Section 1: Algebra of Forms
Addition, Scalar Multiplication and Wedge Product

First, we will define several 1-forms in R^3 for illustration of properties of forms. Below, v, x, y and z are positional vectors while f, g, h and l are 1-forms.

Input := 


v := {v1,v2,v3};
x := {x1,x2,x3};
y := {y1,y2,y3};
z := {z1,z2,z3};
f[x_] := 2 x[[1]] + 3 x[[2]] + x[[3]];
g[x_] := x[[1]] + 4 x[[2]] - x[[3]];
h[x_] := x[[1]] - x[[2]] + 5 x[[3]];
l[x_] := 4 x[[1]] + 7 x[[2]] - 3 x[[3]];

Input := 

Simplify[(f^(3g+2h))[x,y]]

Output =

-5 x2 y1 - 9 x3 y1 + 5 x1 y2 - 11 x3 y2 + 9 x1 y3 + 11 x2 y3

Input := 

Simplify[(3(f^g))[x,y] + (2(f^h))[x,y]]

Output =

-5 x2 y1 - 9 x3 y1 + 5 x1 y2 - 11 x3 y2 + 9 x1 y3 + 11 x2 y3

Input := 

Simplify[((f^(3g+2h)) - (3(f^g) + 2(f^h)))[x,y]]

Output =

0[{x1, x2, x3}, {y1, y2, y3}]

This is to be interpreted as 0. 

Input := 

Simplify[((4f+3g)^h)[x,y]]

Output =

35 x2 y1 - 54 x3 y1 - 35 x1 y2 - 121 x3 y2 + 54 x1 y3 + 
 
  121 x2 y3

Input := 

Simplify[(4(f^h))[x,y] +(3(g^h))[x,y]]

Output =

35 x2 y1 - 54 x3 y1 - 35 x1 y2 - 121 x3 y2 + 54 x1 y3 + 
 
  121 x2 y3

Input := 

Simplify[((4f+3g)^h - (4(f^h) + 3(g^h)))[x,y]]

Output =

0[{x1, x2, x3}, {y1, y2, y3}]

These examples illustrate the linearity of the wedge product wrt both functionals. In general, we have f^( a g + b h) = a f^g + b f^h and (a f + b g)^h = a f^h + b g^h, where a and b are real numbers.

The wedge product of two 1-forms (linear functionals on R) gives a 2-form, which is a bilinear functional on R^2:

Input := 

Simplify[(f^g)[x,y]]

Output =

-5 x2 y1 + 3 x3 y1 + 5 x1 y2 + 7 x3 y2 - 3 x1 y3 - 7 x2 y3

Input := 

Simplify[(g^f)[x,y]]

Output =

5 x2 y1 - 3 x3 y1 - 5 x1 y2 - 7 x3 y2 + 3 x1 y3 + 7 x2 y3

Input := 

Simplify[(f^g)[y,x]]

Output =

5 x2 y1 - 3 x3 y1 - 5 x1 y2 - 7 x3 y2 + 3 x1 y3 + 7 x2 y3

Note the anitisymmetry: (f^g)[x,y] = - (g^f)[x,y]. Also, the wedge product is antisymmetric wrt the arguments of the functionals: (f^g)[x,y] = - (f^g)[y,x]. A direct consequence is therefore: (f^g)[x,y] = (g^f)[y,x]. These will be true for f and g being forms of odd orders. But what if one, or both, form is of even order?

