In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first.
We have that $X$ is a minimal surface, $X(t)$ is a variation (understand here that $X(0) = X$), $g_{ij}(t)$ is the first fundamental form, $g^{ij}(t)$ is its inverse, and $h_{ij}(t)$ is the second fundamental form. We have the variational field $$E(t) = \frac{\partial X(t)}{\partial t},$$ and $E(0) \equiv E = \alpha X_1 + \beta X_2 + \gamma N$.
We also assume isothermal coordinates $g_{ij} = \Lambda^2 \delta_{ij}$. This implies that $g^{ij} = \Lambda^{-2}\delta_{ij}$. He states that: $$\frac{{\rm d}g^{ij}(t)}{{\rm d}t}\Bigg|_{t=0} = -\Lambda^{-4}(E_i \cdot X_j + E_j \cdot X_i),$$ ok. Then he says:
Using $h_{11} = -h_{22}$ and $X_{11}\cdot X_1 = \frac{1}{2}\Lambda_1^2$, $X_{11}\cdot X_2 = -\frac{1}{2}\Lambda_2^2$, etc, we have: $$\frac{1}{2}\sum_{i,j}\frac{{\rm d}g^{ij}(t)}{{\rm d}t}\Bigg|_{t=0}h_{ij} = \gamma \Lambda^{-4}\sum_{i,j}h_{ij}^2 - \Lambda^{-2}(\alpha_1h_{11}+(\alpha_2+\beta_1)h_{12}+\beta_2h_{22})$$
I just don't follow that. We could say that: $$X_1 \cdot X_1 = \Lambda^2 \implies 2 X_{11}\cdot X_1 = 2\Lambda_1\Lambda \implies X_{11}\cdot X_1 = \Lambda_1\Lambda,$$ and I don't see how he got that expression.
Can someone explain to me how to get the highlighted expression please? If you need me to explain some more of the notation please say it.
I think you're correct about the typo. Perhaps he meant $X_{11} \cdot X_1 = \frac{1}{2} (\Lambda^2)_1$ or something. If you use your corrections, then the expression just falls out with a bit of writing.
We calculate $$E_1 = \alpha_1 X_1 + \alpha X_{11} + \beta_1 X_2 + \beta X_{21} + \gamma_1 N + \gamma N_1$$ $$E_2 = \alpha_2 X_1 + \alpha X_{12} + \beta_2 X_2 + \beta X_{22} + \gamma_2 N + \gamma N_2$$ which yields the following: \begin{align*} E_1 \cdot X_1 & = \alpha_1 \Lambda^2 + \alpha X_{11} \cdot X_1 + \beta X_{21} \cdot X_1 + \gamma N_1 \cdot X_1 \\ & = \alpha_1\Lambda^2 + \alpha \Lambda \Lambda_1 + \beta \Lambda \Lambda_2 - \gamma h_{11}\\ E_1 \cdot X_2 & = \alpha X_{11} \cdot X_2 + \beta_1 \Lambda^2 + \beta X_{21} \cdot X_2 + \gamma N_1 \cdot X_2 \\ & = \alpha X_{11} \cdot X_2 + \beta_1 \Lambda^2 + \beta X_{12} \cdot X_2 - \gamma h_{12}\\ E_2 \cdot X_1 & = \alpha_2 \Lambda^2 + \alpha X_{12} \cdot X_1 + \beta X_{22} \cdot X_1 + \gamma N_2 \cdot X_1 \\ & = \alpha_2 \Lambda^2 + \alpha X_{21} \cdot X_1 + \beta X_{22} \cdot X_1 - \gamma h_{21}\\ E_{2} \cdot X_2 & = \alpha X_{12} \cdot X_2 + \beta_2 \Lambda^2 + \beta X_{22} \cdot X_2 + \gamma N_2 \cdot X_2\\ & = \alpha \Lambda \Lambda_1 + \beta_2 \Lambda^2 + \beta \Lambda\Lambda_2 - \gamma h_{22}. \end{align*}
So collecting terms, \begin{align*} \sum_{i,j} E_i \cdot X_j h_{ij}& = \left(\alpha_1 h_{11} + \beta_1 h_{12} + \alpha_2 h_{21} + \beta_2 h_{22} \right) \Lambda^2 + \alpha \Lambda \Lambda_1 (h_{11} + h_{22}) + \beta \Lambda \Lambda_2 (h_{11} + h_{22}) +\alpha \left( X_{11} \cdot X_2 h_{12} + X_{21} \cdot X_1 h_{21} \right) + \beta \left(X_{12} \cdot X_2 h_{12} + X_{22} \cdot X_1 h_{21} \right) - \gamma \sum_{i,j} h_{ij}^2\\ & = \left(\alpha_1 h_{11} + \beta_1 h_{12} + \alpha_2 h_{21} + \beta_2 h_{22} \right) \Lambda^2 - \gamma \sum_{i,j} h_{ij}^2\\ \end{align*}
using $$h_{11} + h_{22} = 0,$$ $$X_{11} \cdot X_2 + X_1 \cdot X_{21} = \left(X_1 \cdot X_2\right)_1 = 0,$$ and $$X_{12} \cdot X_2 + X_1 \cdot X_{22} = \left(X_1 \cdot X_2 \right)_2 = 0.$$