stokes-package.RdProvides functionality for working with tensors, alternating forms, wedge products, Stokes's theorem, and related concepts from the exterior calculus. Uses 'disordR' discipline (Hankin, 2022, <doi:10.48550/arXiv.2210.03856>). The canonical reference would be M. Spivak (1965, ISBN:0-8053-9021-9) "Calculus on Manifolds". To cite the package in publications please use Hankin (2022) <doi:10.48550/arXiv.2210.17008>.
The DESCRIPTION file:
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Generally in the package, arguments that are \(k\)-forms are
denoted K, \(k\)-tensors by U, and spray objects by
S. Multilinear maps (which may be either \(k\)-forms or
\(k\)-tensors) are denoted by M.
M. Spivak 1971. Calculus on manifolds, Addison-Wesley.
R. K. S. Hankin 2022. “Disordered vectors in R: introducing the disordR package.” https://arxiv.org/abs/2210.03856.
R. K. S. Hankin 2022. “Sparse arrays in R: the spray package. https://arxiv.org/abs/2210.03856.”
## Some k-tensors:
U1 <- as.ktensor(matrix(1:15,5,3))
U2 <- as.ktensor(cbind(1:3,2:4),1:3)
## Coerce a tensor to functional form, here mapping V^3 -> R (here V=R^15):
as.function(U1)(matrix(rnorm(45),15,3))
#> [1] -2.150885
## Tensor product is tensorprod() or %X%:
U1 %X% U2
#> A linear map from V^5 to R with V=R^15:
#> val
#> 1 6 11 1 2 = 1
#> 2 7 12 1 2 = 1
#> 4 9 14 3 4 = 3
#> 1 6 11 3 4 = 3
#> 3 8 13 3 4 = 3
#> 5 10 15 3 4 = 3
#> 3 8 13 1 2 = 1
#> 1 6 11 2 3 = 2
#> 2 7 12 3 4 = 3
#> 5 10 15 2 3 = 2
#> 4 9 14 2 3 = 2
#> 3 8 13 2 3 = 2
#> 2 7 12 2 3 = 2
#> 5 10 15 1 2 = 1
#> 4 9 14 1 2 = 1
## A k-form is an alternating k-tensor:
K1 <- as.kform(cbind(1:5,2:6),rnorm(5))
K2 <- kform_general(3:6,2,1:6)
K3 <- rform(9,3,9,runif(9))
## The distributive law is true
(K1 + K2) ^ K3 == K1 ^ K3 + K2 ^ K3 # TRUE to numerical precision
#> [1] FALSE
## Wedge product is associative (non-trivial):
(K1 ^ K2) ^ K3
#> An alternating linear map from V^7 to R with V=R^9:
#> val
#> 1 2 3 4 5 6 8 = -4.2549730
#> 1 2 3 4 5 7 8 = -0.3540383
#> 1 2 3 4 5 6 9 = -1.3075286
#> 1 2 3 5 6 7 8 = -0.7080766
#> 1 2 3 4 6 7 8 = -0.5900639
#> 1 3 4 5 6 7 8 = 1.1734427
K1 ^ (K2 ^ K3)
#> An alternating linear map from V^7 to R with V=R^9:
#> val
#> 1 2 3 5 6 7 8 = -0.7080766
#> 1 2 3 4 5 6 9 = -1.3075286
#> 1 3 4 5 6 7 8 = 1.1734427
#> 1 2 3 4 6 7 8 = -0.5900639
#> 1 2 3 4 5 7 8 = -0.3540383
#> 1 2 3 4 5 6 8 = -4.2549730
## k-forms can be coerced to a function and wedge product:
f <- as.function(K1 ^ K2 ^ K3)
## E is a a random point in V^k:
E <- matrix(rnorm(63),9,7)
## f() is alternating:
f(E)
#> [1] 31.5171
f(E[,7:1])
#> [1] -31.5171
## The package blurs the distinction between symbolic and numeric computing:
dx <- as.kform(1)
dy <- as.kform(2)
dz <- as.kform(3)
dx ^ dy ^ dz
#> An alternating linear map from V^3 to R with V=R^3:
#> val
#> 1 2 3 = 1
K3 ^ dx ^ dy ^ dz
#> An alternating linear map from V^6 to R with V=R^9:
#> val
#> 1 2 3 4 6 9 = 0.6131014
#> 1 2 3 5 6 9 = -0.3961379
#> 1 2 3 4 5 8 = -1.3198595