Section 0: Introduction

Forms are antisymmetric linear functionals. To be specific, let's consider the real vector space R^n (not necessarily equipped with an inner product). A k-form (where k is a natural number) is a linear antisymmetric functional on R^n. It should be noted that the set of all k-forms is a real vector space if we define addition and scalar multiplication pointwisely. The dimension of this space is C(n,k) if k<n+1. If k>n, the condition of antisymmetry will force the k-form to be the zero map and hence the space to become the zero space.

We can define a wedge product (also known as exterior multiplication) of forms, which maps a pair of k- and l- forms vk and vl to a (k+l)-form vk^vl. The definition is as follows, where w1, w2 and w3 are forms of arbitrary order:

Input := 

<<forms.defs.m;

Input := 

??^

x^y gives x to the power y.

Attributes[Power] = {Listable, OneIdentity}
 
(w1_)^(w2_) := Wedge[w1, w2] /; !NumberQ[w2] || w2 == 0
 
Power/: Default[Power, 2] := 1

Input := 

??Wedge

Global`Wedge

Attributes[Wedge] = {Flat, OneIdentity}
 
Wedge[forms__][vectors__] := 
  Signature[{forms}]*Signature /@ Permutations[{forms}] . 
    (ApplyFormList[#1, {vectors}] & ) /@ 
     Permutations[{forms}]
 
Wedge[0, w1_] := 0
 
Wedge[w1_, 0] := 0
 
Wedge[(w1_) + (w2_), w3_] := Wedge[w1, w3] + Wedge[w2, w3]
 
Wedge[w1_, (w2_) + (w3_)] := Wedge[w1, w2] + Wedge[w1, w3]
 
Wedge[(a_)*(w1_), w2_] := a*Wedge[w1, w2] /; NumberQ[a]
 
Wedge[w1_, (a_)*(w2_)] := a*Wedge[w1, w2] /; NumberQ[a]
 
Wedge[forms__] := 0 /; Signature[{forms}] == 0
 
Wedge[forms__] := 
  Signature[{forms}]*Apply[Wedge, Sort[{forms}]] /; 
   !OrderedQ[{forms}]
 
Format[Wedge[w1_, w2__]] := Infix[{w1, w2}, " ^ "]
 
Format[Wedge[w1_]] := w1

Note that the wedge product is linear, associative, distributive and skew-commutative: if wk and wl are k- and l-forms respectively, then wk^wl = (-1)^(k*l) wl^wk.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).