I'm trying to understand how transforming Christoffel symbols works. Specifically I'm thinking about the transformation between Schwarzschild and Eddington-Finkelstein coordiantes. $$\Gamma^v_{\;vv}=\frac{\partial v}{\partial x^m}\frac{\partial x^n}{\partial v}\frac{\partial x^p}{\partial v}\Gamma^m_{\;np}+\frac{\partial^2 x^m}{\partial v^2}\frac{\partial v}{\partial x^m}$$

With $cv=ct+r+r_sln(r-r_s)$, $m$, $n$, and $p$ only get summed through $t$ and $r$. I'm just not sure how to deal with the $\frac{\partial t}{\partial v}$ and $\frac{\partial r}{\partial v}$. I've never done anything like that before. I'm assuming it cannot be the reciprocals of $(\frac{\partial v}{\partial t})$ and $(\frac{\partial v}{\partial r})$, since it is a multivariable function, but beyond that I'm not sure. Just to be clear, I could just find it normally with the new metric, but I want to understand how the Christoffel symbols transform as well.

I'm trying to understand how transforming Christoffel symbols works. Specifically I'm thinking about the transformation between Schwarzschild and Eddington-Finkelstein coordinates, $$\Gamma^v_{\;vv}=\frac{\partial v}{\partial x^m}\frac{\partial x^n}{\partial v}\frac{\partial x^p}{\partial v}\Gamma^m_{\;np}+\frac{\partial^2 x^m}{\partial v^2}\frac{\partial v}{\partial x^m}$$

With $cv=ct+r+r_{S}\ln(r-r_{S})$, $m$, $n$, and $p$ only get summed through $t$ and $r$. I'm just not sure how to deal with the $\frac{\partial t}{\partial v}$ and $\frac{\partial r}{\partial v}$. I've never done anything like that before. I'm assuming it cannot be the reciprocals of $\left(\frac{\partial v}{\partial t}\right)$ and $\left(\frac{\partial v}{\partial r}\right)$, since it is a multivariable function, but beyond that I'm not sure. Just to be clear, I could just find it normally with the new metric, but I want to understand how the Christoffel symbols transform as well.