Essays/SpaceTime2D/SpaceTime2D02
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3 Derivatives
... 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
... 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
... 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
... 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
... 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
... 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
... 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
... 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
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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Contributed by Tom Allen