Line Integral: Meaning of Homotopic

In summary, the conversation is about the term "homeotopic" and its definition in the context of line integrals. The term refers to two curves being continuously deformable into each other through a continuous function. The conversation also mentions the term "related homotopic" which refers to two curves having the same beginning and end points and being continuously deformable through a continuous function. Both terms can be found in sources such as Wikipedia.
  • #1
Karamata
60
0
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.
 
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  • #2
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
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?
 

1. What is a line integral?

A line integral is a mathematical concept used in vector calculus to calculate the total value of a scalar or vector field along a given curve. It measures the accumulated effect of a field over a specific path or line.

2. What does "homotopic" mean in relation to line integrals?

"Homotopic" refers to the idea that two curves can be continuously deformed into one another without being disconnected or intersecting. In the context of line integrals, it means that two curves with the same start and end points can have the same value of the line integral, even if they are different paths.

3. How is homotopy related to the fundamental theorem of calculus?

The fundamental theorem of calculus states that the value of a definite integral can be calculated by evaluating the antiderivative of the integrand at the start and end points of the interval. This is similar to homotopy, where the value of the line integral remains the same as long as the start and end points of the curve are fixed.

4. Can you give an example of two homotopic curves with different line integrals?

Yes, imagine two curves on a sphere, both starting at the North Pole and ending at the South Pole. One curve goes straight down the Prime Meridian, while the other follows the Equator and then curves down to the South Pole. These two curves are homotopic but have different line integrals as the Equator curve covers more distance and thus has a greater accumulated effect of a field along the path.

5. How is homotopy used in real-world applications?

Homotopy is used in various fields of science and engineering, such as physics, biology, and computer graphics. It is used to study the deformations of objects, the behavior of physical systems, and to solve problems involving integration over complex paths. For example, in fluid dynamics, homotopy is used to analyze the flow of a fluid around an obstacle by considering different paths around the obstacle and their accumulated effects on the fluid's velocity and pressure.

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