Cartan's first structure equation proof

[r16]

this is my first post on this site but it looks like the sort of ppl that i would like to associate myself with.

Unfourtanately, I have not had any formal schooling for any mathematics above calculus but i have read a few books and papers and am trying to make due.

I was studying about the cartan's first structure equation and was looking at this proof :

http://www.pzgnet.cc/images/cartan/eq1.png

where \nabla_x is a koszul connection, e_i is a basis and \partial_j A^j_i is a change of basis from e and \omega is a standard connection in the actual equation :

http://www.pzgnet.cc/images/cartan/eq2.png

In step 3 why can the exterior derivitave be applied to A^j_i?

I am no impact no idea on this step and it seems quite important so i don't want to skip it. Any ideas what I am missing?

**nb in equation 2 \omega^i_j should be \omega^j_i

 

[George Jones]

If f is a function, how is the exterior derivative df defined?

It may prove useful to think about vector fields X and/or a coodinate basis

\left\{\frac{\partial}{\partial x_i} \right\}[/itex]<br /> <br /> to answer this.<br /> <br /> Regards,<br /> George

 

[r16]

that may be my biggest problem. I am a visual learner, but I don't have a good physical picture of 'd'.

Algebraically, d is defined a d = \frac{\partial}{\partial x_i} dx^i.
d applied to the 0-form f gives a 1-form df.

Then it would follow that X(f) = df (X) as in the definition of a 1-form.

However is A^j_i equivalent to a scalar because the contravariant and covariant parts of the [\frac{1}{1}] valent tensor cancel out?

This is opposed to \omega^k_i, a [\frac{1}{1}] tensor as well, which acts as a 1-form on X. Is there a standard of how a tensor acts or is it based on the definition of the tensor?

 

[George Jones]

For each i and j, A^{i}_{j} is a component, i.e., a function, while each \omega^{i}_{j} is a 1-form, and, consequently, they are very different animals.

Consider a couple of examples. A vector field X can be expressed in terms a set of basis fields as X = X^{i} e_{i}. Each X^i is is a component, i.e., a scalar-valued function of the base space, while each e_i is a vector field.

Similary, if \omega^{i}_{j} = \omega^{i}_{jk} dx^k, each \omega^{i}_{jk} is a component, while each dx^k is a 1-form.

Bottom line: sometime indices label comonents, and sometimes they label other objects.

Welcome to Physics Forums, and, if I haven't answered all your questions, or if my explanation is not very clear, keep asking questions.

Regards,
George

 

[r16]

thank you very much sir, that clears it up perfectly.

However i do have one more inquery. What is a good physical/geometrical description of an exterior derivitave?

 

[George Jones]

What is a good physical/geometrical description of an exterior derivitave?

Unfortunately, I don't know a good physical/geometrical description of an exterior derivative. Maybe someone else does.

I just work abstractly with its properties.

Regards,
George