Problem Set 6
Differential Geometry


6.1 Differential Forms

Just like group theory is the study of multiplication without the commutative
property and without addition, differential forms are the study of tensors
without the metric tensor "g mu nu".  Anything one can do with forms, one
can do with tensors too.  We study forms for two reasons.  First,
some subjects in physics have no natural metric structure, and the entire
subject can be described in terms of forms.  Classical Hamiltonian mechanics
(as described, for example, by Arnold) is often taught using forms for this
reason; dynamical systems on manifolds is a beautiful, clean subject because
of differential forms.  Second, properties which are derived without the use 
of the metric are usually deeper or more primitive than those which demand 
everything.  In group theory, subgroups and quotient groups are important 
primitive structures in whatever broader context groups arise: vector spaces, 
Lie groups, ...  

Dot products, cross products, div, curl, grad, Gauss' Theorem, Stokes Theorem,
the fundamental theorem of calculus --- all are primitive notions of 
vector spaces which have their origins in differential forms.


I've set up a package with an implementation of differential forms in 
Mathematica.

	<<forms.defs.m

In this package, I do several things in a bludgeoning and clumsy way.

(1) Unfortunately, the symbol wedge "^" is used for Power.  I steal
^ from power, except when the power is a number.  I also make x_^0 = 0
since 0 is a legitimate form.  dx^dy  = Wedge[dx,dy].

(2) Wedge products are linear in each argument, and totally antisymmetric.
Try evaluating w1^w2, w2^(2w1+w2), and (2w2+w3)^w2^(5w2+2w1).

(3) The w's are intended to be one-forms: linear functions on vectors.
For example, if {x,y,z} form a basis for a three-dimensional vector space, 
{dx, dy, dz} form a basis for the dual space of one-forms, where
dx[ax + by + cz] = a, ...

	x = {1,0,0}
	y = ...
	z = ...

	dx[{a_,b_,c_}] := a
	dy[...] := ...
	dz[...] := ...

A general real-valued linear function on vectors now can be given by
a linear combination of dx, dy, and dz.  For example,

	w1 := 4 dx + 2 dy + dz

Try evaluating w1[{3,1,f}].  To avoid confusion later, Clear[w1].

(4) I've defined the operation of n-forms on vectors.  For example, 
w1^w2 (v1,v2) = w1(v1) w2(v2) - w2(v1) w1(v2).  Try (dx^dy)[x,y],
(dx^dy^dz)[x,y], (dx^dy) [x], and (3 dx^dy + dx^dz)[4x + 2y + z, x+4z].
Verify for yourself that this is working.


(5) Given a matrix M mapping vectors to vectors, I've defined the operator 
M* (written Star[M]) which acts on n-forms.  
		(M*) w [v1,...,vn] = w[M v1,...,M vn].
Look at the abstract version Star[Q][w1^w2][v1,v2] (where Q, w1, w2, v1, and
v2 aren't defined).  Let M = {{2,3,1},{3,2,5},{0,0,1}}; verify that
Star[M][dx^dy^dz][x,y,z] = Det[M].  Try this with L* for the general matrix
L =  {{a, b, c}, {d, e, f}, {g, h, i}}.  Set up a four--dimensional vector
space and check that the induced action of a matrix on four-forms also
gives the determinant.  Check that Star[M] acting on some general n-forms and 
sets of n general vectors is equivalent to multiplication by det[M].



______________________________________________________________________________


Now for some new exercises.


(a) How does the operator M* act on (n-1)-forms?  

The space of (n-1) forms is n-dimensional.  For our three-dimensional example,
a basis for the (n-1) forms is 

	TwoFormBasis := {dy^dz,dz^dx,dx^dy}

M* (dy^dz) will be a linear combination of the three basis vectors:
M11 dy^dz + M21 dz^dx + M31 dx^dy.  The component M31 along dx^dy, for 
example, can be discovered by applying M* (dy^dz) to [x,y].  Find the
matrix Mij.  Compare it to the matrix of 2x2 minors of M (see Minor).
Use Star to compute the inverse of M.  Does this also work for 4x4 matrices?


(b) Describe the wedge product of two general one-forms for a three-dimensional
space.  Describe the wedge product of a one-form and a two-form. 
Even without an inner product, forms have analogues of dot and cross products.
Check in two and four dimensions that the analogue of dot products still works,
but the "cross product" analogue doesn't return anything looking like a vector.
What does angular momentum r x p look like in two and four dimensions?
I never liked thinking of torques and angular momentums as vectors anyhow.


