I'm following the book "A Course in Minimal Surfaces" by Colding and Minicozzi. I'm stuck on section 1.3, The first variation formula.
We are given a Riemannian manifold $M$ with metric $g$ and covariant derivative $\nabla$. $\Sigma$ is a $k$-dimensional submanifold.
Let $F: \Sigma \times (-\epsilon, \epsilon) \to M$ be a variation of $\Sigma$ with compact support and fixed boundary. We are trying to evaluate the following: $$ \frac{d}{dt}_{t=0} \mathrm{Vol}(F(\Sigma,t)) = \int_{\Sigma} \nu(t) \sqrt{\det(g^{ij}(0))},$$ where $g_{ij}(t) = g(F_{x_{i}}, F_{x_{j}})$ and $\nu(t) = \sqrt{\det(g_{ij}(t))} \sqrt{\det(g^{ij}(0))}$. $(a^{ij})$ denotes the inverse of the matrix $(a_{ij})$.
To evaluate $d/dt_{t=0} \nu(t)$ at some point $x$, we may choose the coordinate system such that at $x$ it is orthonormal.
Using this and the fact that the $t$ and $x_{i}$ derivatives commute (i.e., $\nabla_{F_{t}} F_{x_{i}} - \nabla_{F_{x_{i}}} F_{t} = [F_{t}, F_{x_{i}}] = 0), \dots$.
Why do the $t$ and $x_{i}$ derivatives commute?
If you calculate within a small neighborhood, $F_t$ is really a smooth map
$$F : V \times I \to U,$$
where $V \subset \mathbb R^k$ and $U \subset \mathbb R^n$ are local coordinates of $\Sigma$ and $M$ respectively. Then
$$F_{x_i} = \frac{\partial F}{\partial x_i},\ \ \ F_t=\frac{\partial F}{\partial t}$$
and they commute because the normal partial differentiation commutes.