According to my textbook, $d{\mathbf r}=dx{\mathbf e_x}+dy{\mathbf e_y}+dz{\mathbf e_z}$ can be written in general orthogonal curvilinear coordinates as $d{\mathbf r}=\frac{\partial\mathbf r}{\partial u}du+\frac{\partial\mathbf r}{\partial v}dv+\frac{\partial\mathbf r}{\partial w}dw$ by considering $x=x(u,v,w)$, $y=y(u,v,w)$ and $z=z(u,v,w)$ and expanding the differential $d\mathbf r$. (where $\frac{\partial \mathbf r}{\partial u},\frac{\partial \mathbf r}{\partial v}$ and $\frac{\partial \mathbf r}{\partial w}$ are vectors tangent, respectively, to coordinate curves along $u, v$ and $w$.
What I've tried so far is:
$dx=\frac{\partial x}{\partial u}du+\frac{\partial x}{\partial v}dv+\frac{\partial x}{\partial w}dw$
$dy=\frac{\partial y}{\partial u}du+\frac{\partial y}{\partial v}dv+\frac{\partial y}{\partial w}dw$
$dz=\frac{\partial z}{\partial u}du+\frac{\partial z}{\partial v}dv+\frac{\partial z}{\partial w}dw$
So $d{\mathbf r}=dx{\mathbf e_x}+dy{\mathbf e_y}+dz{\mathbf e_z}=[\frac{\partial x}{\partial u}du+\frac{\partial x}{\partial v}dv+\frac{\partial x}{\partial w}dw]{\mathbf e_x}+[\frac{\partial y}{\partial u}du+\frac{\partial y}{\partial v}dv+\frac{\partial y}{\partial w}dw]{\mathbf e_y}+[\frac{\partial z}{\partial u}du+\frac{\partial z}{\partial v}dv+\frac{\partial z}{\partial w}dw]{\mathbf e_z}$
But I'm not sure where to go from here, if what I've done so far is even right. Can someone please show me/explain to me how to finish it off?