# wedge v1.0-3

Monthly downloads

## The Exterior Calculus

Provides functionality for working with differentials,
k-forms, wedge products, Stokes's theorem, and related concepts
from the exterior calculus. The canonical reference would be:
M. Spivak (1965, ISBN:0-8053-9021-9). "Calculus on Manifolds",
Benjamin Cummings.

## Readme

# The wedge package: exterior calculus in R

# Overview

The `wedge`

package provides functionality for working with the exterior
calculus. It includes cross 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 -tensors and -forms. A -tensor is a multilinear map , where is considered as a vector space. Given two -tensors the package can calculate their outer product using natural R idiom (see below and the vignette for details).

A -form is an alternating -tensor, that is a -tensor with the property that linear dependence of implies that . Given -forms , the package provides R idiom for calculating their wedge product .

# Installation

You can install the released version of wedge from CRAN with:

```
# install.packages("wedge") # uncomment this to install the package
library("wedge")
```

# The `wedge`

package in use

The package has two main classes of objects, `kform`

and `ktensor`

. We
may define a -tensor as
follows

```
KT <- as.ktensor(cbind(1:4,3:5),1:4)
#> Warning in cbind(1:4, 3:5): number of rows of result is not a multiple of
#> vector length (arg 2)
KT
#> val
#> 1 3 = 1
#> 2 4 = 2
#> 3 5 = 3
#> 4 3 = 4
```

We can coerce `KT`

to a function and then evaluate it:

```
KT <- as.ktensor(cbind(1:4,2:5),1:4)
f <- as.function(KT)
E <- matrix(rnorm(10),5,2)
f(E)
#> [1] -0.1383716
```

Cross products are implemented:

```
KT %X% KT
#> val
#> 3 4 3 4 = 9
#> 2 3 1 2 = 2
#> 2 3 2 3 = 4
#> 3 4 1 2 = 3
#> 4 5 1 2 = 4
#> 1 2 1 2 = 1
#> 1 2 2 3 = 2
#> 2 3 3 4 = 6
#> 3 4 2 3 = 6
#> 4 5 4 5 = 16
#> 4 5 2 3 = 8
#> 1 2 3 4 = 3
#> 4 5 3 4 = 12
#> 1 2 4 5 = 4
#> 2 3 4 5 = 8
#> 3 4 4 5 = 12
```

## Alternating forms

An alternating form (or -form) is an antisymmetric
-tensor; the package can
convert a general -tensor to alternating form using the `Alt()`

function:

```
Alt(KT)
#> val
#> 1 2 = 0.5
#> 2 1 = -0.5
#> 4 3 = -1.5
#> 2 3 = 1.0
#> 3 2 = -1.0
#> 5 4 = -2.0
#> 3 4 = 1.5
#> 4 5 = 2.0
```

However, the package provides a bespoke and efficient representation for
-forms as objects with
class `kform`

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

function:

```
M <- matrix(c(4,2,3,1,2,4),2,3,byrow=TRUE)
M
#> [,1] [,2] [,3]
#> [1,] 4 2 3
#> [2,] 1 2 4
KF <- as.kform(M,c(1,5))
KF
#> val
#> 2 3 4 = 1
#> 1 2 4 = 5
```

We may coerce `KF`

to functional form:

```
f <- as.function(KF)
E <- matrix(rnorm(12),4,3)
f(E)
#> [1] -1.895993
```

# The wedge product

The wedge product of two -forms is implemented as `%^%`

or `wedge()`

:

```
KF2 <- kform_general(6:9,2,1:6)
KF2
#> val
#> 6 7 = 1
#> 6 8 = 2
#> 7 9 = 5
#> 7 8 = 3
#> 6 9 = 4
#> 8 9 = 6
KF %^% KF2
#> val
#> 2 3 4 6 7 = 1
#> 1 2 4 6 8 = 10
#> 1 2 4 6 9 = 20
#> 2 3 4 7 9 = 5
#> 1 2 4 7 9 = 25
#> 2 3 4 6 8 = 2
#> 1 2 4 6 7 = 5
#> 2 3 4 8 9 = 6
#> 2 3 4 6 9 = 4
#> 2 3 4 7 8 = 3
#> 1 2 4 7 8 = 15
#> 1 2 4 8 9 = 30
```

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

```
dx <- as.kform(1)
dy <- as.kform(2)
dz <- as.kform(3)
dx %^% dy %^% dz # element of volume
#> 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)
#> val
#> 1 = 1
#> 2 = 2
#> 3 = 3
#> 4 = 4
#> 5 = 5
#> 6 = 6
```

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

```
grad(1:4) %^% grad(1:6)
#> val
#> 2 5 = 10
#> 3 5 = 15
#> 3 6 = 18
#> 1 5 = 5
#> 2 6 = 12
#> 4 5 = 20
#> 1 6 = 6
#> 4 6 = 24
```

# References

The most concise reference is

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

But an accessible book would be

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

# Further information

For more detail, see the package vignette

`vignette("wedge")`

## Functions in wedge

Name | Description | |

inner | Inner product operator | |

contract | Contractions of \(k\)-forms | |

hodge | Hodge star operator | |

as.1form | Coerce vectors to 1-forms | |

Alt | Alternating multilinear forms | |

issmall | Is a form zero to within numerical precision? | |

keep | Keep or drop variables | |

consolidate | Various low-level helper functions | |

cross | Cross products of \(k\)-tensors | |

Ops.kform | Arithmetic Ops Group Methods for kform and ktensor objects | |

kform | k-forms | |

wedge-package | wedge | |

volume | The volume element | |

symbolic | Symbolic form | |

transform | Linear transforms of \(k\)-forms | |

ktensor | k-tensors | |

zero | Zero tensors and zero forms | |

wedge | Wedge products | |

rform | Random kforms and ktensors | |

scalar | Lose attributes | |

No Results! |

## Vignettes of wedge

Name | ||

wedge.Rmd | ||

No Results! |

## Last month downloads

## Details

Type | Package |

VignetteBuilder | knitr |

License | GPL-2 |

URL | https://github.com/RobinHankin/wedge.git |

BugReports | https://github.com/RobinHankin/wedge/issues |

NeedsCompilation | no |

Packaged | 2019-09-04 02:20:51 UTC; rhankin |

Repository | CRAN |

Date/Publication | 2019-09-04 05:10:03 UTC |

suggests | Deriv , knitr , testthat |

imports | magrittr , methods , partitions , permutations (>= 1.0-4) |

depends | spray (>= 1.0-7) |

Contributors |

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