I am always a little confuse in polar, cylindrical and spherical coordinates in vector calculus vs cylindrical and spherical coordinates in vector fields used in Electromagnetics. I want to clarify what my finding and feel free to correct me and add to it.
A) Vector calculus:
All these are just simply rectangular coordinate presented in polar and spherical coordinate value.
B) Vector field in Spherical and cylindrical coordinates:
This is "true" Spherical or cylindrical coordinates in represents a vector ( vector field ) at a single point P. P can be respect to the origin.
So in conclusion, I think it is very very different between the two, where in the first case A), it is very much like the representation of (x,y,z). The second case B) really about vector fields where you set up a coordinate system at a point P and use the coordinate system to represent the direction and magnitude of the vector field at that point. Therefore the two are not the same.