einstein.RdMultiplies tensors by multiplying over all duplicate names according to Einsteins summing convention by doing an implicit inner product over all dimensions with the same name.
einstein.tensor(...,only=NULL,by=NULL)
## Methods for class tensor
# x %e% y
## Default method
# x %e% ythe tensor product of all the tensors along all duplicate dimensions.
see mul.tensor on details on tensor
multiplication. In einstein.tensor complex operations can be
performed by command and renaming code: The arguments are processed
from left to right and multiplied. Unnamed attributes are regarded as
tensors or scalars and
multiplied with the current result by the Einstein summing convention,
which means an inner product over all dimensions with the same
name. Named attributes can either have the name diag, which performs a
diagmul according to the same-name convention or be of the form
A="B" or "A"="B", for which we have two cases. If both
names are
present in the current result, an inner multiplication (trace) of on
these two dimensions is
performed. If only the first is a name up to this point, the specific
dimension is renamed to the second name. This renaming might be
visible in the result or inducing a multiplication according to the
Einstein convention later.
A <- to.tensor(1:20,c(U=2,V=2,W=5))
B <- to.tensor(1:30,list(U=c("a","b","c"),V=c("B1","B2"),W=1:5))
einstein.tensor(A,U="U'",B)
#> U
#> U' a b c
#> [1,] 1945 2045 2145
#> [2,] 2090 2200 2310
#> attr(,"class")
#> [1] "tensor" "matrix"
einstein.tensor(A,U="U'",mark(B,"k"))
#> , , 1, Uk = a, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 1 3
#> [2,] 2 4
#>
#> , , 2, Uk = a, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 5 7
#> [2,] 6 8
#>
#> , , 3, Uk = a, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 9 11
#> [2,] 10 12
#>
#> , , 4, Uk = a, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 13 15
#> [2,] 14 16
#>
#> , , 5, Uk = a, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 17 19
#> [2,] 18 20
#>
#> , , 1, Uk = b, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 2 6
#> [2,] 4 8
#>
#> , , 2, Uk = b, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 10 14
#> [2,] 12 16
#>
#> , , 3, Uk = b, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 18 22
#> [2,] 20 24
#>
#> , , 4, Uk = b, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 26 30
#> [2,] 28 32
#>
#> , , 5, Uk = b, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 34 38
#> [2,] 36 40
#>
#> , , 1, Uk = c, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 3 9
#> [2,] 6 12
#>
#> , , 2, Uk = c, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 15 21
#> [2,] 18 24
#>
#> , , 3, Uk = c, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 27 33
#> [2,] 30 36
#>
#> , , 4, Uk = c, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 39 45
#> [2,] 42 48
#>
#> , , 5, Uk = c, Vk = B1, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 51 57
#> [2,] 54 60
#>
#> , , 1, Uk = a, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 4 12
#> [2,] 8 16
#>
#> , , 2, Uk = a, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 20 28
#> [2,] 24 32
#>
#> , , 3, Uk = a, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 36 44
#> [2,] 40 48
#>
#> , , 4, Uk = a, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 52 60
#> [2,] 56 64
#>
#> , , 5, Uk = a, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 68 76
#> [2,] 72 80
#>
#> , , 1, Uk = b, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 5 15
#> [2,] 10 20
#>
#> , , 2, Uk = b, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 25 35
#> [2,] 30 40
#>
#> , , 3, Uk = b, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 45 55
#> [2,] 50 60
#>
#> , , 4, Uk = b, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 65 75
#> [2,] 70 80
#>
#> , , 5, Uk = b, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 85 95
#> [2,] 90 100
#>
#> , , 1, Uk = c, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 6 18
#> [2,] 12 24
#>
#> , , 2, Uk = c, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 30 42
#> [2,] 36 48
#>
#> , , 3, Uk = c, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 54 66
#> [2,] 60 72
#>
