# Continuous deformation? Family of curves? So confused!

1. Nov 8, 2011

### 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