Why can't a non-coordinate basis be expressed as derivatives with respect to any coordinates? In Schutz, Geometrical Methods of Mathematical Physics book, pg 44, vector fields on a manifold with coordinates $x^i$ are defined as
$$V=\frac{d}{d\lambda}=V^i\frac{\partial}{\partial x^i}$$
$$W=\frac{d}{d\phi}=W^i\frac{\partial}{\partial x^i}$$
It was said that if ${d}/{d\lambda}$ and $d/d\phi$ are two elements of a basis, then they will not be expressible as derivatives with respect to any coordinates. Such a basis is a non coordinate basis.
What does author mean in the above sentence? Does he mean $$\frac{d}{d\lambda} \neq V^i\frac{\partial}{\partial x^i}$$ $$\frac{d}{d\phi} \neq W^i\frac{\partial}{\partial x^i}~?$$
If yes, doesn't it contradict the fact vector fields can be expressed in the form of the first two equations?
 A: A set of $n$ vector fields do not generally constitute a coordinate basis of an n-dimensional manifold, because you cannot take them all simultaneously to,
$$
V=\frac{d}{d\lambda}=V^i\frac{\partial}{\partial x^i}=\frac{\partial}{\partial u^1}
$$
$$
W=\frac{d}{d\phi}=W^i\frac{\partial}{\partial x^i}=\frac{\partial}{\partial u^2}
$$
etc. Where these $u^i$ are these alleged coordinates in which the basis looks like a bunch of pure partial derivatives.
The reason is that, in general,
$$
\left[\frac{\partial}{\partial u^{i}},\frac{\partial}{\partial u^{j}}\right]=0
$$
Now, in general too, the commutators of $n$ vector fields do not vanish.
I hope that was helpful.
A: Any basis at a point can be expressed by a coordinate system, but this can't be done over a neighbourhood of a point. There are integrability constraints. These constraints are encapsulated in  the Frobenius theorem which says that the Lie brackets of vector fields that constitute the frame field must vanish.
A: I think the correct sentence should be:
"If / and / are two elements of a basis, then they will not be expressible in general as derivatives with respect to any coordinates."
And then, this would be a non-coordinate basis.
Maybe the author avoided the double "in general" as it is just before used.
