A specific example of $F$-related vector fields I need to prove the following:
Let $F:\Bbb{R}\to\Bbb{R}^2$ be the smooth map $F(t)=(\cos t,\sin t)$. Then $d/dt\in\mathcal{T}(\Bbb{R})$ is $F$-related to the vector field $Z\in\mathcal{T}(\Bbb{R}^2)$ defined by $$Z=x\frac{∂}{∂y}-y\frac{∂}{∂x}.$$
So for any smooth $f$ defined on an open subset of $\Bbb{R}^2$ we want $$\frac{d}{dt}(f\circ F)=(x\frac{∂}{∂y}f-y\frac{∂}{∂x}f)\circ F.$$ That is,$$\frac{∂}{∂x}f\frac{d}{dt}F+\frac{∂}{∂y}f\frac{d}{dt}F=(x\frac{∂}{∂y}f-y\frac{∂}{∂x}f)\circ F.$$ What should I do next?

LHS:
$$\frac{d}{dt}(f\circ F)= \frac{d}{dt}(f(\cos t, \sin t))=f_x \cdot (-\sin t)+f_y \cdot \cos t$$
RHS:
$$(x\frac{∂}{∂y}f-y\frac{∂}{∂x}f)\circ F= (x\circ F) f_x-(y\circ F) f_y= \cos t \cdot f_y - \sin t \cdot f_x$$
So they are equal.
