The `stokes`

package provides functionality for working
with the exterior calculus. It includes tensor products and wedge
products and a variety of use-cases. The canonical reference would be
Spivak (see references). A detailed vignette is provided in the
package.

The package deals with

A

You can install the released version of `stokes`

from CRAN with:

```
# install.packages("stokes") # uncomment this to install the package
library("stokes")
set.seed(0)
```

`stokes`

package in
useThe package has two main classes of objects, `kform`

and
`ktensor`

. In the package, we can create a `as.ktensor()`

a matrix of indices and a
vector of coefficents, for example:

```
<- as.ktensor(rbind(1:3,2:4),1:2)
jj
jj#> A linear map from V^3 to R with V=R^4:
#> val
#> 2 3 4 = 2
#> 1 2 3 = 1
```

Above, object `jj`

is equal to

We can coerce tensors to a function and then evaluate it:

```
<- as.ktensor(cbind(1:4,2:5),1:4)
KT <- as.function(KT)
f <- matrix(rnorm(10),5,2)
E f(E)
#> [1] 11.23556
```

Tensor products are implemented:

```
%X% KT
KT #> A linear map from V^4 to R with V=R^5:
#> val
#> 1 2 1 2 = 1
#> 2 3 1 2 = 2
#> 3 4 3 4 = 9
#> 2 3 4 5 = 8
#> 1 2 2 3 = 2
#> 1 2 4 5 = 4
#> 4 5 4 5 = 16
#> 2 3 3 4 = 6
#> 4 5 3 4 = 12
#> 1 2 3 4 = 3
#> 3 4 4 5 = 12
#> 3 4 2 3 = 6
#> 4 5 2 3 = 8
#> 3 4 1 2 = 3
#> 2 3 2 3 = 4
#> 4 5 1 2 = 4
```

Above we see

An alternating form (or `Alt()`

function:

```
Alt(KT)
#> A linear map from V^2 to R with V=R^5:
#> val
#> 5 4 = -2.0
#> 4 5 = 2.0
#> 4 3 = -1.5
#> 3 2 = -1.0
#> 2 3 = 1.0
#> 3 4 = 1.5
#> 2 1 = -0.5
#> 1 2 = 0.5
```

However, the package provides a bespoke and efficient representation
for `kform`

. Such
objects may be created using the `as.kform()`

function:

```
<- matrix(c(4,2,3,1,2,4),2,3,byrow=TRUE)
M
M#> [,1] [,2] [,3]
#> [1,] 4 2 3
#> [2,] 1 2 4
<- as.kform(M,c(1,5))
KF
KF#> An alternating linear map from V^3 to R with V=R^4:
#> val
#> 1 2 4 = 5
#> 2 3 4 = 1
```

Above, we see that `KF`

is equal to `KF`

to functional form:

```
<- as.function(KF)
f <- matrix(rnorm(12),4,3)
E f(E)
#> [1] -5.979544
```

Above, we evaluate `KF`

at a point in `E`

are each interpreted as vectors in

The wedge product of two `^`

or `wedge()`

:

```
<- kform_general(6:9,2,1:6)
KF2
KF2#> An alternating linear map from V^2 to R with V=R^9:
#> val
#> 8 9 = 6
#> 7 9 = 5
#> 6 9 = 4
#> 7 8 = 3
#> 6 8 = 2
#> 6 7 = 1
^ KF2
KF #> An alternating linear map from V^5 to R with V=R^9:
#> val
#> 1 2 4 6 7 = 5
#> 1 2 4 6 8 = 10
#> 2 3 4 6 8 = 2
#> 2 3 4 7 8 = 3
#> 2 3 4 6 9 = 4
#> 1 2 4 6 9 = 20
#> 2 3 4 6 7 = 1
#> 2 3 4 7 9 = 5
#> 1 2 4 7 8 = 15
#> 2 3 4 8 9 = 6
#> 1 2 4 7 9 = 25
#> 1 2 4 8 9 = 30
```

The package can accommodate a number of results from the exterior calculus such as elementary forms:

```
<- as.kform(1)
dx <- as.kform(2)
dy <- as.kform(3)
dz ^ dy ^ dz # element of volume
dx #> An alternating linear map from V^3 to R with V=R^3:
#> val
#> 1 2 3 = 1
```

A number of useful functions from the exterior calculus are provided, such as the gradient of a scalar function:

```
grad(1:6)
#> An alternating linear map from V^1 to R with V=R^6:
#> val
#> 6 = 6
#> 5 = 5
#> 4 = 4
#> 3 = 3
#> 2 = 2
#> 1 = 1
```

The package takes the leg-work out of the exterior calculus:

```
grad(1:4) ^ grad(1:6)
#> An alternating linear map from V^2 to R with V=R^6:
#> val
#> 4 5 = 20
#> 1 5 = 5
#> 2 5 = 10
#> 3 5 = 15
#> 2 6 = 12
#> 4 6 = 24
#> 3 6 = 18
#> 1 6 = 6
```

The most concise reference is

- Spivak 1971.
*Calculus on manifolds*, Addison-Wesley.

But a more leisurely book would be

- Hubbard and Hubbard 2015.
*Vector calculus, linear algebra, and differential forms: a unified approach*. Matrix Editions

For more detail, see the package vignette

`vignette("stokes")`