I have a vector field $$\mathbf V = \mathbf V(X,Y,Z)$$ in Cartesian coordinates that I would like to use to construct a field on a manifold $$T(\xi,\eta) = (X(\xi,\eta),Y(\xi,\eta),Z(\xi,\eta))$$. It is clear I can do this by writing
$$\mathbf U = u^1 \mathbf t_1 + u^2 \mathbf t_2$$ where $$u^1 = \mathbf V\cdot \mathbf t^1$$ and $$u^2 = \mathbf V\cdot \mathbf t^2$$ for covariant vectors $$\mathbf t_1 = T_\xi$$ and $$\mathbf t_2 = T_\eta$$ and related contravariant vectors $$\mathbf t^1$$ and $$\mathbf t^2$$.
What I would now like to do is compute the surface divergence of $$\mathbf U$$. I can do this if I know $$\partial u^1/\partial \xi$$, $$\partial u^1/\partial \eta$$ and so on. This leads to
$$\frac{\partial(\mathbf V \cdot \mathbf t^1)}{\partial \xi} = \mathbf V\cdot \frac{\partial}{\partial \xi} \mathbf t^1 + \frac{\partial \mathbf V}{\partial \xi} \cdot \mathbf t^1$$
I can easily compute the components of $$\partial \mathbf V/\partial \xi$$ as $$\nabla_X V^j \cdot \mathbf t_1$$.
My question is, how can I easily compute $$\partial \mathbf t^1/\partial \xi$$ ? I already compute $$\partial \mathbf t_1/\partial \xi$$ and related Christoffel symbols. Is there a transformation or identity that I can use to get $$\partial \mathbf t^1/\partial \xi$$ from what I already have?