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A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch1, Ch4

  1. 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?

  1. 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.'

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    What is your question? – Arnaud Mortier Aug 08 '18 at 13:32
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    Paths have orientation. Theorem 4.18 and 4.20 above hold regardless of the orientation, and 4.27 requires positive orientation. It looks like you book is precise in that it uses "positively oriented" when it is necessary. – Doug M Aug 08 '18 at 13:49
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    No, a "path" does not necessarily have a direction. A "directed path" or "oriented path" is a path together with one of the two possible orientations specified. Of course, integrating a function, f, along a given path, "from A to B" is different from the integration "from B to A"- in fact one is the negative of the other. In order to talk about integration on a path, that path must be "directed" or "oriented". – user247327 Aug 08 '18 at 13:52
  • @ArnaudMortier Edited to be hopefully clearer or less broad. Thanks! – BCLC Aug 10 '18 at 09:58
  • @DougM Edited question. Do you disagree with user 247327? Thanks! – BCLC Aug 10 '18 at 09:59
  • @user247327 Edited question. Do you disagree with DougM? Thanks! – BCLC Aug 10 '18 at 09:59
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    (1). Yes, but you should understand that definitions is just language, not content. A different author could perfectly not attach orientations to their definition of the word "path". In complex analysis and integration orientation of the path is relevant to the theorems (the content). That is why it is convenient to have it as part of the definition of path. –  Aug 11 '18 at 00:21
  • @fatherBrown Well yeah, that's why I asked about the quote. Thanks! – BCLC Aug 11 '18 at 00:22
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    For a path that is not closed, there may be no "inside" and "outside", so no way to define "positively oriented". For paths that are not simple, there may be several "insides", and no consistent way to define "positively oriented". – Robert Israel Aug 11 '18 at 00:49
  • @RobertIsrael ah thanks! What about simple, closed but not piecewise smooth? – BCLC Aug 11 '18 at 01:03

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For a path that is not closed, there may be no "inside" and "outside", so no way to define "positively oriented". For paths that are not simple, there may be several "insides", and no consistent way to define "positively oriented". – Robert Israel Aug 11 at 0:49

@RobertIsrael ah thanks! What about simple, closed but not piecewise smooth? – BCLC Aug 11 at 1:03

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