Input := 

Simplify[((f^g)^h)[x,y,z]]

Output =

15 (-(x3 y2 z1) + x2 y3 z1 + x3 y1 z2 - x1 y3 z2 - 
 
    x2 y1 z3 + x1 y2 z3)

Input := 

Simplify[(h^(f^g))[x,y,z]]

Output =

15 (-(x3 y2 z1) + x2 y3 z1 + x3 y1 z2 - x1 y3 z2 - 
 
    x2 y1 z3 + x1 y2 z3)

Here, we see that the wedge product is symmetric instead of being antisymmetric if one of the arguments is a 2-form: F^h = h^F if F is a 2-form and h a 1-form.This can be easily deduced from elementary properties of the wedge products including linearity, associativity and antisymmetry when acting on 1-forms:
As ^ is linear, we may assume without loss of generality that F=f^g where f and g are 1-forms. Then F^h = (f^g)^h = f^(g^h)= f^ (-1 h^g) = -1 (f^h)^g = (-1)^2 (h^f)^g = h^(f^g) = h^F.
Generalizing this argument, we deduce that F^H = H^F if F is a form of even order and H one of arbitrary order.

Without the antisymmetry, it is natural for us to doubt whether it is still true that w^w = 0 if w is a form of even order. The answer is negative. Proof: Consider the 2-form F = f^g + h^l. Then
F^F
= (f^g + h^l) ^(f^g + h^l)
= f^g^f^g + h^l^f^g + f^g^h^l + h^l^hl (parentheses can be omitted since we have associativity)
= -1 f^f^g^g - h^f^l^g + f^g^h^l - h^h^l^l
= 0 + f^h^l^g + f^g^h^l + 0
= -f^h^g^l + f^g^h^l = f^g^h^l + f^g^h^l
= 2 f^g^h^l,
which can be nonzero if f,g,h and l are forms in R^n where n>3.

However, since every k-form on R^n with k>n is zero, we see that if F is a form of nonzero even order on R^n with n<4, F^F is a form of 2k>n and so must be zero. Also, if F is a monomial k-form on R^n (ie, F = w1^w2^...^wk where w1,w2,...,wk are 1-forms), then F^F must be zero, since we can rearrange the order of wedging in the monomial F^F and get w1^w1 as a factor.

Now that we understand the effects of permuting the forms in taking the wedge product, a natural question is: what occurs if we permute the vectors that are arguments of the forms instead of the forms themselves? To understand what is happening, we have to first realize that the product of a 2-form and a 1-form gives a 3-form, ie, a linear functional on R^3.

Input := 

Simplify[((f^g)^h)[x,y,z]]

Output =

15 (-(x3 y2 z1) + x2 y3 z1 + x3 y1 z2 - x1 y3 z2 - 
 
    x2 y1 z3 + x1 y2 z3)

Input := 

Simplify[((f^g)^h)[z,x,y]]

Output =

15 (-(x3 y2 z1) + x2 y3 z1 + x3 y1 z2 - x1 y3 z2 - 
 
    x2 y1 z3 + x1 y2 z3)

Input := 

Simplify[((f^g)^h)[y,x,z]]

Output =

15 (x3 y2 z1 - x2 y3 z1 - x3 y1 z2 + x1 y3 z2 + x2 y1 z3 - 
 
    x1 y2 z3)

The cyclic symmetry is clear. In general, for a k-form with argument [x_1, ..., x_k], an even permutation of the arguments leave the result unchanged while an odd permutation will change the sign. (In fact, an analogous results is true for permutation of forms.)

In general, the wedge product of a k-form and an m-form gives a (k+m)-form, ie, a linear functional on R^(k+m).

An alert reader may have probably noticed that so far we have been talking about forms defined on R^3. What about forms in R^n in general? Here let's consider forms in R^4:

Input := 

v := {v1,v2,v3,v4};
x := {x1,x2,x3,x4};
y := {y1,y2,y3,y4};
z := {z1,z2,z3,z4};
f[x_] := 2 x[[1]] + 3 x[[2]] + x[[3]] + 8 x[[4]];
g[x_] := x[[1]] + 4 x[[2]] - x[[3]] + 2 x[[4]];
h[x_] := x[[1]] - x[[2]] + 5 x[[3]] - 3 x[[4]];
l[x_] := 4 x[[1]] + 7 x[[2]] - 3 x[[3]] - x[[4]];

Input := 

Expand[(f^g)[x,y]]

