I'm trying to convert a parametrization of a curve $$\vec \alpha(x,y,z) = \alpha_1(x,y,z) \hat x + \alpha_2(x,y,z) \hat y + \alpha_3(x,y,z) \hat z$$ to $$\vec \alpha(r, \theta, \phi) = \alpha_r(r, \theta, \phi) \hat r + \alpha_\theta (r, \theta, \phi) \hat \theta + \alpha_\phi (r, \theta, \phi) \hat \phi $$.
To do that, let $$x = x(r,\theta, \phi) \\ y = y(r, \theta, \phi) \\ z = z(r, \theta, \phi)$$ be given, and define $$\hat u = \left(\frac{\partial x}{\partial r} \hat x + \frac{\partial y}{\partial r} \hat y + \frac{\partial z}{\partial r} \hat z \right) / a \\ \hat v = \left(\frac{\partial x}{\partial \theta} \hat x + \frac{\partial y}{\partial \theta} \hat y + \frac{\partial z}{\partial \theta} \hat z \right) / b \\ \hat w = \left(\frac{\partial x}{\partial \phi} \hat x + \frac{\partial y}{\partial \phi} \hat y + \frac{\partial z}{\partial \phi} \hat z \right) / c,$$ where $a,b,c$ are the norm of the given expression so that $\vec u, \vec v, \vec w$ are unit vectors.
Now, from this point, how can I write $\alpha$ in terms of $\hat u, \hat v, \hat w$ ?
Note that, as far as I figured, $\hat u$ corresponds to $\hat r$ in a sense, and similarly for the others.
Edit:
I'm particularly interested with the general derivation, so please don't throw the result at me only.
Edit 2:
$u, v, w$ are mutually orthogonal, and $r, \theta, \phi$ are spherical coordinates.