User:John Randall/ExteriorAlgebra

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[Still under construction]

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.