Output =

-5 x2 y1 + 3 x3 y1 + 4 x4 y1 + 5 x1 y2 + 7 x3 y2 + 
 
  26 x4 y2 - 3 x1 y3 - 7 x2 y3 - 10 x4 y3 - 4 x1 y4 - 
 
  26 x2 y4 + 10 x3 y4

Input := 

Expand[(g^f)[x,y]]

Output =

5 x2 y1 - 3 x3 y1 - 4 x4 y1 - 5 x1 y2 - 7 x3 y2 - 
 
  26 x4 y2 + 3 x1 y3 + 7 x2 y3 + 10 x4 y3 + 4 x1 y4 + 
 
  26 x2 y4 - 10 x3 y4

Input := 

Expand[(f^g)[y,x]]

Output =

5 x2 y1 - 3 x3 y1 - 4 x4 y1 - 5 x1 y2 - 7 x3 y2 - 
 
  26 x4 y2 + 3 x1 y3 + 7 x2 y3 + 10 x4 y3 + 4 x1 y4 + 
 
  26 x2 y4 - 10 x3 y4

We see that we still have the antisymmetry properties aforementioned.

A particular 2-form w =p1^q1 + .. +pn^qn in R^(2n) is of particular interest in classical mechanics. (Knowledgeable readers may complain that we should consider d[p1] instead of p1, etc. But since we haven't intoduced the opeartion d and the discussion below does not make use of the differential properties, let's be content with our imprecise notation.) If we wedge w with itself n times, we get the 2n-form (-1)^(r/2) n! p1^...^pn^q1^...^qn, where r is the largest even integer smaller than n. For example,

Input := 

r2 = (p1^q1 + p2^q2);
r2^r2

Output =

-2 p1 ^ p2 ^ q1 ^ q2

Input := 

r3 = (p1^q1 + p2^q2 + p3^q3);
r3^r3^r3

Output =

-6 p1 ^ p2 ^ p3 ^ q1 ^ q2 ^ q3

Input := 


r4 = (p1^q1 + p2^q2 + p3^q3 + p4^q4);
r4^r4^r4^r4

Output =

24 p1 ^ p2 ^ p3 ^ p4 ^ q1 ^ q2 ^ q3 ^ q4

Input := 

r5 = (p1^q1 + p2^q2 + p3^q3 + p4^q4 + p5^q5);
r5^r5^r5^r5^r5

Output =

120 p1 ^ p2 ^ p3 ^ p4 ^ p5 ^ q1 ^ q2 ^ q3 ^ q4 ^ q5

Input := 

r6 = (p1^q1 + p2^q2 + p3^q3 + p4^q4 + p5^q5 + p6^q6);
r6^r6^r6^r6^r6^r6

Output =

-720 p1 ^ p2 ^ p3 ^ p4 ^ p5 ^ p6 ^ q1 ^ q2 ^ q3 ^ q4 ^ q5 ^ 
 
   q6

Input := 

r7 = (p1^q1 + p2^q2 + p3^q3 + p4^q4 + p5^q5 + p6^q6 + p7^q7);
r7^r7^r7^r7^r7^r7^r7

Output =

-5040 p1 ^ p2 ^ p3 ^ p4 ^ p5 ^ p6 ^ p7 ^ q1 ^ q2 ^ q3 ^ 
 
   q4 ^ q5 ^ q6 ^ q7

Input := 

r8 = (p1^q1 + p2^q2 + p3^q3 + p4^q4 + 
      p5^q5 + p6^q6 + p7^q7 + p8^q8);
r8^r8^r8^r8^r8^r8^r8^r8

Output =

40320 p1 ^ p2 ^ p3 ^ p4 ^ p5 ^ p6 ^ p7 ^ p8 ^ q1 ^ q2 ^ 
 
   q3 ^ q4 ^ q5 ^ q6 ^ q7 ^ q8

The 2n-form (-1)^(r/2) n! p1^...^pn^q1^...^qn is an oriented volume element in R^(2n), which can be used as the phase space for a dynamical system with n degrees of freedom.

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