How to find a vector normal to a cylinder in cylindric coordinates?

Question

I'm trying to solve a problem which demands to multiply a vector M and vector normal to a cylinder's surface in cylindric coordinates. Height of the cylinder is infinite and its radius is R. So how do I express normal vector in such coordinates?

Answer 1

Well, if the center of the cylinder coincides with the origin, then the unit vector $\hat{\rho}$ is normal to the cylinder's surface...

Answer 2

Cylindrical coordinates $\rho, \phi, z$ point as follows:

• $\hat\rho$ points straight out from the origin;
• $\hat\phi$ points perpendicular to $\hat\rho$ such that it is parallel to the $x-y$ plane and rotated $90^{\circ}$ in a counterclockwise sense;
• $\hat{z}$ points up.

For a cylinder centered on the origin, and with its axis along the $z$ axis, the normal vector points along $\hat\rho$ along the shaft, and $\pm\hat{z}$ at the two ends.