On the ineptitude of the Lie derivative to define directional derivatives

Question

I'm stuck at question 4-3. from John Lee's Riemannian Manifolds:

There exists a vector field on $\Bbb R^2$ that vanishes along the $x_1$-axis, but whose Lie derivative with respect to $∂_1$ does not vanish on the $x_1$-axis.

Defining: $v=a \partial_1+b \partial_2$ with $a,b \in C^\infty(\Bbb R^2)$ such that $a(x,0)=b(x,0)=0$, the Lie derivative should be: $$\mathcal L_{\partial_1}(v)=[\partial_1,v]=\partial_1 v - v \partial_1= \partial_1 a \partial_1+\partial_1 b \partial_2 - a \partial_1^2-b \partial_2 \partial_1=\frac{\partial a}{\partial x_1}\partial_1+\frac{\partial b}{\partial x_1}\partial_2$$ But now, at the point $p=(x,0)$ $$\mathcal L_{\partial_1}(v)(p)=\frac{\partial a}{\partial x_1}|_p\partial_1+\frac{\partial b}{\partial x_1}|_p\partial_2$$ would have to be $0$, because the vanishing of $a$ and $b$ on the $x_1$-axis implies the vanishing of $\partial_1 a$ and $\partial_1 b$ along the $x_1$-axis.

Where's the error in my above reasoning?

Answer 1

You're right -- that problem is stated incorrectly. The first sentence should read as follows:

Show that there are vector fields $V$ and $W$ on $\mathbf R^2$ such that $V=W=\partial_1$ along the $x^1$-axis, but the Lie derivatives $\mathcal L_V (\partial_2)$ and $\mathcal L_W(\partial_2)$ are not equal on the $x^1$-axis.

You'll want to download a copy of the correction list from my website, which contains a correction for that problem (and for quite a few other mistakes as well).