Problems

01

Curves wrapping around a cylinder

A vertical cylinder is given with axis the $z$-axis. You do not know the radius.

Let $\mathbf{F}=(y^2,x^2,z^2)$. Suppose ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are two simple curves wrapping one time clockwise around the cylinder. Show that:

$${\int }_{{\mathcal{C}}_1}\mathbf{F}\cdot d\mathbf{s}={\int }_{{\mathcal{C}}_2}\mathbf{F}\cdot d\mathbf{s}.$$