From: baez@galaxy.ucr.edu (John Baez)
Subject: Exterior derivative
Date: 11 Mar 2000 05:32:14 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <200003100019.TAA01412@world.std.com>,
Matt McIrvin <mmcirvin@world.std.com> wrote:

>John Baez said:

>>... actually, while you can define the exterior derivative
>>using covariant differentiation, you don't *need* covariant derivatives
>>to define it, nor do you need them for Gauss's theorem. 

>Well, now I'm slightly confused, myself, which is a sign of how long it's
>been since Bott's class.  The exterior derivative doesn't usually involve
>covariant derivatives at all, and Gauss's theorem, expressed in terms of
>the (n-1)-form *E, doesn't involve them.  But when general relativists
>write an expression about divergencelessness (whether spatial or
>spatio-temporal) in terms of indices of a _vector field_, they use
>covariant derivatives.  

Well, it probably boils down to this:

One fun thing about the exterior derivative is that you can
define it either using ordinary partial derivatives, or covariant 
derivatives with respect to the Levi-Civita connection - you 
get the same answer either way.  For example, here's one way
to define the exterior derivative of a 1-form A: 

(dA)_{ab} = d_a A_b - d_b A_a

where the components are taken using a *coordinate basis*, 
and d_a means "partial derivative in the a direction".  But
if we have a metric and thus a Levi-Civita connection D, it's 
also true that

(dA)_{ab} = D_a A_b - D_b A_a !

The reason is that if G_{ab}^c are the Christoffel symbols -
i.e. the components of the Levi-Civita connection in our 
coordinate basis - then we have 

G_{ab}^c = G_{ba}^c

and

D_a A_b = d_a A_b - G_{ab}^c A_c

so the difference between d_a A_b and D_a A_b is symmetric in
a and b, so 

d_a A_b - d_b A_a = D_a A_b - D_b A_a

I remember being very confused about this at one point, because 
some people would use partial derivatives and other people would 
use covariant derivatives, and nobody explained to me why it 
didn't make any difference.  

The key fact is that

G_{ab}^c = G_{ba}^c

which is just an index-ridden way of saying that the connection
D is torsion-free.  So in fact we can use any torsion-free connection
in place of partial derivatives when defining the exterior derivative
operator - it makes no difference.  Sometimes this is actually useful.

(I'm cc'ing this post to you since I know you're off on a business
trip now, and I wouldn't want you to miss this precious piece of
wisdom.)