬─┬─┐
│┌─────────┬─┬───┐│"│1│
││┌─┬─┬───┐│@│cv2││ │ │
│││-│@│sin││ │   ││ │ │
││└─┴─┴───┘│ │   ││ │ │
│└─────────┴─┴───┘│ │ │
└─────────────────┴─┴─┘

   x2R222
┌─────────────────────────┬─┬─┐
│┌───┬─┬─────────────────┐│"│1│
││cv1│*│┌─────────┬─┬───┐││ │ │
││   │ ││┌─┬─┬───┐│@│cv2│││ │ │
││   │ │││-│@│cos││ │   │││ │ │
││   │ ││└─┴─┴───┘│ │   │││ │ │
││   │ │└─────────┴─┴───┘││ │ │
│└───┴─┴─────────────────┘│ │ │
└─────────────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x2R111=:0:"1
x2R112=:0:"1
x2R122=:-@sin@cv2"1
x2R222=:(cv1*-@cos@cv2)"1

x2v1v1v1=:x2R111
x2v1v1v2=:x2R112
x2v1v2v1=:x2R112
x2v1v2v2=:x2R122

x2v2v1v1=:x2R112
x2v2v1v2=:x2R122
x2v2v2v1=:x2R122
x2v2v2v2=:x2R222

x2dvdvdv=:(((x2v1v1v1,x2v1v1v2),:x2v1v2v1,x2v1v2v2),:(x2v2v1v1,x2v2v1v2),:x2v2v2v1,x2v2v2v2)"1


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

   p2bWd1=.0.25;0.25;2;11.62;1
   p2bWd2=.0 0.646447 30,:0 2p1 100

   p2b3d1=.p2bWd1 x2dvdvdv vGen(p2bWd2)
   p2b3d2=.p2bWd1(0|:[:(p2bWd1 x2dvdv])D.1])"1 vGen(p2bWd2)
   ((2^_21)&gXtsz p2b3d1)((2^_17)gXteq[;])((2^_21)&gXtsz p2b3d2)
1
   (p2b3d1=.0),p2b3d2=.0
0 0


GST0207C.jpg


... document ...

   x2R1111
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x2R1112
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x2R1122
┌──┬─┬─┐
│0:│"│1│
└──┴─┴─┘

   x2R1222
┌─────────────────┬─┬─┐
│┌─────────┬─┬───┐│"│1│
││┌─┬─┬───┐│@│cv2││ │ │
│││-│@│cos││ │   ││ │ │
││└─┴─┴───┘│ │   ││ │ │
│└─────────┴─┴───┘│ │ │
└─────────────────┴─┴─┘

   x2R2222
┌───────────────────┬─┬─┐
│┌───┬─┬───────────┐│"│1│
││cv1│*│┌───┬─┬───┐││ │ │
││   │ ││sin│@│cv2│││ │ │
││   │ │└───┴─┴───┘││ │ │
│└───┴─┴───────────┘│ │ │
└───────────────────┴─┴─┘


NB. ... script SpaceTime2D.ijs (continued) ...

x2R1111=:0:"1
x2R1112=:0:"1
x2R1122=:0:"1
x2R1222=:-@cos@cv2"1
x2R2222=:(cv1*sin@cv2)"1

x2v1v1v1v1=:x2R1111
x2v1v1v1v2=:x2R1112
x2v1v1v2v1=:x2R1112
x2v1v1v2v2=:x2R1122

x2v1v2v1v1=:x2R1112
x2v1v2v1v2=:x2R1122
x2v1v2v2v1=:x2R1122
x2v1v2v2v2=:x2R1222


x2v2v1v1v1=:x2R1112
x2v2v1v1v2=:x2R1122
x2v2v1v2v1=:x2R1122
x2v2v1v2v2=:x2R1222

x2v2v2v1v1=:x2R1122
x2v2v2v1v2=:x2R1222
x2v2v2v2v1=:x2R1222
x2v2v2v2v2=:x2R2222

x2dvdvdvdvp1=:((x2v1v1v1v1,x2v1v1v1v2),:x2v1v1v2v1,x2v1v1v2v2),:(x2v1v2v1v1,x2v1v2v1v2),:x2v1v2v2v1,x2v1v2v2v2
x2dvdvdvdvp2=:((x2v2v1v1v1,x2v2v1v1v2),:x2v2v1v2v1,x2v2v1v2v2),:(x2v2v2v1v1,x2v2v2v1v2),:x2v2v2v2v1,x2v2v2v2v2
x2dvdvdvdv  =:(x2dvdvdvdvp1,:x2dvdvdvdvp2)"1


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

   p2bXd1=.0.25;0.25;2;11.62;1
   p2bXd2=.0 0.646447 30,:0 2p1 100

   p2b4d1=.p2bXd1 x2dvdvdvdv vGen(p2bXd2)
   p2b4d2=.p2bXd1(0|:[:(p2bXd1 x2dvdvdv])D.1])"1 vGen(p2bXd2)
   ((2^_22)&gXtsz p2b4d1)((2^_17)gXteq[;])((2^_22)&gXtsz p2b4d2)
1
   (p2b4d1=.0),p2b4d2=.0
0 0


GST0208C.jpg


NB. ... script SpaceTime2D.ijs (continued) ...

xcdv      =:(x1dv,:x2dv)"1
xcdvdv    =:(x1dvdv,:x2dvdv)"1
xcdvdvdv  =:(x1dvdvdv,:x2dvdvdv)"1
xcdvdvdvdv=:(x1dvdvdvdv,:x2dvdvdvdv)"1


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

   p2cUd1=.0.25;0.25;2;11.62;1
   p2cUd2=.0 0.646447 30,:0 2p1 100

   p2c1d1=.p2cUd1 xcdv vGen(p2cUd2)
   p2c1d2=.p2cUd1(0|:[:(p2cUd1 xc])D.1])"1 vGen(p2cUd2)
   ((2^_22)&gXtsz p2c1d1)((2^_17)gXteq[;])((2^_22)&gXtsz p2c1d2)
1
   (p2c1d1=.0),p2c1d2=.0
0 0

   p2c2d1=.p2cUd1 xcdvdv vGen(p2cUd2)
   p2c2d2=.p2cUd1(0|:[:(p2cUd1 xcdv])D.1])"1 vGen(p2cUd2)
   ((2^_21)&gXtsz p2c2d1)((2^_17)gXteq[;])((2^_21)&gXtsz p2c2d2)
1
   (p2c2d1=.0),p2c2d2=.0
0 0

   p2c3d1=.p2cUd1 xcdvdvdv vGen(p2cUd2)
   p2c3d2=.p2cUd1(0|:[:(p2cUd1 xcdvdv])D.1])"1 vGen(p2cUd2)
   ((2^_21)&gXtsz p2c3d1)((2^_17)gXteq[;])((2^_21)&gXtsz p2c3d2)
1
   (p2c3d1=.0),p2c3d2=.0
0 0

   p2c4d1=.p2cUd1 xcdvdvdvdv vGen(p2cUd2)
   p2c4d2=.p2cUd1(0|:[:(p2cUd1 xcdvdvdv])D.1])"1 vGen(p2cUd2)
   ((2^_21)&gXtsz p2c4d1)((2^_17)gXteq[;])((2^_21)&gXtsz p2c4d2)
1
   (p2c4d1=.0),p2c4d2=.0
0 0



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Download: File:LST0206C.txt
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Download: File:LST0208C.txt




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Contributed by Tom Allen