#> , , 4, Uk = c, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 78 90
#> [2,] 84 96
#>
#> , , 5, Uk = c, Vk = B2, Wk = 1
#>
#> V
#> U' [,1] [,2]
#> [1,] 102 114
#> [2,] 108 120
#>
#> , , 1, Uk = a, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 7 21
#> [2,] 14 28
#>
#> , , 2, Uk = a, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 35 49
#> [2,] 42 56
#>
#> , , 3, Uk = a, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 63 77
#> [2,] 70 84
#>
#> , , 4, Uk = a, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 91 105
#> [2,] 98 112
#>
#> , , 5, Uk = a, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 119 133
#> [2,] 126 140
#>
#> , , 1, Uk = b, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 8 24
#> [2,] 16 32
#>
#> , , 2, Uk = b, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 40 56
#> [2,] 48 64
#>
#> , , 3, Uk = b, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 72 88
#> [2,] 80 96
#>
#> , , 4, Uk = b, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 104 120
#> [2,] 112 128
#>
#> , , 5, Uk = b, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 136 152
#> [2,] 144 160
#>
#> , , 1, Uk = c, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 9 27
#> [2,] 18 36
#>
#> , , 2, Uk = c, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 45 63
#> [2,] 54 72
#>
#> , , 3, Uk = c, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 81 99
#> [2,] 90 108
#>
#> , , 4, Uk = c, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 117 135
#> [2,] 126 144
#>
#> , , 5, Uk = c, Vk = B1, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 153 171
#> [2,] 162 180
#>
#> , , 1, Uk = a, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 10 30
#> [2,] 20 40
#>
#> , , 2, Uk = a, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 50 70
#> [2,] 60 80
#>
#> , , 3, Uk = a, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 90 110
#> [2,] 100 120
#>
#> , , 4, Uk = a, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 130 150
#> [2,] 140 160
#>
#> , , 5, Uk = a, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 170 190
#> [2,] 180 200
#>
#> , , 1, Uk = b, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 11 33
#> [2,] 22 44
#>
#> , , 2, Uk = b, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 55 77
#> [2,] 66 88
#>
#> , , 3, Uk = b, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 99 121
#> [2,] 110 132
#>
#> , , 4, Uk = b, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 143 165
#> [2,] 154 176
#>
#> , , 5, Uk = b, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 187 209
#> [2,] 198 220
#>
#> , , 1, Uk = c, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 12 36
#> [2,] 24 48
#>
#> , , 2, Uk = c, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 60 84
#> [2,] 72 96
#>
#> , , 3, Uk = c, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 108 132
#> [2,] 120 144
#>
#> , , 4, Uk = c, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 156 180
#> [2,] 168 192
#>
#> , , 5, Uk = c, Vk = B2, Wk = 2
#>
#> V
#> U' [,1] [,2]
#> [1,] 204 228
#> [2,] 216 240
#>
#> , , 1, Uk = a, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 13 39
#> [2,] 26 52
#>
#> , , 2, Uk = a, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 65 91
#> [2,] 78 104
#>
#> , , 3, Uk = a, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 117 143
#> [2,] 130 156
#>
#> , , 4, Uk = a, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 169 195
#> [2,] 182 208
#>
#> , , 5, Uk = a, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 221 247
#> [2,] 234 260
#>
#> , , 1, Uk = b, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 14 42
#> [2,] 28 56
#>
#> , , 2, Uk = b, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 70 98
#> [2,] 84 112
#>
#> , , 3, Uk = b, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 126 154
#> [2,] 140 168
#>
#> , , 4, Uk = b, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 182 210
#> [2,] 196 224
#>
#> , , 5, Uk = b, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 238 266
#> [2,] 252 280
#>
#> , , 1, Uk = c, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 15 45
#> [2,] 30 60
#>
#> , , 2, Uk = c, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 75 105
