Continuous deformation? Family of curves? So confused

[fleazo]

I understand the concept behind continuous deformations.


Say we have two curves ζ1 and ζ2 from A to B on some domain D and say that Pdx + Qdy is closed. Say we can show n points A=c1,c2,...,cn=B and A=d1,d2,d3,...,dn=B, so that we can first say follow the curve ζ1 from A to c1 then over to d1 by means of straight line l1 (call this modified curve θ1), and then up to B, and then we can also look at following ζ1 from A to c2 then over to d2 by means of line l2 and then up to B (call this θ2), say that if l1 and l2 are contained in a rectangle in the domain D, then we know that because the rectangle is star shaped and Pdx +Qdy is closed on D that ∫Pdx+dy on θ1 and ∫Px+dy on θ2 are both path independent on that rectangle. So those two integrals are the same. We can do this n times and eventually end up with the fact that ∫Pdx + Qdy on ζ1 = ∫Pdx+Qdy on ζ2.


Now, I keep seeing questions that say things like "specify exactly the family of paths that can deform ζ1 continuously to ζ2". I am so lost at things like this. My prof gave me this formula: δ(t,s)=δ₁(t)(1-s) + δ₂(t)s for 0 ≤ s ≤ 1


I am so confused by what exactly this equatino is and just understanding these family of curves. From my understanding of what continuous deformation is it seems like you are picking n points. So wouldn't you be having n curves? I'm so confused

 

[jvicens]

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Thank you for your question about continuous deformations. I can understand your confusion about the concept and the equation your professor gave you. Let me try to explain it in a simpler way.

Continuous deformation is a mathematical concept that describes the gradual change of one curve into another without any sudden breaks or jumps. It is often used in topology, which is the study of properties that are preserved through deformations, such as stretching, bending, and twisting. In your case, ζ1 and ζ2 are two curves that start at point A and end at point B on a certain domain D. The equation your professor gave you, δ(t,s)=δ1(t)(1-s) + δ2(t)s, represents a continuous deformation between ζ1 and ζ2. Here, t represents the parameter that moves along the curves, and s represents the parameter that controls the deformation between the two curves.

To better understand this, let's take a simpler example. Imagine a straight line segment connecting points A and B. If we want to deform this line into a curved line, we can use the equation δ(t,s)=δ1(t)(1-s) + δ2(t)s, where δ1(t) represents the original straight line and δ2(t) represents the curved line. As s increases from 0 to 1, the line gradually deforms from a straight line to a curved line.

Now, going back to your question about the family of paths, it refers to all possible curves that can be obtained by continuously deforming ζ1 into ζ2. These curves can vary depending on the values of t and s, but they all share the same starting and ending points. In other words, they all belong to the same family of curves.

I hope this explanation helps you understand the concept of continuous deformations better. Please let me know if you have any further questions. As scientists, we are always happy to help others understand complex concepts.