I'm reading Jost's Riemannian Geometry and Geometric Analysis and having some questions about the variation formula for submanifolds (page 242 in 7-Ed).
For a local variation $\Phi: M \times (-\epsilon,\epsilon) \to N$ of a submanifold $M$ in $N$, the variation formula is $$ \frac{d}{dt} \text{Vol}(\Phi_t(M))|_{t = 0} = \int_M (e_\alpha \langle X, e_\alpha \rangle - \langle X, \nabla^N_{e_\alpha} e_\alpha \rangle ) \eta$$ where $X = \frac{\partial}{\partial t} \Phi |_{t=0}$, $e_\alpha$ is an oriented orthonormal frame on $M$, and $\eta$ is the volume form on $M$. For the discussion of how it is derived, this site may be helpful: Computing the first variation of volume: all around confusion.
By the Gauss's theorem, the first term of the right hand side vanishes, so $$ \frac{d}{dt} \text{Vol}(\Phi_t(M))|_{t = 0} = - \int_M \langle X, \nabla^N_{e_\alpha} e_\alpha \rangle \eta$$
Then Jost argued that by choosing the nornal coordinate for example, $\nabla^M_{e_\alpha} e_\alpha = 0$ at the point under consideration and thus, $$ \frac{d}{dt} \text{Vol}(\Phi_t(M))|_{t = 0} = - \int_M \langle X^\perp, \nabla^N_{e_\alpha} e_\alpha \rangle \eta \tag{1} $$
My question is that by taking normal coordinate, $\nabla^M_{e_\alpha} e_\alpha = 0$ holds only at one point, say $x_0 \in M$, not the entire $M$ (nor in the support of variation); then how can we conclude equation (1)?