#> [2,] 90 120
#>
#> , , 3, Uk = c, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 135 165
#> [2,] 150 180
#>
#> , , 4, Uk = c, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 195 225
#> [2,] 210 240
#>
#> , , 5, Uk = c, Vk = B1, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 255 285
#> [2,] 270 300
#>
#> , , 1, Uk = a, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 16 48
#> [2,] 32 64
#>
#> , , 2, Uk = a, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 80 112
#> [2,] 96 128
#>
#> , , 3, Uk = a, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 144 176
#> [2,] 160 192
#>
#> , , 4, Uk = a, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 208 240
#> [2,] 224 256
#>
#> , , 5, Uk = a, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 272 304
#> [2,] 288 320
#>
#> , , 1, Uk = b, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 17 51
#> [2,] 34 68
#>
#> , , 2, Uk = b, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 85 119
#> [2,] 102 136
#>
#> , , 3, Uk = b, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 153 187
#> [2,] 170 204
#>
#> , , 4, Uk = b, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 221 255
#> [2,] 238 272
#>
#> , , 5, Uk = b, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 289 323
#> [2,] 306 340
#>
#> , , 1, Uk = c, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 18 54
#> [2,] 36 72
#>
#> , , 2, Uk = c, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 90 126
#> [2,] 108 144
#>
#> , , 3, Uk = c, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 162 198
#> [2,] 180 216
#>
#> , , 4, Uk = c, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 234 270
#> [2,] 252 288
#>
#> , , 5, Uk = c, Vk = B2, Wk = 3
#>
#> V
#> U' [,1] [,2]
#> [1,] 306 342
#> [2,] 324 360
#>
#> , , 1, Uk = a, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 19 57
#> [2,] 38 76
#>
#> , , 2, Uk = a, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 95 133
#> [2,] 114 152
#>
#> , , 3, Uk = a, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 171 209
#> [2,] 190 228
#>
#> , , 4, Uk = a, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 247 285
#> [2,] 266 304
#>
#> , , 5, Uk = a, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 323 361
#> [2,] 342 380
#>
#> , , 1, Uk = b, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 20 60
#> [2,] 40 80
#>
#> , , 2, Uk = b, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 100 140
#> [2,] 120 160
#>
#> , , 3, Uk = b, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 180 220
#> [2,] 200 240
#>
#> , , 4, Uk = b, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 260 300
#> [2,] 280 320
#>
#> , , 5, Uk = b, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 340 380
#> [2,] 360 400
#>
#> , , 1, Uk = c, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 21 63
#> [2,] 42 84
#>
#> , , 2, Uk = c, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 105 147
#> [2,] 126 168
#>
#> , , 3, Uk = c, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 189 231
#> [2,] 210 252
#>
#> , , 4, Uk = c, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 273 315
#> [2,] 294 336
#>
#> , , 5, Uk = c, Vk = B1, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 357 399
#> [2,] 378 420
#>
#> , , 1, Uk = a, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 22 66
#> [2,] 44 88
#>
#> , , 2, Uk = a, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 110 154
#> [2,] 132 176
#>
#> , , 3, Uk = a, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 198 242
#> [2,] 220 264
#>
#> , , 4, Uk = a, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 286 330
#> [2,] 308 352
#>
#> , , 5, Uk = a, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 374 418
#> [2,] 396 440
#>
#> , , 1, Uk = b, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 23 69
#> [2,] 46 92
#>
#> , , 2, Uk = b, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 115 161
#> [2,] 138 184
#>
#> , , 3, Uk = b, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 207 253
#> [2,] 230 276
#>
#> , , 4, Uk = b, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 299 345
#> [2,] 322 368
#>
#> , , 5, Uk = b, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 391 437
