The vector field ${\bf B}({\bf x})$ is defined in cylindrical polar coordinates $\rho ,\phi, z$ by ${\bf B}({\bf x})={\rho}^{-1}e_{\phi}, \rho \ne 0$. Evaluate $\oint_C {\bf B}\cdot d{\bf x}$, where $C$ is the circle $z=0, \rho=1$ and $0\le \phi \le 2\pi$.
I am able to do it by converting to Cartesian, but I think I should be able to do it leaving in Cylindrical coordinates, so I would ask that someone complete the question without converting to Cartesian.
Thank you
Note $d\mathbf{x}$ points in the direction of $\mathbf{e}_\phi$ and has magnitude of $d\phi$. So $d\mathbf{x} = \mathbf{e}_\phi d\phi$ and $\oint_C \mathbf{B}\cdot d\mathbf{x} = \int \mathbf{e}_\phi\cdot \mathbf{e}_\phi d\phi = \int d\phi = 2\pi$.