The right proof of Cramer is this. Write $\vec b = x\vec a_1+y\vec a_2+z\vec a_3$. Do the determinant $\left|\begin{matrix} \vec b_1 & \vec a_2 & \vec a_3\end{matrix}\right|$. Out pops Cramer's rule for $x$.
At any rate, I discussed the geometric interpretation of these coefficients in affine geometry in the middle of that lecture that's linked.
How does one show that the determinant of the jacobian of the $n$-dimensional cartesian to polar coordinate transformation $(x_1,\cdots,x_n)\mapsto(r,\theta_1,\cdots,\theta_{n-1})\in(0,\infty)\times(-\pi,\pi)\times(0,\pi)\times\cdots\times(0,\pi)$ is $r^{n-1}\sin\theta_2\cdots\sin\theta_{n-1}$?
Yeah, this presumes familiarity with forms, anyhow, @Semiclassic. You can read through a few sections of my book if your library has it. There are fancier treatments available, of course.
No, my question is "under which conditions can one take a laplacian out of an integral" (stated a bit more precisely), because I need to do that to prove a property of the fundamental solutions to the Laplace equation
@TedShifrin I do but I'm not sure how that works. I need the derivative bounded by an integrable function and then do some dominated convergence style stuff?
@Alessandro: Or you can do it classically where the partial derivatives are continuous. You can play games by puncturing your space where the singularities are and then arguing convergence of the integrals.
@Balarka, @Danu, @MikeM: I see (approximately) why the fiber is torsion in the unit circle bundle. Take a generic vector field with (nondegenerate) zeroes at $p_i$. Take out little disks around $p_i$. Normalizing the vector field gives a section and hence a $2$-chain except over those little disks, and presumably the boundary of this $2$-chain is the sum of the indices times the fiber.
Including the derivatives, @Alessandro? Then it's just classic Riemann integral stuff. Are you trying to prove the legitimacy of differentiation under the integral? If so, I can give you my exercise on that from my book :)
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@Ted I mean to say, why can you give a section outside those little disks about $p_i$? Precisely because the unit tangent bundle is trivial on the 1-skeleton.
If you delete those disks that defo retrs to the 1-skeleton of the surface.
I decided to not do the $d_2$ computation in the middle of the night. I'll perhaps try it out tomorrow and inform you accordingly. Maybe I'll understand Lie groups better tonight instead :)