#> [2,] 414 460
#>
#> , , 1, Uk = c, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 24 72
#> [2,] 48 96
#>
#> , , 2, Uk = c, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 120 168
#> [2,] 144 192
#>
#> , , 3, Uk = c, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 216 264
#> [2,] 240 288
#>
#> , , 4, Uk = c, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 312 360
#> [2,] 336 384
#>
#> , , 5, Uk = c, Vk = B2, Wk = 4
#>
#> V
#> U' [,1] [,2]
#> [1,] 408 456
#> [2,] 432 480
#>
#> , , 1, Uk = a, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 25 75
#> [2,] 50 100
#>
#> , , 2, Uk = a, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 125 175
#> [2,] 150 200
#>
#> , , 3, Uk = a, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 225 275
#> [2,] 250 300
#>
#> , , 4, Uk = a, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 325 375
#> [2,] 350 400
#>
#> , , 5, Uk = a, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 425 475
#> [2,] 450 500
#>
#> , , 1, Uk = b, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 26 78
#> [2,] 52 104
#>
#> , , 2, Uk = b, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 130 182
#> [2,] 156 208
#>
#> , , 3, Uk = b, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 234 286
#> [2,] 260 312
#>
#> , , 4, Uk = b, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 338 390
#> [2,] 364 416
#>
#> , , 5, Uk = b, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 442 494
#> [2,] 468 520
#>
#> , , 1, Uk = c, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 27 81
#> [2,] 54 108
#>
#> , , 2, Uk = c, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 135 189
#> [2,] 162 216
#>
#> , , 3, Uk = c, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 243 297
#> [2,] 270 324
#>
#> , , 4, Uk = c, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 351 405
#> [2,] 378 432
#>
#> , , 5, Uk = c, Vk = B1, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 459 513
#> [2,] 486 540
#>
#> , , 1, Uk = a, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 28 84
#> [2,] 56 112
#>
#> , , 2, Uk = a, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 140 196
#> [2,] 168 224
#>
#> , , 3, Uk = a, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 252 308
#> [2,] 280 336
#>
#> , , 4, Uk = a, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 364 420
#> [2,] 392 448
#>
#> , , 5, Uk = a, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 476 532
#> [2,] 504 560
#>
#> , , 1, Uk = b, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 29 87
#> [2,] 58 116
#>
#> , , 2, Uk = b, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 145 203
#> [2,] 174 232
#>
#> , , 3, Uk = b, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 261 319
#> [2,] 290 348
#>
#> , , 4, Uk = b, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 377 435
#> [2,] 406 464
#>
#> , , 5, Uk = b, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 493 551
#> [2,] 522 580
#>
#> , , 1, Uk = c, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 30 90
#> [2,] 60 120
#>
#> , , 2, Uk = c, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 150 210
#> [2,] 180 240
#>
#> , , 3, Uk = c, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 270 330
#> [2,] 300 360
#>
#> , , 4, Uk = c, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 390 450
#> [2,] 420 480
#>
#> , , 5, Uk = c, Vk = B2, Wk = 5
#>
#> V
#> U' [,1] [,2]
#> [1,] 510 570
#> [2,] 540 600
#>
#> attr(,"class")
#> [1] "tensor"
einstein.tensor(A,U="U'",mark(B,"k"),V="Vk",W="Wk")
#> Uk
#> U' a b c
#> [1,] 1945 2045 2145
#> [2,] 2090 2200 2310
#> attr(,"class")
#> [1] "tensor" "matrix"
einstein.tensor(A,U="U'",mark(B,"k"),V="Vk",W="Wk",1/10)
#> Uk
#> U' a b c
#> [1,] 194.5 204.5 214.5
#> [2,] 209.0 220.0 231.0
#> attr(,"class")
#> [1] "tensor" "matrix"
einstein.tensor(A,U="U'",mark(B,"k"),V="Vk",W="Wk",diag=to.tensor(c(1,1/10,1/100),c(Uk=3)))
#> Uk
#> U' a b c
#> [1,] 1945 204.5 21.45
#> [2,] 2090 220.0 23.10
#> attr(,"class")
#> [1] "tensor" "matrix"
ftable(einstein.tensor(A,U="U'",B))
#> U a b c
#> U'
#> U'1 1945 2045 2145
#> U'2 2090 2200 2310
ftable(einstein.tensor(A,U="U'",mark(B,"k")))
#> Wk 1 2 3 4 5
#> U' V W Uk Vk
#> U'1 V1 W1 a B1 1 7 13 19 25
