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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?

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