Let $(x,y) \in S^1$. Then we can calculate $$T_{(x,y)} S^1 = \{(a,b) \in \mathbb{R}^2 : ax + by = 0\} = span\{(y,-x)\}$$ But I read somewhere the tangent space is spanned by $y \partial_x – x \partial_y$. How does this follow from the above? I know that $\{\partial_x, \partial_y\}$ is a basis for $T_{(x,y)}\mathbb{R}^2$ but I don't know how to relate this to the calculation above.
Thank you.
Best Answer
The vector $(y, -x)$ is being written “in coordinates” with respect to the basis $\{\partial_x, \partial_y\}$; it corresponds to the linear combination is $y \partial_x - x \partial_y$.