In a book I see the following

" The concept of the winding number s usefull to characterize what is meant by the inside(interior) and the outside(exterior) of a closed curve $\gamma$, respectively, in the following way $$Int(\gamma)=\{z\notin\gamma:n(\gamma;z)\neq 0\}$$ and $$Ext(\gamma)=\{z\notin\gamma:n(\gamma;z)=0\}$$ Moreover, a closed curve $\gamma:[a,b]\rightarrow\mathbb{C}$ is said to be positively oriented if $n(\gamma;z)>0$ for every $z$ inside $\gamma$ and negatively oriented if $n(\gamma;z)<0$ for every $z$ outside $\gamma$"

my question is: "negatively oriented if $n(\gamma;z)<0$ for every $z$ outside $\gamma$" makes no sense to me as the author already defines $Ext(\gamma)=\{z\notin\gamma:n(\gamma;z)=0\}$, I mean when $z$ is outside gamma by definition $n(\gamma;z)=0$, I am confused please teach me.

  • 2
    $\begingroup$ I think it is a typo. Replace outside by inside and you'll be fine. $\endgroup$ – Gregor Botero Aug 25 '12 at 21:34
  • $\begingroup$ Yes, it sounds like a slip-of-tongue (finger?) to me. $\endgroup$ – anon Aug 25 '12 at 21:38

You're right, this is a typo. See wolfram's definition (two images with values $\pm 1$), it is very clear that $z\in Int(\gamma)$.


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