I would very much appreciate it if someone could explain or at least indicate a proof of the following assertion in Spivak's ''Compr. Intro. to Diff. Geom.'', vol. 2, p. 171 (3rd ed., 2nd printing):
''It is easy to see that if we pick a different Riemannian normal coordinate system at $p$, then the resulting function on the 2-dimensional subspaces of $M_p$ will change by $(\det B)^2$, where $B\in O(n)$, so that $\det B=\pm 1$.''
The function is given on $M_p\times M_p$ by:
$$ Q=c_{ij,kl}(dx^i\wedge dx^k)\cdot (dx^j\wedge dx^l), $$ where $c_{ij,kl}=\frac{\partial^2 g_{ij}}{\partial x^k\partial x^l}$.
My main difficulty is that I do not know how to relate $\frac{\partial^2 g_{ij}}{\partial x^k\partial x^l}$ to $\frac{\partial^2 \bar{g}_{\alpha \beta}}{\partial y^\alpha \partial y^\beta}$, where $\bar g_{\alpha \beta}$ are the coefficients of the metric in the coordinate system $y$.
Thanks in advance.