#> B2 4 10 16 22 28
#> b B1 2 8 14 20 26
#> B2 5 11 17 23 29
#> c B1 3 9 15 21 27
#> B2 6 12 18 24 30
#> W2 a B1 5 35 65 95 125
#> B2 20 50 80 110 140
#> b B1 10 40 70 100 130
#> B2 25 55 85 115 145
#> c B1 15 45 75 105 135
#> B2 30 60 90 120 150
#> W3 a B1 9 63 117 171 225
#> B2 36 90 144 198 252
#> b B1 18 72 126 180 234
#> B2 45 99 153 207 261
#> c B1 27 81 135 189 243
#> B2 54 108 162 216 270
#> W4 a B1 13 91 169 247 325
#> B2 52 130 208 286 364
#> b B1 26 104 182 260 338
#> B2 65 143 221 299 377
#> c B1 39 117 195 273 351
#> B2 78 156 234 312 390
#> W5 a B1 17 119 221 323 425
#> B2 68 170 272 374 476
#> b B1 34 136 238 340 442
#> B2 85 187 289 391 493
#> c B1 51 153 255 357 459
#> B2 102 204 306 408 510
#> V2 W1 a B1 3 21 39 57 75
#> B2 12 30 48 66 84
#> b B1 6 24 42 60 78
#> B2 15 33 51 69 87
#> c B1 9 27 45 63 81
#> B2 18 36 54 72 90
#> W2 a B1 7 49 91 133 175
#> B2 28 70 112 154 196
#> b B1 14 56 98 140 182
#> B2 35 77 119 161 203
#> c B1 21 63 105 147 189
#> B2 42 84 126 168 210
#> W3 a B1 11 77 143 209 275
#> B2 44 110 176 242 308
#> b B1 22 88 154 220 286
#> B2 55 121 187 253 319
#> c B1 33 99 165 231 297
#> B2 66 132 198 264 330
#> W4 a B1 15 105 195 285 375
#> B2 60 150 240 330 420
#> b B1 30 120 210 300 390
#> B2 75 165 255 345 435
#> c B1 45 135 225 315 405
#> B2 90 180 270 360 450
#> W5 a B1 19 133 247 361 475
#> B2 76 190 304 418 532
#> b B1 38 152 266 380 494
#> B2 95 209 323 437 551
#> c B1 57 171 285 399 513
#> B2 114 228 342 456 570
#> U'2 V1 W1 a B1 2 14 26 38 50
#> B2 8 20 32 44 56
#> b B1 4 16 28 40 52
#> B2 10 22 34 46 58
#> c B1 6 18 30 42 54
#> B2 12 24 36 48 60
#> W2 a B1 6 42 78 114 150
#> B2 24 60 96 132 168
#> b B1 12 48 84 120 156
#> B2 30 66 102 138 174
#> c B1 18 54 90 126 162
#> B2 36 72 108 144 180
#> W3 a B1 10 70 130 190 250
#> B2 40 100 160 220 280
#> b B1 20 80 140 200 260
#> B2 50 110 170 230 290
#> c B1 30 90 150 210 270
#> B2 60 120 180 240 300
#> W4 a B1 14 98 182 266 350
#> B2 56 140 224 308 392
#> b B1 28 112 196 280 364
#> B2 70 154 238 322 406
#> c B1 42 126 210 294 378
#> B2 84 168 252 336 420
#> W5 a B1 18 126 234 342 450
#> B2 72 180 288 396 504
#> b B1 36 144 252 360 468
#> B2 90 198 306 414 522
#> c B1 54 162 270 378 486
#> B2 108 216 324 432 540
#> V2 W1 a B1 4 28 52 76 100
#> B2 16 40 64 88 112
#> b B1 8 32 56 80 104
#> B2 20 44 68 92 116
#> c B1 12 36 60 84 108
#> B2 24 48 72 96 120
#> W2 a B1 8 56 104 152 200
#> B2 32 80 128 176 224
#> b B1 16 64 112 160 208
#> B2 40 88 136 184 232
#> c B1 24 72 120 168 216
#> B2 48 96 144 192 240
#> W3 a B1 12 84 156 228 300
#> B2 48 120 192 264 336
#> b B1 24 96 168 240 312
#> B2 60 132 204 276 348
#> c B1 36 108 180 252 324
#> B2 72 144 216 288 360
#> W4 a B1 16 112 208 304 400
#> B2 64 160 256 352 448
#> b B1 32 128 224 320 416
#> B2 80 176 272 368 464
#> c B1 48 144 240 336 432
#> B2 96 192 288 384 480
#> W5 a B1 20 140 260 380 500
#> B2 80 200 320 440 560
#> b B1 40 160 280 400 520
#> B2 100 220 340 460 580
#> c B1 60 180 300 420 540
#> B2 120 240 360 480 600
ftable(einstein.tensor(A,U="U'",mark(B,"k"),V="Vk",W="Wk"))
#> Uk a b c
#> U'
#> U'1 1945 2045 2145
#> U'2 2090 2200 2310
ftable(einstein.tensor(A,U="U'",mark(B,"k"),V="Vk",W="Wk",1/10))
#> Uk a b c
#> U'
#> U'1 194.5 204.5 214.5
#> U'2 209.0 220.0 231.0
ftable(einstein.tensor(A,U="U'",mark(B,"k"),V="Vk",W="Wk",diag=to.tensor(c(1,1/10,1/100),c(Uk=3))))
#> Uk a b c
#> U'
#> U'1 1945.00 204.50 21.45
#> U'2 2090.00 220.00 23.10
dim(A[[U=~M]])
#> M V W
#> 2 2 5
A[[U=~M]]
#> , , 1
#>
#> V
#> M [,1] [,2]
#> [1,] 1 3
#> [2,] 2 4
#>
#> , , 2
#>
#> V
#> M [,1] [,2]
#> [1,] 5 7
#> [2,] 6 8
#>
#> , , 3
#>
#> V
#> M [,1] [,2]
#> [1,] 9 11
#> [2,] 10 12
#>
#> , , 4
#>
#> V
#> M [,1] [,2]
#> [1,] 13 15
#> [2,] 14 16
#>
#> , , 5
#>
#> V
#> M [,1] [,2]
#> [1,] 17 19
#> [2,] 18 20
#>
#> attr(,"class")
#> [1] "tensor"
A[[U=~M,V=~"L"]]
#> , , 1
#>
#> L
#> M [,1] [,2]
#> [1,] 1 3
#> [2,] 2 4
#>
#> , , 2
#>
#> L
#> M [,1] [,2]
#> [1,] 5 7
#> [2,] 6 8
#>
#> , , 3
#>
#> L
#> M [,1] [,2]
#> [1,] 9 11
#> [2,] 10 12
#>
#> , , 4
#>
#> L
#> M [,1] [,2]
#> [1,] 13 15
#> [2,] 14 16
#>
#> , , 5
#>
#> L
#> M [,1] [,2]
#> [1,] 17 19
#> [2,] 18 20
#>
#> attr(,"class")
#> [1] "tensor"