I've been recently been going back over the basics of GR, differential geometry in particular. I was watching one of Susskind's lectures and did not understand the argument made here (26:33 - 35:40).
In short, the argument goes as follows (I think): we have some generic metric ${ g }_{ m n }^{ ' }\left( y \right)$. Suppose we have a coordinate transformation that takes ${ g }_{ m n }^{ ' }\left( y\right) \rightarrow { g }_{ m n }\left( x\right)$ such that ${ g }_{ m n }\left( X\right) ={ \delta }_{ m n }$ for a particular point $x=X$.
Susskind wants to show that, in general, the first derivatives, ${ \partial }_{ r }{ g }_{ m n }\left( x\right)$, can be chosen to be zero, but the second derivatives, ${ \partial }_{ r }{ \partial }_{ s }{ g }_{ m n }\left( x \right) $, can not (that is, at $x=X$).
He does this by looking at the expansion of $x$ in terms of $y$ about the point $x=X$. For simplicity, he chooses ${ X }^{ m }=0$ and for the $x$ and $y$ coordinate systems to have the same origin: $${ x }^{ m }={ a }_{ r }^{ m }{ y }^{ r }+{ b }_{ rs }^{ m }{ y }^{ r }{ y }^{ s }+{ c }_{ rst }^{ m }{ y }^{ r }{ y }^{ s }{ y }^{ t }+\dots$$
The argument is (again, I think) that because (for the case of a four-dimensional space) ${ \partial }_{ r }{ g }_{ m n }\left( x\right)=0$ is 40 equations and ${ b }_{ rs }^{ m }$ consists of 40 variables, we can always choose values of ${ b }_{ rs }^{ m }$ that satisfy the equations. Meanwhile, ${ \partial }_{ r }{ \partial }_{ s }{ g }_{ \mu \nu }\left( x \right) =0$ is 160 equations, but ${ c }_{ rst }^{ m }$ consists of only 80 variables, so we do not have enough free parameters to force the second derivatives to all vanish.
The problem is that I simply don't see why the existence of 40 variables in that expansion means that we can satisfy the 40 equations. Is the connection a simple one or do I just have to do something like grind out the values of the derivatives at $x=X$ using the series expansion?
