creativity or something like that I can't really lol
There was only one question I think I didn't answer as well as I should have though...so I wonder if it was all based on that one answer I gave...unless maybe all the answers I gave were inadequate, but I just don't know. Again, unknown unknowns. XD
you guys know of a good book that goes over Einstein-Cartan theory?
hmmm...it is not at all apparent to me why the integral of a n-form over a n-Manifold is independent of the choice of covering sets and the partition of unity subordinate to that covering...
Wald just says "it can be shown"...without at least giving an intuitive motivation why that would be the case...
$f_\alpha$ is a smooth function on $O_\alpha$ so I guess the expression $\Sigma_\alpha f_\alpha = 1$ is really saying the $f$'s have to add up to 1 at each point
I kept thinking of it as $f_\alpha$ is one number that weights the different $O_\alpha$
I think it's to make the integrand well-defined on the whole manifold because the $f_{\alpha}$'s are zero off an open set and $1$ on it so $\sum_{\alpha} \int f_{\alpha} \omega$ is now well-defined everywhere or something
then where the $O_\alpha$ do intersect you have $\Sigma_\alpha f_\alpha = 1$...but in points where the $O_\alpha$ do intersect the integral must match between the intersecting subsets so it doesn't matter how you slice that piece using the $f_\alpha$ you will always get the same answer.
Say you have $N$ strings fluctuating in a box with $N/2$ strings fixed at $(0,0)$ and at $(1,1)$ and the other $N/2$ strings fixed at $(1,0)$ and at $(0,1).$ All these strings vibrate back and forth to create patterns. Has anyone studied these patterns before?
