# First fundamental form and Christoffel symbols

I am currently taking a course in the geometry of curves and surfaces in $\mathbb R^3$ (elementary differential geometry) and I am trying to memorize the expression for the Christoffel symbols. In the textbook we use (Pressley) they are defined in terms of the first fundamental form (i.e. in terms of $E, F$ and $G$ where $E=\sigma_u\cdot\sigma_u$, $F=\sigma_u\cdot\sigma_v$, $G=\sigma_v\cdot\sigma_v$). Unfortunately we are just given six individual equations which are quite monstrous. I know that there are general expressions for these (e.g. here) but unfortunately I can't am not familiar with tensor calculus so I don't really understand the notation used. If anyone knows a simpler way to remember these than just brute force memorization I would be very thankful!

Note: Here is a screenshot from my textbook that includes the said equations.

There is a nice way to simplify this. Take the dot product of the above with $\varphi_u$ and $\varphi_v$ to get: \begin{equation*} \begin{split} <\varphi_{uu}, \varphi_u> &= \boldsymbol\Gamma_{11}^1\cdot<\varphi_u, \varphi_u>+\>\>\boldsymbol\Gamma_{11}^2\cdot<\varphi_v, \varphi_u> + \>\>\boldsymbol e\cdot<\boldsymbol{\vec n}, \varphi_u>\\[2ex] &= \boldsymbol\Gamma_{11}^1\cdot \boldsymbol E + \boldsymbol\Gamma_{11}^2\cdot\boldsymbol F\\[4ex] <\varphi_{uv}, \varphi_u> &= \boldsymbol\Gamma_{12}^1\cdot<\varphi_u, \varphi_u> +\>\> \boldsymbol\Gamma_{12}^2\cdot<\varphi_v, \varphi_u> + \>\>\boldsymbol f\cdot <\boldsymbol{\vec n}, \varphi_u>\\[2ex] &= \boldsymbol\Gamma_{12}^1\cdot\boldsymbol E + \boldsymbol\Gamma_{12}^2\cdot\boldsymbol F\\[4ex] <\varphi_{vv}, \varphi_u> &= \boldsymbol\Gamma_{22}^1\cdot<\varphi_u, \varphi_u> +\>\> \boldsymbol\Gamma_{22}^2\cdot<\varphi_v, \varphi_u> + \>\>\boldsymbol g\cdot<\boldsymbol{\vec n}, \varphi_u>\\[2ex] &= \boldsymbol\Gamma_{22}^1\cdot \boldsymbol E+\boldsymbol\Gamma_{22}^2\cdot\boldsymbol F\\[4ex] <\varphi_{uu}, \varphi_v> &= \boldsymbol\Gamma_{11}^1\cdot<\varphi_u, \varphi_v> +\>\> \boldsymbol\Gamma_{11}^2 \cdot<\varphi_v, \varphi_v> + \>\>\boldsymbol e\cdot<\boldsymbol{\vec n}, \varphi_v>\\[2ex] &= \boldsymbol\Gamma_{11}^1\cdot\boldsymbol F + \boldsymbol\Gamma_{11}^2\cdot \boldsymbol G\\[4ex] <\varphi_{uv}, \varphi_v> &= \boldsymbol\Gamma_{12}^1\cdot<\varphi_u, \varphi_v>+\>\>\boldsymbol\Gamma_{12}^2\cdot<\varphi_v, \varphi_v>+\>\>\boldsymbol f\cdot<\boldsymbol{\vec n}, \varphi_v>\\[2ex] &=\boldsymbol\Gamma_{12}^1\cdot\boldsymbol F + \boldsymbol\Gamma_{12}^2\cdot\boldsymbol G\\[4ex] <\varphi_{vv}, \varphi_v> &= \boldsymbol\Gamma_{22}^1\cdot<\varphi_u, \varphi_v>+\>\>\boldsymbol\Gamma_{22}^2\cdot<\varphi_v, \varphi_v>+\>\>\boldsymbol g\cdot<\boldsymbol {\vec n}, \varphi_v>\\[2ex] &=\boldsymbol\Gamma_{22}^1\cdot\boldsymbol F + \boldsymbol\Gamma_{22}^2\cdot\boldsymbol G\\[4ex] \end{split} \end{equation*}
• That is indeed interesting! Unfortunately it doesn't really simplify the task of obtaining the individual $\Gamma_{i,j}^k$ coefficients from $E, F$ and $G$ as far as I can see. Apr 2, 2017 at 21:25
• What I had in mind was something similar to what is on the Wikipedia article (i.e. a single formula with a bunch of indices), maybe something directly in terms of $\sigma_u$ and $\sigma_v$. Apr 2, 2017 at 21:36