Coefficients of connection under diffeomorphism and metric change. Let $\varphi_t :M\rightarrow M $ is a family of diffeomorphism. $\widehat{g}_{ij}(x,t)$ is a solution of $$\frac{\partial}{\partial t}g_{ij}=-2R_{ij} ,\ 
y(x,t)=\varphi_t(x)=\{y^1(x,t),...,y^n(x,t)\},\ g_{ij}(x,t)=\varphi_t^*\widehat{g}_{ij}(x,t)$$
How to show that :
$$
\Gamma_{jl}^k(x,t)=
\frac{\partial y^a}{\partial x^j}  \frac{\partial y^b}{\partial x^l}
\frac{\partial x^k}{\partial y^r}\widehat{\Gamma}_{ab}^r(y,t)
+\frac{\partial x^k}{\partial y^a}
\frac{\partial^2y^a}{\partial x^j\partial x^l}
$$
I really can't compute it out , so I really need a detail answer ,so thanks.
 A: Assume that $ f(x)=y$ is a diffeomorphism and $$h:=f^\ast g $$
Define
$$ g_{\alpha\beta }:= g(\partial_{y_\alpha},\partial_{y_\beta}) $$
$$  h_{ij}:= h(\partial_{x_i},\partial_{x_j}) $$
$$ f^\alpha_i:= \frac{\partial y^\alpha}{\partial x_i  } $$
$$ h^{jk}=f_\zeta^j g^{\eta\zeta } f_\eta^k $$
So $$ h_{ij,l}:= \partial_{x_l} h_{ij} =\partial_{x_l} \{
g_{\alpha\beta } f^\alpha_i f^\beta_j\} $$
$$ = g_{\alpha\beta,\gamma}f^\gamma_lf^\alpha_i f^\beta_j +
g_{\alpha\beta } f^\alpha_{il} f^\beta_j +g_{\alpha\beta }
f^\alpha_i f^\beta_{jl}
$$ That is $$ \Gamma(h)_{ij}^k = \frac{1}{2} h^{kl} (h_{il,j} +
h_{jl,i} - h_{ij,l}) $$
$$ = \frac{1}{2}f_\zeta^l g^{\eta\zeta } f_\eta^k
\{g_{\alpha\beta,\gamma}f^\gamma_jf^\alpha_i f^\beta_l +
g_{\alpha\beta } f^\alpha_{ij} f^\beta_l +g_{\alpha\beta }
f^\alpha_i f^\beta_{jl}
$$ $$+
g_{\alpha\beta,\gamma}f^\gamma_if^\alpha_j f^\beta_l +
g_{\alpha\beta } f^\alpha_{ij} f^\beta_l +g_{\alpha\beta }
f^\alpha_j f^\beta_{il}$$ $$ -(
g_{\alpha\beta,\gamma}f^\gamma_lf^\alpha_i f^\beta_j +
g_{\alpha\beta } f^\alpha_{il} f^\beta_j +g_{\alpha\beta }
f^\alpha_i f^\beta_{jl} ) \}$$
$$ = \frac{1}{2}f_\zeta^l g^{\eta\zeta } f_\eta^k
\{g_{\alpha\beta,\gamma}f^\gamma_jf^\alpha_i f^\beta_l +
2g_{\alpha\beta } f^\alpha_{ij} f^\beta_l
$$ $$+
g_{\gamma\beta,\alpha}f^\alpha_if^\gamma_j f^\beta_l  -
g_{\alpha\gamma,\beta}f^\beta_lf^\alpha_i f^\gamma_j \}$$
$$ = \Gamma(g)_{\alpha\gamma}^\tau
 f^\gamma_jf^\alpha_i f_\tau^k + f^\alpha_{ij}
f_\alpha^k$$
