S is subset of $\Bbb R^3 $ consisting of the union of
1) $z$ axis
2) the unit circle $x^2+y^2=1,z=0$
3) the points $(0,y,0)$ with $y \ge1 $
Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, D_1, D_2,D_3$ be the oriented 1-manifolds in $A$ like in the picture. $F$ is a vector field in $A, curlF=0$ in $A$, and $$\int_{C_1}F\cdot T=3$$
$$\int_{C_2}F\cdot T=7$$
What can you say about $$\int_{D_i} F\cdot T$$?
Can someone please give a pedagogical answer for this?
Thanks a lot!
The idea is that you can move around a curve inside of $ \mathbb{R}^3 - S $ ("homotope") and preserve the value of the line integral. Secondly, if you pinch a curve, then you can break it at the pinch into two curves and the line integral of the initial curve will be the sum of the line integrals of the two. And of course, reversing the orientation of the curve will negate the line integral. I hope you will understand why these are true.
So $D_3 $ is obviously zero, you can contract it down to a point. $ D_1 $ I can pinch right in the middle, and write it as a sum/difference of $ C_1 $ and $ C_2 $ (depending on how the orientations will match up). And similarly $D_2 $ is two copies of $ C_1 $, if you will pull it to the other side of the circle.
I hope this will help!