I'm studying the proof of this theorem on Plateau's problem in the textbook "Elementary Differential Geometry" by Andrew Presley, but I don't understand, being more specific, why $\beta^0=\gamma^0=0$ in the final part. For first, let me enunciate the theorem:
Theorem: with the above notation, assumme that the surface $\varphi^{\tau}$ vanishes along the boundary curve $\pi$. then \begin{align} (\mathcal{A}(0))' = -2\int_{\operatorname{int}(\pi)} H(EG - F^2)^{1/2}\alpha du dv, \label{1} \end{align} where $H$ is the mean curvature of $\sigma$, $E$, $F$ and $G$ are the coefficients of its first fundamental form, and $\alpha = \varphi\cdot N$, where $N$ is the standar unit normal of $\sigma$.
And this is the proof:
I think that reason is related with the following fact, $\varphi^{\tau}(u,v) = 0$ when $(u,v)$ is a point on the curve $\pi$ where $\varphi^{\tau}= (\sigma^{\tau})^{\prime}$, but i didn't get it to, any help will be really appreciate.
Your suspicion is correct.
The theorem assumes that the surface variation $\varphi^\tau(u,v) = \mathbf{0}$ for every point $(u,v)$ on the boundary curve $\pi$. Now, the equation $$\varphi^\tau = \alpha^\tau \mathbf{N}^\tau + \beta^\tau\sigma^\tau_u + \gamma^\tau \sigma^\tau_v$$ holds for all points $(u,v)$ in the interior region $\text{Int}(\pi)$ and on the boundary curve $\pi$. In particular, if $(u,v)$ is on the boundary, then: $$\mathbf{0} = \alpha^\tau(u,v) \mathbf{N}^\tau + \beta^\tau(u,v) \sigma^\tau_u + \gamma^\tau(u,v) \sigma^\tau_v.$$ Since $\{\mathbf{N}^\tau, \sigma^\tau_u, \sigma^\tau_v\}$ is linearly independent, it follows that for every point $(u,v)$ on the boundary curve $\pi$: $$\alpha^\tau(u,v) = 0, \ \ \ \beta^\tau(u,v) = 0, \ \ \ \gamma^\tau(u,v) = 0.$$ Setting $\tau = 0$, we arrive at the claim.