# Proof of first Bianchi identity

The proof is often simplified by using the following theorem:

"If the metric tensor $$(g_{ij})$$ is positive definite, then, at the origin of a Riemannian coordinate system $$(y^i)$$, all $$\partial g_{ij}/\partial y^k,\partial g^{ij}/\partial y^k, \Gamma_{ijk}$$, and $$\Gamma^i_{jk}$$ are zero."

It follows that at $$O$$, the origin of normal coordinates: $$R_{ijkl}=\frac{\partial \Gamma_{lji}}{\partial y^k}-\frac{\partial \Gamma_{kji}}{\partial y^l}$$ (1.0)

The whole proof will not be shown here.

I understand that the above expression comes directly from the definition of the Riemann Tensor of the first kind $$R_{ijkl}=\frac{\partial \Gamma_{lji}}{\partial y^k}-\frac{\partial \Gamma_{kji}}{\partial y^l}+\Gamma_{ilr}\Gamma^{r}_{jk}-\Gamma_{ikj}\Gamma^r_{jl}$$ but if all the Christoffel symbols vanish at the origin of the Riemannian(normal) coordinates then why do the first two terms (i.e. the RHS of (1.0)) remain?

• PS: I think my question deserves to be asked here at Physics SE since it is used so often in GR Commented Jun 9, 2019 at 15:02

For the same reason that given a function $$f(x)$$ which zero at some $$x=x_0$$, it does not imply that $$f'(x_0)=0$$.