Managing covariate and contravariate indices

Each index of a tensor can be covariate or contravariate. The is.* routines check the state of the individual indices based on the tensor, its dimension or its index names. drag.tensor can change the state for the tensor and contraname for the names of the tensor.

drag.tensor(x,g,d)
contraname(x)
is.covariate(x,...)
# S3 method for class 'tensor'
is.covariate(x,...)
# S3 method for class 'numeric'
is.covariate(x,...)
# S3 method for class 'character'
is.covariate(x,...)
as.covariate(x,...)
# S3 method for class 'character'
as.covariate(x,...)
is.contravariate(x,...)
# S3 method for class 'numeric'
is.contravariate(x,...)
# S3 method for class 'character'
is.contravariate(x,...)
as.contravariate(x,...)
# S3 method for class 'character'
as.contravariate(x,...)

Arguments

x

the tensor, its dimension (for *.numeric) or its index-names (for *.character and contraname)

g

The geometry tensor \(g_{ij}\) giving the transformation between covariate and contravariate. It needs to have either covariate and or contravariate indices.

d

a vector (or list) of indices that should be dragged, i.e. multiplied with \(g_i^{\;j}\) in the right way such that it changes from covariate to contravariate or vice versa. The name of the index is kept, only its state changes. The index is thus dragged from one state to the other. Indices can given in covariate or contravariate form.

...

only for generic use

Value

drag.tensor

returns a tensor like x but with the dimension

is.covariate

returns a boolean vector giving true for every covariate index

is.contravariate

returns a boolean vector giving true for every contravariate index

as.*

changes the state of the indices

contraname

returns the names with opposite the opposite covariate and contravariate state

Details

The covariate and contravariate state of a dimension corresponds to column and row vectors. The transformation between these type is done by a linear mapping give by the geometry tensor g, which is the identity matrix if the enclosing the geometry is represented by the orthonormal basis and ordinary scalar product.

See also

riemann.tensor, to.tensor, Tensor

Author

K. Gerald van den Boogaart

Examples

g <- to.tensor(c(1,2,0,1),c(i=2,j=2))
A <- to.tensor(rnorm(8),c(a=2,b=2,c=2))
A2 <- drag.tensor(A,g,c("b","c"))
A2
#> , , 1
#> 
#>       ^b
#> a            [,1]       [,2]
#>   [1,] 0.01917759  0.5114724
#>   [2,] 0.02956075 -2.3332364
#> 
#> , , 2
#> 
#>       ^b
#> a            [,1]      [,2]
#>   [1,]  2.6442020 -6.174703
#>   [2,] -0.4203428  6.463261
#> 
#> attr(,"class")
#> [1] "tensor"
names(A2)
#> [1] "a"  "^b" "^c"
as.covariate(names(A2))
#> [1] "a" "b" "c"
as.contravariate(names(A2))
#> [1] "^a" "^b" "^c"
is.covariate(A2)
#> [1]  TRUE FALSE FALSE
is.contravariate(A2)
#> [1] FALSE  TRUE  TRUE
riemann.tensor(A2,g)
#> , , 1, 1, 1
#> 
#>       ^b
#> a            [,1]       [,2]
#>   [1,] 0.01917759  0.5114724
#>   [2,] 0.02956075 -2.3332364
#> 
#> , , 2, 1, 1
#> 
#>       ^b
#> a            [,1]      [,2]
#>   [1,]  2.6442020 -6.174703
#>   [2,] -0.4203428  6.463261
#> 
#> , , 1, 2, 1
#> 
#>       ^b
#> a            [,1]      [,2]
#>   [1,] 0.03835518  1.022945
#>   [2,] 0.05912151 -4.666473
#> 
#> , , 2, 2, 1
#> 
#>       ^b
#> a            [,1]      [,2]
#>   [1,]  5.2884040 -12.34941
#>   [2,] -0.8406855  12.92652
#> 
#> , , 1, 1, 2
#> 
#>       ^b
#> a      [,1] [,2]
#>   [1,]    0    0
#>   [2,]    0    0
#> 
#> , , 2, 1, 2
#> 
#>       ^b
#> a      [,1] [,2]
#>   [1,]    0    0
#>   [2,]    0    0
#> 
#> , , 1, 2, 2
#> 
#>       ^b
#> a            [,1]       [,2]
#>   [1,] 0.01917759  0.5114724
#>   [2,] 0.02956075 -2.3332364
#> 
#> , , 2, 2, 2
#> 
#>       ^b
#> a            [,1]      [,2]
#>   [1,]  2.6442020 -6.174703
#>   [2,] -0.4203428  6.463261
#> 
#> attr(,"class")
#> [1] "tensor"