This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW.
My question is, referring to the answer provided, what closed surfaces are used in the proof for both $$ \oint_{C_1} \vec{F} \,dr , \oint_{C_2} \vec{F} \,dr $$
What does the curve $C_1 - C_2$ in $$\oint_{C_1 - C_2} \vec{F} \,dr $$ even mean? How does Stoke's theorem apply here, and why does it mean the surface in between the two curves?
In basic geometry you take by definition that (for $a,b$ numbers and $C,D$ curves): $$ \int_{aC+bD}\omega\,dx := a\int_{C}\omega\,dx + b\int_{D}\omega\,dx. $$
The motivation for this comes from algebraic topology.
So: $$ \oint_{C_1-C_2}F\,dr =\oint_{C_1}F\,dr - \oint_{C_2}F\,dr. $$
You can think of the part $S$ of the cylinder between the two curves as "going" or "flowing" from $C_2$ to $C_1$, which in math means simply: $$ C_2 + \partial S = C_1 $$ so that: $$ \oint_{C_1-C_2}F\,dr =\oint_{\partial S}F\,dr. $$
The surfaces are not closed! The curves are.