1. 1:50, since metric components are defined to be dot products of basis vectors, I don't think you need another arbitrary vector r=xi+yj+zk. If you already have a coordinate system, just find the coordinate basis and take dot products, am I right? 2.4:20, to be precise, g^{km} means the components of the inverse matrix of the metric, which aren't guaranteed to be reciprocals of g_{km}. 3. 12:22, you used "a" for the radius when you first plugged in the spherical coordinates. So, what you have shown in the end is, up to a scalar, the scalar curvature of a sphere is inverse proportional to its radius squared, yes?
First of all, thanks for paying so much attention. 1. Not sure how to define r, theta, and phi if I don't project them onto a Cartesian frame. Would be interesting to learn. 2. You are correct, only because the off diagonals are zero can I get away with that, else I would have to take the proper tensor inverse. 3. Yes
