I have recently started a course on differential geometry (from a physicists perspective) and I am having trouble with vector fields.
We defined vector fields on a manifold M as a map: $ \eta: M \to TM $ , where TM is a tangent bundle. In a given coordinate system, one ca write $\eta$ as follows: $\eta = \sum_{i=1}^n \eta^i \frac{\partial}{\partial x^i}$
Using this definition, how can i construct a vectorfield $\eta$ on Manifold $M=(0,\infty) \times (-\pi,\pi) \subset R^2$ and express this in a coordinatesystem (polar coordinates for example).
Thanks for your time.