Gradient in general co-ordinate system

[sebb1e]

I know that for a general co-ordinate system, the gradient can be expressed as it is at the bottom of this page:

http://en.wikipedia.org/wiki/Orthogonal_coordinates#Differential_operators_in_t hree_dimensions

However, the book I am working from (A First Course in Continuum Mechanics by Gonzalez and Stuart) defines it as:

"Let {ei} be an arbitrary frame. Then grad phi(x)= (partial d phi by d xi)(x) ei where (x1,x2,x3) are the coordinates of x in ei"

I don't understand how this relates to the actual definition, for example given the definition in my book, how do I show that the gradient in cylindricals contains a 1/r in front of the unit theta term?

I presume that these ei are completely general so they are not unit vectors?

I need to understand this notation properly so I can apply it to deformation gradients etc