I am confused about the difference between a vector operating on another (like $u(v)$, used in Lie Brackets) and covariant derivatives ($\nabla\_u v$). Can we not use Christoffel symbols in the vector derivative? Let us have a metric $ds^2 = g\_{xx} dx^2 + g\_{yy} dy^2$. Let us have two vectors in this system: $u = u^x e\_x + u^y e\_y$ and $v = v^x e\_x + v^y e\_y$. We assume that neither the coefficients nor the basis vectors remain constant.
What should be $u(v)$ and $\nabla\_u v$ in this case?