# User:John Randall/ExteriorAlgebra

[Still under construction]

## Contents

## Introduction

We define , the vector space of -vectors in a real vector space of dimension . These are essentially lists for with an equivalence relation. More precisely, we will defined a -vector as the formal symbol , the wedge product of .

The wedge product generalizes several ideas. Essentially, the determinant is an -vector, the wedge product of its rows, and in the case , is . The interpretation of the latter as a vector is misleading: this is an accident of dimension.

The reader who is interested in the concrete formulation can skip the next section.

## Formal description

We define , the space of -vectors on , for as follows. Each is a vector space. The bottom dimension spaces are given by and . For , we define , as follows: Every element of can be written (non uniquely) as a formal symbol where is multilinear, antisymmetric and associative which satisfies the conditions:

1. is linear in each of the . 2. If and are vector in , then , and (antisymmetry). 3. If are linearly dependent, then .

A basis for can be given as follows. Let be an ordered basis for (the ordering corresponds to a choice of handedness). Then each can be written as a linear combination , and a basis for is given by of ordered -fold wedge products of the basis for given by , where . This gives the dimension of as . For , any set of vectors must be linearly dependent, so .

The *exterior algebra* of is
.

Regardless of the basis, there is a natural homomorphism .
If has an inner product, there is also a natural isomorphism
(*Hodge duality*).
This is important when interpreting cross product in .

## Concrete interpretation

A -vector in can be represented (non uniquely) as a matrix, so

]A=:3 5 $ ? 15#10 6 4 0 6 4 9 8 5 7 4 0 1 2 7 9

represents a 3-vector in .

There is a product, called *wedge product*, that given a -vector and a
-vector, produces a -vector. In terms of matrices, this is just
given by concatenation.

wedge=:, ]B=:2 5 $ ? 10#10 3 2 1 2 9 5 3 9 6 9 A wedge B 6 4 0 6 4 9 8 5 7 4 0 1 2 7 9 3 2 1 2 9 5 3 9 6 9

This representation is not unique. If is a -vector represented by a matrix , then

(a) =0 if the rows of are linearly dependent

(b) swapping two rows of replaces by (antisymmetry)

(c) multiplying a row of by a constant multiplies by that constant (multilinearity)

(d) adding a multiple of one row of to another does not change

A basis for the space of -vectors is given by -fold wedge
products of distinct vectors of size with exactly ones,
lexicographically ordered.
Consequently the vector space has
dimension ` k!n `.
For example,

0 1 0 1 0 1 1 0

is one of the basis vector for 2-vectors in .

With respect to this basis, the coordinates of a -vector represented by a matrix are obtained by for a basis vector can be calculated by using to select columns of and then taking the determinant of the resulting matrix.

det=:-/ .* cols=:{&.|: comb=: 4 : 0 k=. i.>:d=.y-x z=. (d$<i.0 0),<i.1 0 for. i.x do. z=. k ,.&.> ,&.>/\. >:&.> z end. ; z ) coords=:det@cols"1 _~ comb/@: $ ]C=: 2 5 $ i.10 0 1 2 3 4 5 6 7 8 9 coords C _5 _10 _15 _20 _5 _10 _15 _5 _10 _5

### Determinant

The space of -vectors in has a single coordinate. If represents an -vector , then its coordinate is .

### Levi-Civita symbol

The *Levi-Civita symbol* or *complete tensor* gives the wedge product
of basis vectors in .
These need not be distinct, and order matters.
If ` I ` is a list of numbers in ` i.n ` of length ` n `, then
` M =:I {=@. i. n ` is a matrix representing an -vector
the wedge product of the corresponding basis elements, and so has one coordinate .
By the rules for equivalence of k-vectors:

If ` I `has repeated indices, then .

Otherwise ` I ` is a permutation of ` i.n `, and is the parity of
` I { i.n. `

The calculation is given by the complete tensor CT.

CT =: C.!.2 @ (#:i.) @ $~ I=:0 2 1 ]M=:I { =@i.3 1 0 0 0 0 1 0 1 0 coords M _1 (< I) { CT 3 _1 J=:0 2 2 coords J { =@i.3 0 (< J) { CT 3 0

### Cross product

If and are vectors in , is a 2-vector, while is a 1-vector. We can reconcile these by considering the standard ordered orthonormal basis for , The ordering corresponds to the right hand rule.

We can then identify a 2-vector with a 1-vector by defining an operation given by its action on the basis:

, , .

i=:,: 1 0 0 j=:,: 0 1 0 k=:,: 0 0 1 star=:1 _1 1 * |. @ coords cross=:star @ wedge (i cross j),(j cross k),:(k cross i) 0 0 1 1 0 0 0 1 0 (,:1 2 3) cross (,: 4 5 6) _3 6 _3

### Area and volume

Coming soon.