I am having trouble doing this problem. I know that we need to use Stokes' Theorem. $D_3$ is easy. The problem is I can't seem to picture a shape whose boundary is $D_1$ and $C_1$ and/or $D_1$ and $C_2$ (the way the lines are drawn seems to make a connected smooth surface impossible). Similarly, creating a smooth surface that has $D_2$ and $C_1$ and/or $D_2$ and $C_2$ as boundaries also seems impossible.
Question: Let S be the subset of $\mathbb{R^3}$ consisting of the union of:
i) the z-axis
ii) the unit circle $x^2+y^2=1, z=0$
iii) the points $(0, y, 0)$ with $y \geq 1$
Let $A$ be the open set $\mathbb{R}^3-S$ of $\mathbb{R}^3$. Let $C_1, C_2, D_1, D_2, D_3$ be the oriented 1-manifolds in $A$ that are pictured in Figure 38.3. Suppose that $F$ is a vector field in $A$, with $curl F = 0$ in $A$ that $\int_{C_1} \langle F,T \rangle ds = 3$ and $\int_{C_2} \langle F,T \rangle ds = 7$.
What can you say about the integral $\int_{D_i} \langle F,T \rangle ds$ for $i=1,2,3$? Justify your answers.
