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Line integral - term

  1. Mar 19, 2012 #1
    Hi,

    can someone tell me where I can find the term: "homeotopic" (or, something like that, I don't know how to write in English).

    My professor mentioned that term in the line integral, here it is:

    Let [itex]\Omega \subseteq \mathbb{R}^k[/itex] be area (open and connected set). Curves [itex]\varphi, \psi: [\alpha, \beta]\longrightarrow \Omega[/itex] are continuous.

    [itex]\varphi[/itex] and [itex]\psi[/itex] are homotopic if there is continuous function [itex]H:[\alpha, \beta]\times[0,1]\longrightarrow \Omega[/itex] such that valid [itex]H(t,0)=\varphi(t)[/itex] and [itex]H(t,1)=\psi(t)[/itex].

    Sorry for bad English.
     
  2. jcsd
  3. Mar 19, 2012 #2

    Office_Shredder

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    homotopic is the correct word. What you wrote down is usually the definition.... wikipedia has a good animated graphic of homotopic curves if you're confused by the definition
    http://en.wikipedia.org/wiki/Homotopy

    at the top you see a picture of one curve morphing into another one via a homotopy... at each step the curve is the image of H(t,s) for some fixed value of s.
     
  4. Mar 19, 2012 #3
    Thank you Office_Shredder.

    No, no, I'm not confused with definition. Problem was that I couldn't find this phrase in a book for Analysis. Do you perhaps know where can I find him? (except Wikipedia)


    And, he mentioned term: related homotopic, something like
    [itex]\varphi(\alpha)=\psi(\alpha), \varphi(\beta)=\psi(\beta), \forall s \hspace{4mm} H(s,0)=\varphi(s) \hspace{4mm} \text{and} \hspace{4mm} H(s,1)=\psi(s)[/itex] (beginning and end are the same of this two function). What would be correct word for this?
     
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