# Line integral - term

1. Mar 19, 2012

### Karamata

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

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

2. Mar 19, 2012

### Office_Shredder

Staff Emeritus
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.

3. Mar 19, 2012

### Karamata

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
$\varphi(\alpha)=\psi(\alpha), \varphi(\beta)=\psi(\beta), \forall s \hspace{4mm} H(s,0)=\varphi(s) \hspace{4mm} \text{and} \hspace{4mm} H(s,1)=\psi(s)$ (beginning and end are the same of this two function). What would be correct word for this?