Vector Fields on Real Numbers

Question

I'm looking at vector fields on the manifold $\mathbb{R}$, in the sense that a vector field $v$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}\times T_p\mathbb{R}$. These seem so simple that there is not much to talk about. Can anyone tell me if the following assertions are correct?

1: $[v,w]=0$ for all vector fields $v$, $w$ since the Lie Bracket is defined by $v\frac{\partial w}{\partial x}-w\frac{\partial v}{\partial x}$, which is 0 .

2: There is only one linearly independent vector field in $\mathbb{R}$, since no two vectors in $\mathbb{R}$ can be linearly independent.

I think I might have some base misunderstanding here. Can anyone tell me if this is right?

Answer 1

1. That is not correct. For example, the vector fields $v = \frac{d}{dx}$ and $w = x \frac{d}{dx}$ don't commute: their bracket is

$$[v, w] = \left( \frac{d}{dx} + x \frac{d^2}{dx^2} \right) - x \frac{d^2}{dx^2} = \frac{d}{dx}.$$

2. That depends on what you mean. Do you mean linear dependence over $\mathbb{R}$ or over smooth functions $C^{\infty}(\mathbb{R})$?

(In 1. above I am thinking of a vector field on $\mathbb{R}$ as a derivation on smooth functions $C^{\infty}(\mathbb{R})$. This point of view may seem indirect but I think it is ultimately less confusing. Among other things, the Lie bracket just becomes the commutator bracket.)