(c) w^w == 0 for any one-form w, because of the antisymmetry of the
wedge product.  However, w1^w2 = w2^w1 for two-forms w1 and w2 (try it).
In four dimensions, find a form whose wedge product with itself is not zero.

Consider the two--form w arising in classical mechanics

	dp1^dq1 + dp2^dq2 + ... dpn^dqn.

If we wedge w with itself n times, we get a multiple of the phase volume form

	dp1^...^dpn ^ dq1^...^dqn.

By trying a few cases, find the multiplicative factor.


(d) We've been mysterious about using dx for the one form returning the x
component of a vector.  Really, d is a central operator in differential
forms; it sends m-forms into (m+1)-forms by differentiating.  Functions
are zero--forms.  


There are four rules defining the action of d on forms.  I copy these
straight out of Flanders.  They are a bit abstract, but that makes them ideal
for Mathematica.  The first rule is that d is distributive over addition:


	d[w1_ + w2_] := d[w1] + d[w2]

The second rule is a wedge product rule: d[w1^w2] = d[w1]^w2 +- w1^d[w2],
where the sign is (-1) to the length of the forms w1.  Formally,
functions are zero forms, so this rule becomes three

	d[Wedge[w1_,w2__]] := d[w1] ^ w2 - w1 ^ d[w2]
	d[f_ d[w_]] := d[f] ^ d[w]
	d[f_ Wedge[w1__] ] := d[f] ^ Wedge[w1] + f d[Wedge[w1]]

The third rule is that d squared is zero.

	d[d[w_]] := 0

The fourth rule defines d acting on a function to be the gradient.  This
depends on the coordinates we use.  In three dimensions, ... 

	Clear[x]; Clear[y]; Clear[z]; Clear[dx]; Clear[dy]; Clear[dz]
	d[f_] := D[f,x] d[x] + D[f,y] d[y] + D[f,z] d[z] /; !MemberQ[{x,y,z},f]

Finally, we can recognize forms as d[something], so we can pull out
functions from wedge products.

	Wedge[f_ d[x_],w2_] := f Wedge[d[x],w2]
	Wedge[w1_,f_ d[x_]] := f Wedge[w1,d[x]]


To demonstrate, try d[x^2 + y^3 - Sin[z]], and then d[%].  Try
d[x^2 d[y] + (3x-4+z^2) d[z]].  Show that d[f[x,y,z]] gives a gradient,
that d[p dx + q dy + r dz] gives a curl, and that 
d[a dy^dz + b dz^dx + c dx^dy] gives a divergence.

Show (by hand) using d^2 = 0 (property 3) that curl[grad[f]] = 0 and that 
div[curl[v]] = 0. 


(e) Differential forms should be thought of as integrands.  One forms
f[x,y] dx + g[x,y] dy are integrands for line integrals, two forms
dx ^ dy are integrands for surface integrals, and so forth.  (We showed
in class that even if u and v are not orthogonal, du^dv still represents
the oriented area spanned by the parallelogram.)

One of the key results in differential forms is the Generalized Stokes Theorem,
which states in words that the integral of an n-form w over the boundary of
an n-surface S is equal to the integral of dw over S.

		/	     /
		| w	=    | dw
		/	     /
		 Del S	      S

Show (by hand) in three dimensions that this reduces to Gauss' theorem, Stokes' 
theorem, and the fundamental theorem of calculus, respectively, for two-forms,
one-forms, and zero-forms w. 

(f) Maxwell's Equations!  Define the two--form F to be
F = (Ex dx + Ey dy + Ez dz) ^ (c dt) + (Hx dy^dz + Hy dz^dx + Hz dx^dy)
(This should look familiar.)  Two of Maxwell's equations are given by
dF == 0.

	EditDef[d]
	d[f_] := D[f,x] d[x] + D[f,y] d[y] + D[f,z] d[z] + D[f,ct] d[ct]/; 
						!MemberQ[{x,y,z,ct},f]

 F := 
  (Ex[x,y,z,ct] d[x] + Ey[x,y,z,ct] d[y] + Ez[x,y,z,ct] d[z]) ^ (d[ct]) +
  (Hx[x,y,z,ct] d[y]^d[z] + Hy[x,y,z,ct] d[z]^d[x] + Hz[x,y,z,ct] d[x]^d[y])

	d[F]
	Collect[%,{...}]