- Question 1: Do all paths have an orientation?
The following is a quote from Ch1 of the book: 'Figure 1.8 shows two examples. We remark that each path comes with an orientation, i.e. a sense of direction.'
Should I interpret as that the paths in Fig 1.8 have orientations, but not necessarily other paths? Or that the paths in Fig 1.8 are examples to say that any path has an orientation? Or what?
- Question 2: In re (Q1) above and the definition below, are there definitions for a path to be positively oriented even though it is not simple, piecewise smooth or closed?
Definition of positively oriented: 'A piecewise smooth simple closed path $\gamma$ is positively oriented if it is parametrised s.t. its inside is on the left as the parametrisation traverses $\gamma$. An example is a counter-clockwise oriented circle.'