Please explain about general coordiante

[merrypark3]

Hello.

In Arfken(6rd edi.), p.104 around eqn(2.3),

~~In general, these unit vectors will depend on the position in space. Then a vector may be written,
$$V=\hat {q_{1}}V_{1}+ \hat q_{2} V_{2}+ \hat q_{3} V_{3}$$
$$\underline{\mbox{but the coordinate or position vector is different in general,}}\\ r \neq \hat q_{1} V_{1}+ \hat q_{2} V_{2}+ \hat q_{3} V_{3} ,$$
$$\mbox{as the special case}$$
$$r=r\hat r \\\\\\\ \mbox{for spherical polar coordinates and}\\\\\\\\ r= \rho \hat \rho +z \hat z \\\\\\\\\\ \mbox {for cylindrical coordinates demonstrate. \cdots}$$
What's the meaning of the underlined sentence?

 

[tiny-tim]

Hello merrypark3!

It means that the vectors from the point V have nothing to do with the vector of the point V.

For example, in spherical coordinates, the vector of V is simply r in the e_r direction (θ and φ don't matter, and indeed e_θ and e_φ aren't even defined at the origin), and in cylindrrical coordinates, the vector of V is simply ρ in the e_ρ direction and z in the e_z direction (φ doesn't matter).